Review Article On the Relationship between Variational ...eprints.imtlucca.it/3125/1/109029.pdf · Review Article On the Relationship between Variational Level Set-Based and SOM-Based

Review ArticleOn the Relationship between Variational Level Set-Based andSOM-Based Active Contours

Mohammed M Abdelsamea12 Giorgio Gnecco2

Mohamed Medhat Gaber3 and Eyad Elyan3

1Department of Mathematics Faculty of Science University of Assiut Assiut 71516 Egypt2IMT Institute for Advanced Studies Piazza S Francesco 19 55100 Lucca Italy3Robert Gordon University Garthdee House Garthdee Road Aberdeen AB10 7QB UK

Correspondence should be addressed to Mohamed Medhat Gaber mgaber1rguacuk

Received 17 December 2014 Revised 27 March 2015 Accepted 29 March 2015

Academic Editor Dominic Heger

Copyright copy 2015 Mohammed M Abdelsamea et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Most Active Contour Models (ACMs) deal with the image segmentation problem as a functional optimization problem as theywork on dividing an image into several regions by optimizing a suitable functional Among ACMs variational level set methodshave been used to build an active contour with the aim of modeling arbitrarily complex shapes Moreover they can handle alsotopological changes of the contours Self-OrganizingMaps (SOMs) have attracted the attention of many computer vision scientistsparticularly in modeling an active contour based on the idea of utilizing the prototypes (weights) of a SOM to control the evolutionof the contour SOM-based models have been proposed in general with the aim of exploiting the specific ability of SOMs to learnthe edge-map information via their topology preservation property and overcoming some drawbacks of other ACMs such astrapping into local minima of the image energy functional to be minimized in such models In this survey we illustrate the mainconcepts of variational level set-based ACMs SOM-based ACMs and their relationship and review in a comprehensive fashion thedevelopment of their state-of-the-art models from a machine learning perspective with a focus on their strengths and weaknesses

1 Introduction

Image segmentation is the problem of partitioning thedomain Ω of an image 119868(119909) where 119909 isin Ω is the pixel loca-tion within the image into different subsets Ω

119894 where each

subset has a different characterization in terms of colorintensity texture andor other features used as similaritycriteria Segmentation is a fundamental component of imageprocessing and plays a significant role in computer visionobject recognition and object tracking

Traditionally image segmentation methods can be clas-sified into five categories The first category is made up ofthreshold-based segmentation methods [1] These methodsare pixel-based and usually divide the image into two subsetsthat is the foreground and the background using a thresholdon the value of some feature (eg gray level and color value)These methods assume that the foreground and backgroundin the image have different ranges for the values of the features

to be thresholded Over the years many different thresh-olding techniques have been developed including Minimumerror thresholding Moment-preserving thresholding andOtsursquos thresholding just to mention a few The most popularthresholding method Otsursquos algorithm [2] improves theimage segmentation performance over other threshold-basedsegmentation methods in the following way The thresholdused in Otsursquos algorithm is chosen in such a way to optimize atrade-off between the maximization of the interclass variance(ie between pairs of pixels belonging to the foreground andthe background resp) and the minimization of the intraclassvariance (ie between pairs of pixels belonging to the sameregion) Otsursquos thresholding algorithm and its extension tothe case of multiple thresholds [3] are good for thresholdingan image whose intensity histogram is either bimodal ormultimodal (eg they provide a satisfactory solution in thecase of the segmentation of large objects with nearly uniformintensities significantly different from the intensity of the

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2015 Article ID 109029 19 pageshttpdxdoiorg1011552015109029

2 Computational Intelligence and Neuroscience

background) However they have not the ability to segmentimages with a unimodal distribution (eg images containingsmall objects with different intensities) and their outputs aresensitive to noiseThus postprocessing operations are usuallyrequired to obtain a final satisfactory segmentation

The second category of methods is called boundary-based segmentation [4] These methods detect boundariesand discontinuities in the image based on the assumptionthat the intensity values of the pixels linking the foregroundand the background are distinct The firstsecond orderderivatives of the image intensity are usually used to highlightthose pixels (eg this is the case of Sobel and Prewitt edgedetectors [4] as first-order methods and the Laplace edgedetector [1] as a second-order method resp) The differencebetween first and second order methods is that the latter canalso localize the local displacement and orientation of theboundary By far the most accurate technique of detectingboundaries and discontinuities in an image is the Cannyedge detector [5] The Canny edge detector is less sensitiveto noise than other edge detectors as it convolves the inputimage with a Gaussian filter The result is a slightly blurredversion of the input image This method is also very easyto be implemented However it is still sensitive to noiseand leads often to a segmentation result characterized by adiscontinuous detection of the object boundaries

The third category of methods is called region-basedsegmentation [6] Region-based segmentation techniquesdivide an image into subsets based on the assumption thatall neighboring pixels within one subset have a similar valueof some feature for example the image intensity Regiongrowing [7] is the most popular region-based segmentationtechnique In region growing one has to identify at first aset of seeds as initial representatives of the subsets Thenthe features of each pixel are compared to the features ofits neighbor(s) If a suitable predefined criterion is satisfiedthen the pixel is classified as belonging to the same subsetassociated with its ldquomost similarrdquo seed Accordingly regiongrowing relies on the prior information given by the seedsand the predefined classification criterion A second popularregion-based segmentation method is region ldquosplitting andmergingrdquo In suchmethod the input image is first divided intoseveral small regionsThen on the regions a series of splittingand merging operations are performed and controlled by asuitable predefined criterion As region-based segmentationis an intensity-based method the segmentation result ingeneral leads to a nonsmooth and badly shaped boundary forthe segmented object

The fourth category of methods is learning-based seg-mentation [8]There are two general strategies for developinglearning-based segmentation algorithms namely generativelearning and discriminative learning Generative learning[9] utilizes a data set of examples to build a probabilisticmodel by finding the best estimate of its parameters for someprespecified parametric form of a probability distributionOne problem with these methods is that the best estimate ofthe parametersmay not provide a satisfactorymodel becausethe parametricmodel itselfmaynot be correct Another prob-lem is that the classificationclustering framework associatedwith a parametric probabilistic model may not provide an

accurate description of the data due to the limited numberof parameters in the model even in the case in whichits training is well performed Techniques following thegenerative approach include K-means [10] the Expectation-Maximization algorithm [11] and Gaussian Mixture Models[12] Discriminative learning [13 14] ignores probability andattempts to construct a good decision boundary directly Suchan approach is often extremely successful especially when noreasonable parametric probabilistic model of the data existsDiscriminative learning assumes that the decision boundarycomes from another class of nonparametric solutions andchooses the best element of that class according to a suitableoptimality criterion Techniques following the discriminativeapproach include Linear Discriminative Analysis [15] NeuralNetworks [16] and Support Vector Machines [17] The mainproblems with the application of these methods to image seg-mentation are their sensitivity to noise and the discontinuityof the resulting object boundaries

The last category of methods is energy-based segmenta-tion [18 19]This class ofmethods is based on an energy func-tional and deals with the segmentation problem as a func-tional optimization problem whose goal is to partition theimage into regions based on the maximizationminimizationof the energy functional (Loosely speaking a functional isdefined as a function of a function that is a functional takesa function as its input argument and returns a scalar) Themost well-known energy-based segmentation techniques arecalled ldquoactive contoursrdquo or Active Contour Models (ACMs)The main idea of active contours is to choose an initialcontour inside the image domain to be segmented and thenmake such a contour evolve by using a series of shrinking andexpanding operations Some advantages of the active con-tours over the aforementioned methods are that topologicalchanges of the objects to be segmented can be often handledimplicitlyMore importantly complex shapes can bemodeledwithout the need of prior knowledge about the image Finallyrich information can be inserted into the energy functionalitself (eg boundary-based and region-based information)

More specifically ACMs usually deal with the segmen-tation problem as an optimization problem formulated interms of a suitable ldquoenergyrdquo functional constructed in sucha way that its minimum is achieved in correspondence witha contour that is a close approximation of the actual objectboundary Starting from an initial contour the optimizationis performed iteratively evolving the current contour withthe aim of approximating better and better the actual objectboundary (hence the denomination ldquoactive contourrdquo modelswhich are used also formodels that evolve the contour but arenot based on the explicit minimization of a functional [20])In order to guide efficiently the evolution of the current con-tour ACMs allow to integrate various kinds of informationinside the energy functional such as local information (egfeatures based on spatial dependencies among pixels) globalinformation (eg features which are not influenced by suchspatial dependencies) shape information prior informationand a-posteriori information learned from examples (Dueto the possible lack of a precise prior information on theshape of the objects to be segmented in this respect mostACMs make only the assumption that it is preferable to have

Computational Intelligence and Neuroscience 3

a smooth boundary [21] This goal is achieved by incor-porating a suitable regularization term into their energyfunctionals [19]) As a consequence depending on the kindof information used one can divide ACMs into severalcategories for example edge-based ACMs [22ndash25] globalregion-based ACMs [26 27] edgeregion-based ACMs [28ndash30] local region-based ACMs [31ndash33] and globallocalregion-based ACMs [34 35] In particular edge-based ACMsmake use of an edge-detector (in general the gradient of theimage intensity) to stop the evolution of the active contouron the true boundaries of the objects of interest Insteadregion-based ACMs use with the same purpose statisticalinformation about the regions to be segmented (eg intensitytexture and color distribution) Depending on how the activecontour is represented one can also distinguish betweenparametrized [36] and variational level set-based ACMs [26]One important advantage of the latter is that they can handleimplicitly topological changes of the objects to be segmented

Although ACMs often provide an effective and efficientmeans to extract smooth and well-defined contours trappinginto local minima of the energy functional may still occurbecause such a functional may be constructed on the basisof simplified assumptions on properties of the images tobe segmented (eg the assumption of Gaussian intensitydistributions for the setsΩ

119894in the case of theChan-Vese active

contourmodel [21 26])Motivated by this observation and bythe specific ability of SOMs to learn via their topology preser-vation property [37] information about the edge map of theimage (ie the set of points obtained by an edge-detectionalgorithm) a new class of ACMs named SOM-based ACMs[38 39] has been proposed with the aim of modelling andcontrolling effectively the evolution of the active contour bya Self-Organizing Map (SOM) often without relying on anexplicit energy functional to be minimized In this paper wereview some concepts of ACMs with a focus on SOM-basedACMs illustrating both their strengths and limitations Inparticular we focus on variational level set-based ACMs andSOM-based ACMs and on their relationship The paper is asubstantial extension of the short survey about SOM-basedACMs that we presented in [40] A summary of the mainstrengths and drawbacks of theACMspresented in the surveyis reported in Table 1 Illustrating the motivations for suchstrengths and drawbacks is the main focus of this paper

The paper is organized as follows Section 2 provides asummary of variational level set-based ACMs In Section 3we review the state of the art of SOM-based ACMs not usedin combination with variational level set methods Section 4describes a recent class of SOM-based ACMs combined withsuch methods Finally Section 5 provides some conclusions

2 Variational Level Set-Based ACMs

To build an active contour there are mainly two methodsThe first one is an explicit or Lagrangian method whichresults in parametric active contours also called ldquoSnakesrdquofrom the name of one of the models that use such a kindof parametrization [19] The second one is an implicit orEulerian method which results in geometric active contoursknown also as variational level set methods

s = 0

s = 02

s = 1

Figure 1 The parametric representation of a contour

Zero level set120601(x) gt 0120601(x) = 0

120601(x) lt 0

Figure 2 The geometric representation of a contour

In parametric ACMs the contour 119862 see Figure 1 isrepresented as

119862 = 119909 isin Ω 119909 = (1199091 (119904) 1199092 (119904)) 0 le 119904 le 1 (1)

where 1199091(119904) and 119909

2(119904) are functions of the scalar parameter

119904 A representative parametric ACM is the Snakes modelproposed by Kass et al [19] (see also [36] for successive devel-opments)

The main drawbacks of parametric ACMs are the fre-quent occurrence of local minima in the image energy func-tional to be optimized (which is mainly due to the presenceof a gradient energy term inside such a functional) and thefact that topological changes of the objects (eg mergingand splitting) cannot be handled during the evolution of thecontour

The difference between parametric active contour andgeometric (or variational level set-based) Active ContourModels is that in geometric active contours the contouris implemented via a variational level set method Such arepresentation was first proposed by Osher and Sethian [41]In such methods the contour 119862 see Figure 2 is implicitlyrepresented by a function 120601(119909) called ldquolevel set functionrdquowhere 119909 is the pixel location inside the image domainΩ Thecontour 119862 is then defined as the zero level set of the function120601(119909) that is

119862 = 119909 isin Ω 120601 (119909) = 0 (2)

A common and simple expression for 120601(119909) which is usedby most authors is

120601 (119909) =

+120588 for119909 isin inside (119862)

0 for119909 isin 119862

minus120588 for119909 isin outside (119862)

(3)


Table 1 A summary of the Active Contour Models (ACMs) reviewed in the paper

ACM Reference Regional informationMain strengthsadvantages Main limitationsdisadvantages

Local Global

GAC [24] No NoMakes use of boundary information Hardly converges in the presence of

ill-defined boundariesIdentifies accurately well-definedboundaries

Very sensitive to the contourinitialization

CV [26] No Yes

Can handle objects with blurredboundaries in a global way Makes strong statistical assumptions

Can handle noisy objects Only suitable for Gaussian intensitydistributions of the subsets

SBGFRLS [68] No Yes

Very efficient computationally androbust to the contour initialization Makes strong statistical assumptions

Gives efficient and effective solutionscompared to CV and GAC It is hard to adjust its parameters

LBF [71] Yes No

Can handle complex distributions withinhomogeneities Computationally expensive

Can handle foregroundbackgroundintensity overlap


LIF [32] Yes No Behaves likewise LBF but iscomputationally more efficient

Very sensitive to noise and contourinitialization

LRCV [31] Yes No Computationally very efficientcompared to LBF and LIF


LSACM [73] Yes NoRobust to the initial contour Computationally expensiveCan handle complex distributions withinhomogeneities Relies on a probabilistic model

GMM-AC [75] No YesExploits prior knowledge Makes strong statistical assumptions

Very efficient and effective Requires a huge amount of supervisedinformation

SISOM [38] No No

Localizes the salient contours using aSOM

Topological changes cannot behandled

No statistical assumptions arerequired

Computationally expensive andsensitive to parameters

TASOM [39] No No

Adjusts automatically the number ofSOM neurons

No topological changes can behandled

Less sensitive to the model parameterscompared to SISOM

Sensitive to noise and blurredboundaries

BSOM [93] No Yes

Exploits regional information Topological changes cannot behandled

Deals better with ill-definedboundaries compared to SISOM andTASOM

Computationally expensive andproduces discontinuities

eBSOM [94] No YesProduces smooth contours Topological changes cannot be

handledControls the smoothness of thedetected contour better than BSOM Computationally expensive

FTA-SOM [96] No YesConverges quickly Topological changes cannot be

handledIs more efficient than SISOM TASOMand eBSOM Sensitive to noise

CFBL-SOM [97] No YesExploits prior knowledge Topological changes cannot be

handledDeals well with supervisedinformation Sensitive to the contour initialization


Table 1 Continued


Local Global

CAM-SOM [98] No Yes

Can handle objects with concavitiessmall computational cost


More efficient than FTA-SOM High computational cost compared tolevel set-based ACMs

CSOM-CV [102] No YesVery robust to the noise Supervised information is requiredRequires a small amount of supervisedinformation

Suitable only for handling images in aglobal way

SOAC [103] Yes No

Can handle complex images in a localand supervised way Supervised information is required

Can handle inhomogeneities andforegroundbackground intensityoverlap

Sensitive to the contour initialization

SOMCV [104] No YesReduces the intervention of the user Is easily trapped into local minimaCan handle multimodal intensitydistributions Deals with images in a global way

SOM-RAC [105] Yes Yes

Robust to noise scene changes andinhomogeneities Very expensive computationallyRobust to the contour initialization

where 120588 is a positive real number (possibly dependent on 119909and 119862 in such case it is denoted by 120588(119909 119862))

In the variational level setmethod expressing the contour119862 in terms of the level set function 120601 the energy functionalto be minimized can be expressed as follows

119864 (120601) = 119864in (120601) + 119864out (120601) + 119864119862 (120601) (4)

where 119864in(120601) and 119864out(120601) are integral energy terms inside andoutside the contour and 119864

119862(120601) is an integral energy term for

the contour itself More precisely the three terms are definedas

119864in (120601) = int120601(119909)gt0

119890 (119909) 119889119909 = intΩ

119867(120601 (119909)) sdot 119890 (119909) 119889119909

119864out (120601) = int120601(119909)lt0

119890 (119909) 119889119909 = intΩ

(1 minus 119867 (120601 (119909))) sdot 119890 (119909) 119889119909

119864119862(120601) = int

Ω

1003817100381710038171003817nabla119867 (120601 (119909))1003817100381710038171003817 119889119909 = int

Ω

120575 (120601 (119909)) sdot1003817100381710038171003817nabla120601 (119909)

1003817100381710038171003817 119889119909

(5)

where 119890(119909) is a suitable loss function and 119867 and 120575 arerespectively the Heaviside function and the Dirac deltadistribution that is

119867(119911) =

1 if 119911 ge 0

0 if 119911 lt 0

120575 (119911) =119889

119889119911119867 (119911)

(6)

Accordingly the evolution of the level set function 120601

provides the evolution of the contour 119862 In the variational

level set framework the (local) minimization of the energyfunctional 119864(120601) can be obtained by evolving the level setfunction 120601 according to the following Euler-Lagrange PartialDifferential Equation (in the following when writing partialdifferential equations in general we do not write explicitlythe arguments of the involved functions which are describedeither in the text or in the references from which suchequations are reported) (PDE)

120597120601

120597119905= minus

120597119864 (120601)

120597120601 (7)

where 120601 is now considered as a function of both the pixellocation 119909 and time 119905 and the term 120597119864(120601)120597120601 denotes thefunctional derivative of 119864 with respect to 120601 (ie looselyspeaking the generalization of the gradient to an infinite-dimensional setting) So (7) represents the application tothe present functional optimization problem of an extensionto infinite dimension of the classical gradient method forunconstrained optimization According to the specific kind ofPDE (see (7)) that models the contour evolution variationallevel set methods can be divided into several categories suchas Global Active Contour Models (GACMs) [42ndash46] whichuse global information and Local Active Contour Models(LACMs) [47ndash51] which use local information

21 Unsupervised Models In order to guide efficiently theevolution of the current contour ACMs allow to integratevarious kinds of information inside the energy functionalsuch as local information (eg features based on spatialdependencies among pixels) global information (eg fea-tures that are not influenced by such spatial dependencies)shape information prior information and also a-posterioriinformation learned from examples As a consequence


depending on the kind of information used one can furtherdivide ACMs into several subcategories for example edge-based ACMs [22ndash25 52 53] global region-based ACMs[26 27 45 54 55] edgeregion-based ACMs [28 30 56ndash58]and local region-based ACMs [34 35 59ndash62]

In particular edge-based ACMs make use of an edge-detector (in general the gradient of the image intensity) totry to stop the evolution of the active contour on the trueboundaries of the objects of interest One of themost popularedge-based active contours is the Geodesic Active Contour(GAC) model [24] which is described in the following

Geodesic Active Contour (GAC) Model [24] The level setformulation of the GAC model can be described as follows

120597120601

120597119905= 119892

1003817100381710038171003817nabla1206011003817100381710038171003817 (nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

) + 120572) + nabla119892 sdot nabla120601 (8)

where 120601 is the level set function nabla is the gradient operatornablasdot is the divergence operator 120572 gt 0 is a ldquoballoonrdquo force term(controlling the rate of expansion of the level set function)and 119892 is an Edge Stopping Function (ESF) defined as follows

119892 =1

1 +1003817100381710038171003817nabla119892120590 lowast 119868

1003817100381710038171003817

2=

1

1 +1003817100381710038171003817119892120590 lowast nabla119868

1003817100381710038171003817

2 (9)

where 119892120590is a Gaussian kernel function with width 120590 lowast is the

convolution operator and 119868 is the image intensity Hence theESF function provides information related to the gradient ofthe image intensity

For images with a high level of noise the presence ofthe Edge Stopping Function may not be enough to stop thecontour evolution at the right boundaries Motivated by thisissue a novel edge-based ACM has been proposed in [63]with the aim of improving the robustness of the segmentationto the noise This has been achieved by regularizing theLaplacian of the image through an anisotropic diffusion termwhich also preserves edge information

Since edge-basedmodels make use of an edge-detector tostop the evolution of the initial guess of the contour on theactual object boundaries they can handle only images withwell-defined edge information Indeed when images have ill-defined edges the evolution of the contour typically does notconverge to the true object boundaries

An alternative solution consists in using statistical infor-mation about a region (eg intensity texture and color) toconstruct a stopping functional that is able to stop the contourevolution on the boundary between two different regions asit happens in region-based models (see also the survey paper[64] for the recent state of the art of region-based ACMs)[26 27] An example of a region-based model is illustratedin the following

Chan-Vese (CV) Model [26] The CV model is a well-knownrepresentative global region-based ACM (at the time ofwriting it has receivedmore than 4000 citations according toScopus) After its initialization the contour in the CV modelis evolved iteratively in an unsupervised fashion with the aimof minimizing a suitable energy functional constructed insuch a way that its minimum is achieved in correspondence

with a close approximation of the actual boundary betweentwo different regions The energy functional 119864CV of the CVmodel for a scalar-valued image has the expression

119864CV (119862) = 120583 sdot Length (119862) + V sdot Area (in (119862))

+ 120582+intin(119862)

(119868 (119909) minus 119888+(119862))2

119889119909

+ 120582minusintout(119862)

(119868 (119909) minus 119888minus(119862))2

119889119909

(10)

where 119862 is a contour 119868(119909) isin R denotes the intensity of theimage indexed by the pixel location 119909 in the image domainΩ 120583 ge 0 is a regularization parameter which controls thesmoothness of the contour in(119862) (foreground) and out(119862)(background) represent the regions inside and outside thecontour respectively and V ge 0 is another regularizationparameter which penalizes a large area of the foregroundFinally 119888+(119862) and 119888minus(119862) which are defined respectively as

119888+(119862) = mean (119868 (119909) | 119909 isin in (119862))

119888minus(119862) = mean (119868 (119909) | 119909 isin out (119862))

(11)

represent the mean intensities of the foreground and thebackground respectively and 120582

+ 120582minus

ge 0 are parameterswhich control the influence of the two image energy termsintin(119862)(119868(119909) minus 119888

+(119862))2119889119909 and intout(119862)(119868(119909) minus 119888

minus(119862))2119889119909 respec-

tively inside and outside the contour The functional isconstructed in such a way that when the regions in(119862) andout(119862) are smooth and ldquomatchrdquo the true foreground and thetrue background respectively 119864CV(119862) reaches its minimum

Following [65] in the variational level set formulation of(10) the contour 119862 is expressed as the zero level set of anauxiliary function 120601 Ω rarr R

119862 = 119909 isin Ω 120601 (119909) = 0 (12)

Note that different functions 120601(119909) can be chosen to expressthe same contour 119862 For instance denoting by 119889(119909 119862) theinfimumof the Euclidean distances of the pixel119909 to the pointson the curve 119862 120601(119909) can be chosen as a signed distancefunction defined as follows

120601 (119909) =

119889 (119909 119862) 119909 isin in (119862)

0 119909 isin 119862

minus119889 (119909 119862) 119909 isin out (119862)

(13)

This variational level set formulation has the advantage ofbeing able to deal directly with the case of a foreground anda background that are not necessarily connected internally

After replacing119862with120601 and highlighting the dependenceof 119888+(119862) and 119888minus(119862) on 120601 in the variational level set formula-tion of the CVmodel the (local) minimization of the cost (10)is performed by applying the gradient-descent technique inan infinite-dimensional setting (see (7) and also the reference


[26]) leading to the following PDE which describes theevolution of the contour

120597120601

120597119905= 120575 (120601) [120583nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

) minus V minus 120582+ (119868 minus 119888+ (120601))2

+ 120582minus(119868 minus 119888

minus(120601))2

]

(14)

where 120575(sdot) is the Dirac generalized function The first termin 120583 of (14) keeps the level set function smooth the secondone in V controls the propagation speed of the evolvingcontour while the third and fourth terms in 120582+ and 120582minus can beinterpreted respectively as internal and external forces thatdrive the contour toward the actual object boundary Then(14) is solved iteratively in [26] by replacing the Dirac delta bya smooth approximation and using a finite difference schemeSometimes also a reinitialization step is performed in whichthe current level set function 120601 is replaced by its binarization(ie for a constant 120588 gt 0 a level set function of the form (3)representing the same current contour)

The CVmodel can also be derived in a Maximum Likeli-hood setting by making the assumption that the foregroundand the background follow Gaussian intensity distributionswith the same variance [21] Then the model approximatesglobally the foreground and background intensity distri-butions by the two scalars 119888

+(120601) and 119888

minus(120601) respectively

which are their mean intensities Similarly Leventon et alproposed in [66] to use Gaussian intensity distributions withdifferent variances inside a parametric density estimationmethod Also Tsai et al in [67] proposed to use insteaduniform intensity distributions to model the two intensitydistributions However such models are known to performpoorly in the case of objects with inhomogeneous intensities[21]

Compared to edge-based models region-based modelsusually perform better in images with blurred edges and areless sensitive to the contour initialization

Hybrid models that combine the advantages of bothedge and regional information are able to control better thedirection of evolution of the contour than the previouslymentioned models For instance the Geodesic-Aided Chan-Vese (GACV) model [28] is a popular hybrid model whichincludes both region and edge information in its formulationAnother example of a hybrid model is the following one

Selective Binary and Gaussian Filtering Regularized(SBGFRLS) Model [68] The SBGFRLS model combinesthe advantages of both the CV and GAC models It utilizesthe statistical information inside and outside the contourto construct a region-based Signed Pressure Force (SPF)function which is used in place of the edge stoppingfunction (ESF) used in the GAC model (recall (9)) The SPFfunction is so called because it tends to make the contour 119862shrink when it is outside the object of interest and expandotherwise The evolution of the contour in the variational

level set formulation of the SBGFRLS model is described bythe following PDE

120597120601

120597119905= spf (119868 (119909)) sdot 120572 1003817100381710038171003817nabla120601

1003817100381710038171003817 (15)

where 120572 is a balloon force parameter and the SPF function spfis defined as

spf (119868 (119909)) =119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2)

max119909isinΩ (119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2))

(16)

where 119888+(119862) and 119888minus(119862) are defined likewise in the CV modelabove One can observe that compared to the CV modelin (14) the Dirac delta term 120575(120601) has been replaced by nabla120601which according to [68] has an effective range on the wholeimage rather than the small range of the former Also thebracket in (14) is replaced by the function spf defined in (16)To regularize the curve 119862 the authors of [68] (following thepractice consolidated in other papers eg [22 68 69]) ratherthan relying on the computationally costly 120583nabla sdot (nabla120601nabla120601)

term convolve the level set curve with a Gaussian kernel 119892120590

that is

120601 larr997888 119892120590lowast 120601 (17)

where the width 120590 of the Gaussian 119892120590has a role similar to the

one of 120583 in (14) of the CV model If the value of 120590 is smallthen the level set function is sensitive to the noise and such asmall value does not allow the level set function to flow intothe narrow regions of the object to be segmented

Overall this model is faster is computationally moreefficient and performs better than the conventional CVmodel as pointed out in [68] However it still has similardrawbacks as the CV model such as its inefficiency inhandling images with several intensity levels its sensitivity tothe contour initialization and its inability to handle imageswith intensity inhomogeneity (arising eg as an effect of slowvariations in object illumination possibly occurring duringthe image acquisition process)

In order to deal with images with intensity inhomo-geneity several authors have introduced in the SPF functionterms that relate to local and global intensity information[34 35 59 70] However these models are still sensitive tocontour initialization and additive noise Furthermore whenthe contour is close to the object boundary the influence ofthe global intensity force may distract the contour from thereal object boundary leading to object leaking [31] that isthe presence of a final blurred contour

In general global models cannot segment successfullyobjects that are constituted by more than one intensity classOn the other hand sometimes this is possible by usinglocal models which rely on local information as their maincomponent in the associated variational level set framework


However suchmodels are still sensitive to the contour initial-ization andmay lead to object leaking Some examples of suchlocal region-based ACMs are illustrated in the following

Local Binary Fitting (LBF) Model [71] The evolution of thecontour in the LBF model is described by the following PDE

120597120601

120597119905= minus 120575

120576(120601) (120582

11198901minus 12058221198902) + V120575

120576(120601) nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

)

+ 120583(nabla2120601 minus nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

))

(18)

where V and 120583 are nonnegative constants nabla2 is the Laplacianoperator 120576 gt 0 and the functions 119890

1and 1198902are defined as

follows

1198901 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

1 (119909))2119889119910

1198902 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

2 (119909))2119889119910

(19)

where 1198911and 119891

2are respectively internal and external gray-

level fitting functions and119892120590(119909) is aGaussian kernel function

of width120590 Also for 120576 gt 0 120575120576(120601) is a suitable regularizedDirac

delta function defined as follows

120575120576 (119909) =

1

120587

120576

1205762 + 1199092 (20)

In more details the functions 1198911and 119891

2are defined as

1198911 (119909) =

119892120590 (119909) [119867 (120601 (119909)) 119868 (119909)]

119892120590 (119909)119867 (120601 (119909))

1198912 (119909) =

119892120590 (119909) [(1 minus 119867 (120601 (119909))) 119868 (119909)]

119892120590 (119909) (1 minus 119867 (120601 (119909)))

(21)

In general the LBF model can produce good segmenta-tions of objects with intensity inhomogeneities Furthermoreit has a better performance than the well-known PiecewiseSmooth (PS) model [33 72] for what concerns segmentationaccuracy and computational efficiency However the LBFmodel only takes into account the local gray-level informa-tion Thus in this model it is easy to be trapped into a localminimum of the energy functional and the model is alsosensitive to the initial location of the active contour Finallyoversegmentation problems may occur

Local Image Fitting (LIF) Energy Model [32] Zhang et alproposed in [32] the LIF energy model to insert local imageinformation in their energy functional The evolution of thecontour in the LIF model is described by the following PDE

120597120601

120597119905= (119868 minus 119868

LFI) (1198981minus 1198982) 120575120576(120601) (22)

where the intensity 119868LFI of the local fitted image LFI is definedas follows

119868LFI

= 1198981119867120576(120601) + 119898

2(1 minus 119867

120576(120601)) (23)

where 1198981and 119898

2are the average local intensities inside and

outside the contour respectivelyThe main idea of this model is to use the local image

information to construct an energy functional which takesinto account the difference between the fitted image and theoriginal one to segment an image with intensity inhomo-geneities The complexity analysis and experimental resultsshowed that the LIF model is more efficient than the LBFmodel while yielding similar results

However the models above are still sensitive to thecontour initialization and to high levels of additive noiseCompared to the two above-mentioned models a modelthat has shown higher accuracy when handling images withintensity inhomogeneity is the following one

Local Region-Based Chan-Vese (LRCV)Model [31]TheLRCVmodel is a natural extension of the already-mentioned Chan-Vese (CV) model Such an extension is obtained by insertinglocal intensity information into the objective functionalThisis the main feature of the LRCV model which provides to itthe capability of handling images with intensity inhomogene-ity which is missing instead in the CV model

The objective functional 119864LRCV of the LRCV model hasthe expression

119864LRCV (119862) = 120582+intin(119862)

(119868 (119909) minus 119888+(119909 119862))

2

119889119909


(119868 (119909) minus 119888minus(119909 119862))

2

119889119909

(24)

where 119888+(119909 119862) and 119888minus(119909 119862) are functions which represent

the local weighted mean intensities of the image aroundthe pixel 119909 assuming that it belongs respectively to theforegroundbackground

119888+(119909 119862) =

intin(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intin(119862) 119892120590 (119909 minus 119910) 119889119910

119888minus(119909 119862) =

intout(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intout(119862) 119892120590 (119909 minus 119910) 119889119910

(25)

where 119892120590is a Gaussian kernel function with int

R2119892120590(119909)119889119909 = 1

and width 120590 gt 0The evolution of the contour in the LRCV model is

described by the following PDE

120597120601

120597119905= 120575 (120601) [minus120582

+(119868 minus 119888

+(119909 120601))

2

+ 120582minus(119868 minus 119888

minus(119909 120601))

2

]

(26)

Equation (26) can be solved iteratively by replacingthe Dirac delta by a smooth approximation and using afinite difference scheme Moreover one can perform also aregularization step in which the current level set function120601 isreplaced by its convolution by a Gaussian kernel with suitablewidth 1205901015840 gt 0


A drawback of the LRCV model is that it relies only onthe local information coming from the current location of thecontour so it is sensitive to the contour initialization

Locally Statistical Active Contour Model (LSACM) [73] Thismodel has been proposed with the aim of handling imagescharacterized by intensity inhomogeneity and of being robustto the contour initialization It can be considered as ageneralization of the Local Intensity Clustering (LIC) modelproposed in [74] which is applicable for both simultaneoussegmentation and bias correction

The evolution of the level set function in the LSACMmodel is controlled by the following gradient descent formu-lation

120597120601

120597119905= (1198892minus 1198891) 120575 (120601) (27)

where 1198891and 119889

2are two functions (one related to the fore-

ground the other one to the background) having suit-able integral representations Due to this fact the LSACMmodel is able to combine the information about the spatialdependencies between pixels belonging to the same classand yields a soft segmentation However like the previousmodel also this one is characterized by a high computationalcost in addition to the limitation of relying on a particularprobabilistic model

22 Supervised Models From a machine learning perspec-tive ACMs for image segmentation can use both super-vised and unsupervised information Both kinds of ACMsrely on parametric andor nonparametric density estima-tion methods to approximate the intensity distributions ofthe subsets to be segmented (eg foregroundbackground)Often in such models one makes statistical assumptionson the image intensity distribution and the segmentationproblem is solved by a Maximum Likelihood (ML) or Maxi-mumA-Posteriori (MAP) probability approach For instancefor scalar-valued images in both parametricnonparametricregion-based ACMs the objective energy functional hasusually an integral form (see eg [75]) whose integrands areexpressed in terms of functions 119890

119894(119909) having the form

119890119894 (119909) = minus log (119901

119894 (119868 (119909))) forall119894 isin I (28)

where I is the number of objects (subsets) Ω119894to be seg-

mented Here 119901119894(119868(119909)) = 119901(119868(119909) | 119909 isin Ω

119894) is the conditional

probability density of the image intensity 119868(119909) conditionedon119909 isin Ω

119894 so the log-likelihood term log (119901

119894(119868(119909))) quantifies

how much an image pixel is likely to be an element ofthe subset Ω

119894 In the case of supervised ACMs the models

119901119894(119868(119909)) are estimated from a training set one for each

subset Ω119894 Similarly for a vector-valued image I(119909) with 119863

components the terms 119890119894(119909) have the form

119890119894 (119909) = minus log (119901

119894 (I (119909))) forall119894 isin I (29)

where 119901119894(I(119909)) = 119901(I(119909) | 119909 isin Ω

119894)

Now we briefly discuss some supervised ACMs whichtake advantage of the availability of labeled training data

As an example Lee et al proposed in [75] a supervised ACMwhich is formulated in a parametric form In the followingwe refer to such a model as a Gaussian Mixture Model(GMM-) based ACM since it exploits supervised trainingexamples to estimate the parameters of multivariate Gaussianmixture densities In such a model the level set evolutionPDE is given for example in the case of multispectral imagesI(119909) by

120597120601

120597119905= 120575 (120601) [120573120581 (120601) + log 119901in (I) minus log 119901out (I)] (30)

where 120573 ge 0 is a regularization parameter and 120581(120601) is theaverage curvature of the level set function 120601

The two terms 119901in(I(119909)) and 119901out(I(119909)) in (30) are thenexpressed in [75] as

119901in (I (119909)) 119901out (I (119909)) =119870

sum

119896=1

120572119896N (120583119896 Σ119896 I (119909)) (31)

where 119870 is the number of computational units N(120583119896 Σ119896 sdot)

119896 = 1 119870 are Gaussian functions with centers 120583119896and

covariance matrices Σ119896 and the 120572

119896rsquos are the coefficients of

the linear combination All the parameters (120572119896 120583119896 Σ119896) are

then estimated from the training examples Besides GMM-based ACMs also nonparametric Kernel Density Estimation(KDE-) based models with Gaussian computational unitshave been proposed in [76 77] with the same aim In the caseof scalar images they have the form

119901in (119868 (119909)) =1

|119871+|

|119871+|

sum

119894=1

K(119868 (119909) minus 119868 (119909

+

119894)

120590KDE)

119901out (119868 (119909)) =1

|119871minus|

|119871minus|

sum

119894=1

K(119868 (119909) minus 119868 (119909

minus

119894)

120590KDE)

(32)

where the pixels 119909+119894and 119909

minus

119894belong respectively to given

sets 119871+ and 119871

minus of training pixels inside the true fore-groundbackground (with cardinalities |119871+| and |119871minus| resp)120590KDE gt 0 is the width of theGaussian kernel used in theKDE-based model and

K (119906) =1

radic2120587exp(minus119906

2

2) (33)

Of course suchmodels can be extended to the case of vector-valued images (in particular replacing 1205902KDE by a covariancematrix)

23 Other Variational Level Set-Based ACMs

Supervised Boundary-Based GAC (sBGAC) Model [78] ThesBGAC model is a supervised level set-based ACM whichwas proposed by Paragios and Deriche with the aim ofproviding a boundary-based framework that is derived by theGAC for texture image segmentation Itsmain contribution isthe connection between theminimization of a GAC objectivewith a contour propagation method for supervised texture


segmentation However sBGAC is still limited to boundary-based information which results in a high sensitivity to thenoise and to the initial contour

Geodesic Active Region (GARM) Model [79] GARM wasproposed with the aim of reducing the sensitivity of sBGACto the noise and to the contour initialization by integratingthe region-based information along with the boundary infor-mation GARM is a supervised texture segmentation ACMimplemented by a variational level set method

The inclusion of supervised examples in ACMs canimprove significantly their performance by constructing aKnowledge Base (KB) to be used as a guide in the evo-lution of the contour However state-of-the-art supervisedACMs often make strong statistical assumptions on theimage intensity distribution of each subset to be modeledSo the evolution of the contour is driven by probabilitymodels constructed based on given reference distributionsTherefore the applicability of such models is limited by howaccurate the probability models are

3 SOM-Based ACMs

Before discussing SOM-based ACMs we shortly review theuse of SOMs as a tool in pattern recognition (hence in imagesegmentation as a particular case)

31 Self-Organizing Maps (SOMs) The SOM [37] which wasproposed by Kohonen is an unsupervised neural networkwhose neurons update concurrently their weights in a self-organizing manner in such a way that during the learningprocess the weights of the neurons evolve adaptively intospecific detectors of different input patterns A basic SOMis composed of an input layer an output layer and anintermediate connection layer The input layer contains aunit for each component of the input vector The outputlayer consists of neurons that are typically located eitheron a 1-119863 or a 2-119863 grid and are fully connected with theunits in the input layer The intermediate connection layer iscomposed of weights (also called prototypes) connecting theunits in the input layer and the neurons in the output layer(in practice one has one weight vector associated with eachoutput neuron where the dimension of the weight vector isequal to the dimension of the input) The learning algorithmof the SOM can be summarized by the following steps

(1) Initialize randomly the weights of the neurons in theoutput layer and select suitable learning rate andneighborhood size around a ldquowinnerrdquo neuron

(2) For each training input vector find the winner neu-ron also called Best Matching Unit (BMU) neuronusing a suitable rule

(3) Update the weights on the selected neighborhood ofthe winner neuron

(4) Repeat Steps (2)-(3) above selecting another traininginput vector until learning is accomplished (ie asuitable stopping criterion is satisfied)

More precisely after its random initialization the weight119908119895of each neuron 119895 is updated at each iteration 119905 through the

following self-organization learning rule

119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)

where 119909(119905) is the input of the SOM at time 119905 120578(119905) is a learningrate and ℎ

119895119887(119905) is a neighborhood kernel around the BMU

neuron 119887 (ie the neuron whose weight vector 119908119887is the

closest to the input 119909(119905)) Both functions 120578(119905) and ℎbn(119905)are designed to be time-decreasing in order to stabilize theweights 119908

119895(119905) for 119905 sufficiently large Usual choices of the

functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)

where 1205780gt 0 is the initial learning rate and 120591

120578gt 0 is a time

constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887

is the distance between the neurons 119895 and 119887and 120590

119887(119905) is a suitable choice for the width of the Gaussian

function in (36)SOMs have been used extensively for image segmenta-

tion but often not in combination with ACMs [80 81] In thefollowing subsection we review in brief some of the existingSOM-based segmentation models which are not related toACMs

311 SOM-Based Segmentation Models Not Related to ACMsIn [82] a SOM-based clustering technique was used as athresholding technique for image segmentation The ideawas to apply the intensity histogram of the image to feeda SOM that divides the histogram into regions Huang etal in [83] proposed to use a two-stages SOM system insegmenting multispectral images (specifically made of threecomponents or channels) In the first stage the goal was toidentify a large initial set of color classes while the secondstage aimed to identify a final batch of segmented clustersIn [84] Jiang et al used SOMs to segment multispectralimages (specifically made of five components) by clusteringthe pixels based on their color and on other spatial featuresThen those clustered regions were merged into a predefinednumber of regions by the application of some morphologicaloperations Concluding SOMs have been extensively usedin the field of segmentation and as stated in [85ndash90] theSOM-based segmentation models proposed in the literatureyielded improved segmentation results compared to thedirect application of the classical SOM

Although SOMs are traditionally associated with unsu-pervised learning in the literature there exist also supervisedSOMs A representative model of a supervised SOM isthe Concurrent Self-Organizing Map (CSOM) [91] whichcombines several SOMs to deal with the pattern classifi-cation problem (hence the image segmentation problemas a particular case) in a parallel processing way with


the aim of minimizing a suitable objective function usuallythe quantization error of the maps In a CSOM each SOM isconstructed and trained individually on a subset of examplescoming only from its associated classThe aim of this trainingis to increase the discriminative capability of the system Sothe training of the CSOM is supervised for what concerns theassigment of the training examples to the various SOMs buteach individual SOM is trained with the SOM specific self-organizing (hence unsupervised) learning rule

We conclude mentioning that when SOMs are usedas supervisedunsupervised image segmentation techniquesthe application of the resulting model usually produces seg-mented objects characterized by disconnected boundariesand the segmentation result is often sensitive to the noise

312 SOM-Based Segmentation Models Related to ACMs Inorder to improve the robustness of edge-based ACMs tothe blur and to ill-defined edge information SOMs havebeen also used in combination with ACMs with the explicitaim of modelling the active contour and controlling itsevolution adopting a learning scheme similar to Kohonenrsquoslearning algorithm [37] resulting in SOM-based ACMs [3839] (which belong in the case of [38 39] to the class ofedge-based ACMs) The evolution of the active contour in aSOM-based ACM is guided by the feature space constructedby the SOM when learning the weights associated with theneurons of themapMoreover other kinds of neural networkshave been used with the aim of approximating the edgemap for example multilayer perceptrons [92] One reasonto prefer SOMs to other neural network models consists inthe specific ability of SOMs to learn for example the edge-map information via their topology preservation property Areview of SOM-based ACMs belonging to the class of edge-based ACMs is provided in the two following subsectionswhereas Section 4 presents a more recent class of SOM-basedACMs combined with variational level set methods

32 An Example of a SOM-Based ACM Belonging to the Classof Edge-Based ACMs The basic idea of existing SOM-basedACMs belonging to the class of edge-based ACMs is tomodeland implement the active contour using a SOM relying inthe training phase on the edge map of the image to updatethe weights of the neurons of the SOM and consequentlyto control the evolution of the active contour The points ofthe edge map act as inputs to the network which is trainedin an unsupervised way (in the sense that no supervisedexamples belonging to the foregroundbackground resp areprovided) As a result during training the weights associatedwith the neurons in the output map move toward pointsbelonging to the nearest salient contour In the following weillustrate the general ideas of using a SOM in modelling theactive contour by describing a classical example of a SOM-based ACM belonging to the class of edge-based ACMswhich was proposed in [38] by Venkatesh and Rishikesh

Spatial Isomorphism Self-Organizing Map (SISOM-) BasedACM [38]This is the first SOM-based ACMwhich appearedin the literature It was proposed with the aim of localizing

the salient contours in an image using a SOM to model theevolving contour The SOM is composed of a fixed numberof neurons (and consequently a fixed number of ldquoknotsrdquoor control points for the evolving curve) and has a fixedstructure The model requires a rough approximation of thetrue boundary as an initial contour Its SOM network isconstructed and trained in an unsupervisedway based on theinitial contour and the edge-map information The contourevolution is controlled by the edge information extractedfrom the image by an edge detector The main steps of theSISOM-based ACM can be summarized as follows

(1) Construct the edgemap of the image to be segmented(2) Initialize the contour to enclose the object of interest

in the image(3) Obtain the horizontal and vertical coordinates of the

edge points to be presented as inputs to the network(4) Construct a SOM with a number of neurons equal to

the number of the edge points of the initial contourand two scalar weights associated with each neuronthe points on the initial contour are used to initializethe SOM weights

(5) Repeat the following steps for a fixed number of iter-ations

(a) Select randomly an edge point and feed its coor-dinates to the network

(b) Determine the best-matching neuron(c) Update the weights of the neurons in the net-

work by the classical unsupervised learningscheme of the SOM [37] which is composed ofa competitive phase and a cooperative one

(d) Compute a neighborhood parameter for thecontour according to the updated weights anda threshold

Figure 3 illustrates the evolution procedure of the SISOM-based ACM On the left-side of the figure the neurons of themap are represented by gray circles while the black circlerepresents the winner neuron associated with the currentinput to the map (in this case the gray circle on the right-hand side of the figure which is connected by the graysegments to all the neurons of the map) On the right-handside instead the positions of the white circles represent theinitial prototypes of the neurons whereas the positions ofthe black circles represent their final values at the end oflearning The evolution of the contour is controlled by thelearning algorithm above which guides the evolution of theprototypes of the neurons of the SOM (hence of the activecontour) using the points of the edge map as inputs to theSOM learning algorithm As a result the final contour isrepresented by a series of prototypes of neurons located nearthe actual boundary of the object to be segmented

We conclude by mentioning that in order to producegood segmentations the SISOM-based ACM requires theinitial contour (which is used to initialize the prototypes ofthe neurons) to be very close to the true boundary of theobject to be extracted and the points of the initial contour


SOM

Initial prototypesConverged prototypes

Figure 3 The architecture of the SISOM-based ACM proposed in[38]

have to be assigned to the neurons of the SOM in a suitableorder if such assumptions are satisfied then the contourextraction process performedby themodel is generally robustto the noise Moreover differently from other ACMs thismodel does not require a particular energy functional to beoptimized

33 Other SOM-Based ACMs Belonging to the Class of Edge-Based ACMs In this subsection we describe other SOM-based ACMs belonging also to the class of edge-based ACMsand highlight their advantages and disadvantages

Time Adaptive Self-Organizing Map (TASOM-) Based ACM[39] The TASOM-based ACM was proposed by Shah-Hosseini and Safabakhsh as a development of the SISOM-based ACM with the aim of inserting neurons incrementallyinto the SOM map or deleting them incrementally thusdetermining automatically the required number of controlpoints of the extracted contour The addition and deletionprocesses are based on the closeness of any two adjacentneurons 119895 and 119895+1 More precisely if the distance 119889(119908

119895 119908119895+1)

between the corresponding weights 119908119895and 119908

119895+1is smaller

than a given threshold 120579119897gt 0 then the two neurons are

merged whereas a new neuron is inserted between the twoneurons if 119889(119908

119895 119908119895+1) is larger than another given threshold

120579ℎgt 0 Moreover at each time 119905 each neuron 119895 is provided

with its specific dynamic learning rate 120578119895(119905) which is defined

as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)

where 120572 isin (0 1) is a constant 119904119891is a positive constant which

controls the slope of 119891 119909(119905) is the input at time 119905 and 119904119897(119905) isa suitable scaling function which makes the SOM networkinvariant to scaling transformations Finally at each time119905 each neuron 119895 is also associated with the neighborhoodfunction ℎ

119895119887(119905) which has the form of (36)

The TASOM-based ACM can overcome one of the mainlimitations of the SISOM-based ACM that is its sensitivityto the contour initialization in the sense that for a successfulsegmentation the initial guess of the contour in the TASOM-based ACM can be even far from the actual object boundaryLikewise in the case of the SISOM-based ACM topologicalchanges of the objects (eg splitting and merging) cannot behandled by the TASOM-based ACM since both models relycompletely on the edge information (instead than on regionalinformation) to drive the contour evolution

Batch Self-Organizing Map (BSOM-) Based ACM [20 93]Thismodel is a modification of the TASOM-based ACM andwas proposed by Venkatesh et al with the aim of dealingbetter with the leaking problem (ie the presence of a finalblurred contour) which often occurs when handling imageswith ill-defined edges Such a problem is due to the explicituse by the TASOM-based ACM of only edge information tomodel and control the evolution of the contour The BSOM-basedACM instead relies on the fact that the image intensityvariation inside a local region can be used in a way to increasethe robustness of the model during the movements of thecontour As a consequence the BSOM-based ACM associatesa region boundary term 119906(119895) with each neuron 119895 in order tocontrol better the movements of the neurons Such a term isdefined as

119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)

where 119873119903is the number of neighborhood points of the

neuron 119895 that are taken into account for the local analysis ofthe region boundary 119868(sdot) is the image intensity function sgnis the signum function and119901

119886119896

119901119887119896

are suitable neighborhoodpoints of the neuron 119895 outside and inside the contourrespectively Now the sign of the difference in (38) betweenthe image intensities at the points 119901

119886119896

and 119901119887119896

should be thesame for all 119896 if the neuron 119895 is near a true region boundaryIn this way the robustness of the model is increased inhandling images with blurred edges At the same time theBSOM-based ACM is less sensitive to the initial guess of thecontour when compared to parametric ACMs like Snakesand to the SOM-based ACMs described above However likeall such models the BSOM-based ACM has not the ability tohandle topological changes of the objects to be segmentedAn extension of the BSOM-based ACM was proposed in[94 95] and applied therein to the segmentation of pupilimages Such a modified version of the basic BSOM-basedACM increases the smoothness of the extracted contour andprevents the extracted contour from being extended over thetrue boundaries of the object

Fast Time Adaptive Self-Organizing Map (FTA-SOM-) BasedACM [96]This is anothermodification of the TASOM-basedACM and it was proposed by Izadi and Safabakhsh withthe aim of decreasing the computational complexity of themethod by using an adaptive speed parameter instead of theone used in [39] which was fixed instead Such an adaptive


SOM

Initial prototypesConverged prototypesSupervised prototypes

Figure 4The architecture of the CFBL-SOM-basedACMproposedin [97]

speed parameterwas also proposedwith the aimof increasingthe speed of convergence and accuracy The FTA-SOM-based ACM is also based on the observation that choosingthe learning rate parameters of the prototypes of the neuronsof the SOM in such a way that they are equal to a largefixed value when they are far from the boundary and to asmall value when they are near the boundary can lead toa significant increase of the convergence speed of the activecontour Accordingly in each iteration the FTA-SOM-basedACM finds the minimum distance of each neuron from theboundary then it sets the associated learning rate as a fractionof that distance

Coarse to Fine Boundary Location Self-Organizing Map(CFBL-SOM-) Based ACM [97] The above SOM-basedACMs work in an unsupervised way as the user is requiredonly to provide an initial contour to be evolved automaticallyIn [97] Zeng et al proposed the CFBL-SOM-based ACMas the first supervised SOM-based ACM that is a model inwhich the user is allowed to provide supervised points (super-vised ldquoseedsrdquo) from the desired boundaries Starting from thiscoarse information the SOM neurons are then employed toevolve the contour to the desired boundaries in a ldquocoarse-to-finerdquo approach The CFBL-SOM-based ACM follows such astrategy when controlling the evolution of the contour Soan advantage of the CFBL-SOM-based ACM over the SOM-based ACMs described above is that it allows to integrateprior knowledge about the desired boundaries of the objectsto be segmented which comes from the user interactionwith the SOM-based ACM segmentation framework Whencompared to such SOM-based ACMs this property providesthe CFBL-SOM-based ACM with the ability of handlingobjects with more complex shapes inhomogeneous intensitydistributions and weak boundaries

Figure 4 illustrates the evolution procedure of the CFBL-SOM-based ACM which is similar to the one of the SISOM-based ACMThe only difference is represented by the dashedcircles which are used as supervised pixels to increase therobustness of the model to the initialization of the contourDue to this reason for a successful segmentation the white

circles in Figure 4 can be initialized even far away fromthe actual boundary of the object differently from Figure 3Finally due to the presence of the supervision this methodalso allows one to handle more complex images

Conscience Archiving andMean-Movement Mechanisms Self-Organizing Map (CAM-SOM-) Based ACM [98] The CAM-SOM-based ACM was proposed by Sadeghi et al as anextension of the BSOM-ACM by introducing three mech-anisms called Conscience Archiving and Mean-movementThe main achievement of the CAM-SOM-based ACM is toallow more complex boundaries (such as concave bound-aries) to be captured and to provide a reduction of thecomputational cost By the Conscience mechanism theneurons are not allowed to ldquowinrdquo too much frequently whichmakes the capture of more complex boundaries possibleThe Archiving mechanism allows a significant reduction inthe computational cost By such mechanism neurons whoseprototypes are close to the boundary of the object to besegmented and whose values have not changed significantlyin the last iterations are archived and eliminated from subse-quent computations Finally in order to ensure a continuousmovement of the active contour toward concave regions theMean-movement mechanism is used in each epoch to forcethe winner neuron to move toward the mean of a set offeature points instead of a single feature point Together theConscience and Mean-movement mechanisms prevent thecontour from stopping the contour evolution at the entranceof object concavities

Extracting Multiple Objects The main limitation of variousof the SOM-based ACMs reviewed above is their inability todetect multiple contours and to recognize multiple objectsA similar problem arises in parametric ACMs such asSnakes To dealwith themultiple contour extraction problemVenkatesh et al proposed in [93] to use a splitting criterionHowever if the initial contour is outside the objects contoursinside an object still cannot be extracted by using such acriterion Sadeghi et al proposed in [98] another splittingcriterion (to be checked at each epoch) such that the maincontour can be divided into several subcontours wheneverthe criterion is satisfied The process is repeated until eachof the subcontours encloses one single object Howeverthe merging process is still not handled implicitly by themodel which reduces its scope especially when handlingimages containing multiple objects in the presence of noiseor ill-defined edges Moreover Ma et al proposed in [99]to use a SOM to classify the edge elements in the imageThis model relies first on detecting the boundaries of theobjectsThen for each edge pixel a feature vector is extractedand normalized Finally a SOM is used as a clustering toolto detect the object boundaries when the feature vectorsare supplied as inputs to the map As a result multiplecontours can be recognized However the model sharesthe same limitations of other models that use a SOM as aclustering tool for image segmentation [80 100 101] resultingin disconnected boundaries and a high sensitivity to thepresence of the noise


Training session Online session

Figure 5 The architecture of the CSOM-CV ACM proposed in [102]

4 SOM-Based ACMs Combined withVariational Level Set Methods

Recently a new class of SOM-based ACMs combined withvariational level set methods has been proposed in [102ndash105] with the aim of taking advantage of both SOMs andvariational level set methods in order to handle imagespresenting challenges in computer vision in an efficienteffective and robust way In this section we describe themaincontributions of such approaches by comparing them withthe above-mentioned active contour models

Concurrent Self-Organizing Map-Based Chan-Vese (CSOM-CV) Model CSOM-CV [102] is a novel regional ACM whichrelies on a CSOM made of two SOMs to approximate theforeground and background image intensity distributionsin a supervised fashion and to drive the evolution of theactive contour accordingly The model integrates such aninformation inside the framework of the Chan-Vese (CV)model hence the name of such a model is ConcurrentSelf-Organizing Map-based Chan-Vese (CSOM-CV) modelThe main idea of the CSOM-CV model is to concurrentlyintegrate the global information extracted by a CSOM froma few supervised pixels into the level-set framework ofthe CV model to build an effective ACM The proposedmodel integrates the advantages of the CSOM as a powerfulclassification tool and of the CVmodel as an effective tool forthe optimization of a global energy functional The evolutionof the contour in theCSOM-CVmodel (which is a variationallevel set method) is described by the following PDE

120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)

where 119890+(119909 119862) and 119890minus(119909 119862) are two energy terms which areused to determine the forces acting inside and outside thecontour respectively They are defined respectively as

119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+

119887(119862) is the prototype of the neuron of the first

SOM that is the BMU neuron to the mean intensity inside

the current contour while 119908minus

119887(119862) is the prototype of the

neuron of the second SOM that is the BMU neuron to themean intensity outside it

Figure 5 illustrates the off-line (ie training session) andon-line components of the CSOM-CV model In the off-line session the foreground supervised pixels are representedin light gray while the background ones are represented indark gray The first SOM is trained using the intensity ofthe foreground supervised pixels whereas the second oneis trained using the intensity of the background supervisedpixels In such a session the neurons of the two SOMsare arranged in such a way that the topological structureof the foreground and background intensity distributionsare preserved Finally in the online session the learnedprototypes of the ldquoforegroundrdquo and ldquobackgroundrdquo neuronsassociated respectively with the two SOMs (and representedin light and dark gray resp in Figure 5) are used implicitlyto control the evolution of the contour toward the true objectboundary

Self-Organizing Active Contour (SOAC)Model [103] Likewisethe CSOM-CV model also the SOAC model combines avariational level set method with the prototypes associatedwith the neurons of a SOM which are learned during the off-line phaseThen in the online phase the contour evolution isimplicitly controlled by the minimization of the quantizationerror of the organized neuronsThe SOACmodel can handleimages withmultiple intensity classes intensity inhomogene-ity and complex distributions with a complicated foregroundand background overlap Compared to CSOM-CV the SOACmodel makes the following important improvement itsregional descriptors 119908+

119887(119909 119862) and 119908minus

119887(119909 119862) (which are used

in a similar way as the ones 119908+119887(119862) and 119908

minus

119887(119862) in (4) and

(41) resp) depend on the pixel location 119909 while CSOM-CVuses the regional descriptors 119908+

119887(119862) and 119908

minus

119887(119862) which are

constant functions So CSOM-CV is a global ACM (ie thespatial dependencies of the pixels are not taken into accountin such a model since it just considers only the averageintensities inside and outside the contour) whereas the SOACmodel makes also use of local information which providesit the ability of handling more complex images Finally


the experimental results reported in [102] have shown thehigher accuracy of the segmentation results obtained bySOACon several synthetic and real images compared to somewell-known ACMs

SOM-Based Chan-Vese (SOMCV) Model [104]This is similarto the CSOM-CV model with the difference that now thetraining of the model is completely unsupervised differentlyfrom the two previous models Likewise for the CSOM-CVmodel the prototypes of the trained neurons encode globalintensity information also in this case The SOMCV modelcan handle images with many intensity levels and complexintensity distributions and it is robust to additive noiseExperimental results reported in [104] have shown the higheraccuracy of the segmentation results obtained by the SOMCVmodel on several synthetic and real images when comparedto the CV active contour model A significant differencewith the CSOM-CV model is that the intervention of thefinal user is significantly reduced in the SOMCV modelsince no supervised information is used Finally SOMCVhasa Self-Organizing Topology Preservation (SOTP) propertywhich allows to preserve the topological structures of theforegroundbackground intensity distributions during theactive contour evolution Indeed SOMCV relies on a setof self-organized neurons by automatically extracting theprototypes of selected neurons as global regional descriptorsand iteratively in an unsupervised way integrates them in theevolution of the contour

SOM-Based Regional Active Contour (SOM-RAC) Model[105] Finally likewise the SOMCV model also the SOM-RAC model relies on the global information coming fromselected prototypes associated with a SOM which is trainedoff-line in an unsupervised way to model the intensitydistribution of an image and used on-line to segmentan identical or similar image In order to improve therobustness of the model global and local information arecombined in the on-line phase differently from the threemodels above The main motivation for the SOM-RACmodel is to deal with the sensitivity of local ACMs tothe contour initialization (which arise eg when intensityinhomogeneity and additive noise occur in the images)through the combination of global and local informationby a SOM-based approach Indeed global information playsan important role to improve the robustness of ACMsagainst the contour initialization and the additive noisebut if used alone it is usually not sufficient to handleimages containing intensity inhomogeneity On the otherhand local information allows one to deal effectively withthe intensity inhomogeneity but if used alone it producesusually ACMs very sensitive to the contour initializationThe SOM-RAC model combines both kinds of informa-tion relying on global regional descriptors (ie suitablyselected weights of a trained SOM) on the basis of localregional descriptors (ie the local weighted mean intensi-ties) In this way the SOM-RAC model is able to integratethe advantages of global and local ACMs by means of aSOM

5 Conclusions and Future Research Directions

In this paper a survey has been provided about the cur-rent state of the art of Active Contour Models (ACMs)with an emphasis on variational level set-based ACM Self-Organizing Map (SOM-) based ACMs and their relation-ships (see Figure 6)

Variational level set-based ACMs have been proposedin the literature with the aim of handling implicitly topo-logical changes of the objects to be segmented Howeversuch methods are usually trapped into local minima of theenergy functional to be minimizedThen SOM-based ACMshave been proposed with the aim of exploiting the specificability of SOMs to learn the edge-map information via theirtopology preservation property and reducing the occurrenceof local minima of the functional to be minimized whichis also typical of parametric ACMs such as Snakes This ispartly due to the fact that such SOM-based ACMs do notrely on an explicit gradient energy term Although SOM-based ACMs belonging to the class of edge-based ACMscan effectively outperform other ACM models in handlingcomplex images most of such SOM-based ACMs are stillsensitive to the contour initialization compared to variationallevel set-based ACMs especially when handling compleximages with ill-defined edges Moreover such SOM-basedACMs have not usually the ability to handle topologicalchanges of the objects For this reason we have concludedthe paper presenting a recently proposed class of SOM-basedACMs which takes advantage of both SOMs and variationallevel set methods with the aims of preserving topologicallythe intensity distribution of the foreground and backgroundin a supervisedunsupervised way and at the same time ofallowing topological changes of the objects to be handledimplicitly

Among future research directions we mention (1) thepossibility of combining inside SOM-based ACMs otheradvantages of variational level set methods in handling thetopological changes in order to obtain a new class of modelswhich are able to handle the topological changes implicitlyand at the same time to avoid trapping into local minima(2) the development of more sophisticated supervisedsemi-supervised SOM-based ACMs based for example on the useof Concurrent Self-Organizing Maps (CSOMs) [91] relyingon regional-based information (eg localglobal statisticalinformation about the intensity texture and color distribu-tion) to guide the evolution of the active contour in a morerobust way (3) the possibility of extending current SOM-based ACMs in such a way that the underlying neuronsare incrementally addedremoved in an automatic way andsuitably trained with the aim of overcoming the limitationof manually adapting the topology of the network and ofreducing the sensitivity of the model to the choice of theparameters (4) the inclusion of other kinds of prior informa-tion (eg shape information) in the models reviewed in thepaper with the aim of handling complex images presentingchallenging problems such as occlusion and (5) possible fur-ther developments of the machine-learning components ofthe reviewed models from a streaming-learning perspectivewhich could lead to a better understanding of video contents


prior info

Regional info

Boundary info

Shape info

Imageshape

Level set-based models

(eg GAC CV LIF and LRCV)

SOM set-based models

(eg SISOM TASOM eBSOM and CAM-SOM)

Level set-SOM based models

(eg CSOM-CV SOAC SOMCV and SOM-RAC)

SOM-based priorinfo

Figure 6 Some relationships between variational level set-based ACMs and Self-Organizing Map (SOM-) based ACMs

through real-time segmentations Such developments couldbe obtained by integrating streaming-learning algorithmsinto the segmentation framework of SOM-based ACMs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

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[2] N Otsu ldquoA threshold selection method from gray-level his-togramsrdquo IEEE Transactions on Systems Man and Cyberneticsvol 9 no 1 pp 62ndash66 1979

[3] P-S Liao T-S Chen and P-C Chung ldquoA fast algorithm formultilevel thresholdingrdquo Journal of Information Science andEngineering vol 17 no 5 pp 713ndash727 2001

[4] Z Musoromy S Ramalingam and N Bekooy ldquoEdge detectioncomparison for license plate detectionrdquo inProceedings of the 11thInternational Conference on Control Automation Robotics andVision (ICARCV rsquo10) pp 1133ndash1138 December 2010

[5] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol8 no 6 pp 679ndash698 1986

[6] I Karoui R Fablet J-M Boucher and J-M Augustin ldquoVaria-tional region-based segmentation using multiple texture statis-ticsrdquo IEEE Transactions on Image Processing vol 19 no 12 pp3146ndash3156 2010

[7] R Adams and L Bischof ldquoSeeded region growingrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol16 no 6 pp 641ndash647 1994

[8] M P Pathegama and O Gol ldquoEdge-end pixel extractionfor edge-based image segmentationrdquo International Journal ofComputer Information Systems and Control Engineering vol 1no 2 pp 434ndash437 2007

[9] L E Baum and T Petrie ldquoStatistical inference for probabilisticfunctions of finite state Markov chainsrdquo The Annals of Mathe-matical Statistics vol 37 pp 1554ndash1563 1966

[10] J B MacQueen ldquoSome methods for classification and analysisof multivariate observationsrdquo in Proceedings of the 5th BerkeleySymposium onMathematical Statistics and Probability vol 1 pp281ndash297 1967

[11] D M Titterington A F Smith and U E Makov StatisticalAnalysis of FiniteMixtureDistributions vol 198 ofWiley series inprobability and mathematical Wiley New York NY USA 1985

[12] A Declercq and J H Piater ldquoOnline learning of Gaussianmixture modelsmdasha two-level approachrdquo in Proceedings of the3rd International Conference on Computer Vision Theory andApplications (VISAPP rsquo08) pp 605ndash611 January 2008

[13] P Singla and P Domingos ldquoDiscriminative training of Markovlogic networksrdquo in Proceedings of the 20th National Conferenceon Artificial Intelligence pp 868ndash873 AAAI Press July 2005

[14] Y N Andrew and I J Michael ldquoOn discriminative vs gener-ative classifiers a comparison of logistic regression and naiveBayesrdquo in Proceedings of the Advances in Neural InformationProcessing Systems 14 (NIPS rsquo01) 2001

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[19] MKass AWitkin andD Terzopoulos ldquoSnakes active contourmodelsrdquo International Journal of Computer Vision vol 1 no 4pp 321ndash331 1988

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[21] S Chen and R J Radke ldquoLevel set segmentation with bothshape and intensity priorsrdquo in Proceedings of the 12th IEEEInternational Conference on Computer Vision (ICCV rsquo09) pp763ndash770 IEEE Kyoto Japan October 2009

[22] G Zhu S Zhang Q Zeng and C Wang ldquoBoundary-basedimage segmentation using binary level set methodrdquo OpticalEngineering vol 46 no 5 Article ID 050501 2007

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[26] T F Chan and L A Vese ldquoActive contours without edgesrdquo IEEETransactions on Image Processing vol 10 no 2 pp 266ndash2772001

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[30] Y Tian F Duan M Zhou and Z Wu ldquoActive contour modelcombining region and edge informationrdquo Machine Vision andApplications vol 24 no 1 pp 47ndash61 2013

[31] S Liu and Y Peng ldquoA local region-based Chan-Vese modelfor image segmentationrdquo Pattern Recognition vol 45 no 7 pp2769ndash2779 2012

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[83] H-Y Huang Y-S Chen and W-H Hsu ldquoColor image seg-mentation using a self-organizing map algorithmrdquo Journal ofElectronic Imaging vol 11 no 2 pp 136ndash148 2002

[84] Y Jiang K Chen and Z Zhou ldquoSOM based image segmen-tationrdquo in Rough Sets Fuzzy Sets Data Mining and GranularComputing G Wang Q Liu Y Yao and A Skowron Edsvol 2639 of Lecture Notes in Computer Science pp 640ndash643Springer Berlin Germany 2003

[85] G Dong and M Xie ldquoColor clustering and learning for imagesegmentation based on neural networksrdquo IEEE Transactions onNeural Networks vol 16 no 4 pp 925ndash936 2005

[86] N C Yeo K H Lee Y V Venkatesh and S H Ong ldquoColourimage segmentation using the self-organizingmap and adaptiveresonance theoryrdquo Image and Vision Computing vol 23 no 12pp 1060ndash1079 2005

[87] A R F Araujo and D C Costa ldquoLocal adaptive receptive fieldself-organizing map for image color segmentationrdquo Image andVision Computing vol 27 no 9 pp 1229ndash1239 2009

[88] A Soria-Frisch ldquoUnsupervised construction of fuzzy measuresthrough self-organizing feature maps and its application incolor image segmentationrdquo International Journal of Approxi-mate Reasoning vol 41 no 1 pp 23ndash42 2006

[89] M Y Kiang M Y Hu and D M Fisher ldquoAn extended self-organizing map network for market segmentationmdasha telecom-munication examplerdquo Decision Support Systems vol 42 no 1pp 36ndash47 2006

[90] D Tian and L Fan ldquoA brain MR images segmentation methodbased on SOMneural networkrdquo in Proceedings of the 1st Interna-tional Conference on Bioinformatics and Biomedical Engineering(ICBBE rsquo07) pp 686ndash689 IEEE Wuhan China July 2007

[91] V-E Neagoe and A-D Ropot ldquoConcurrent self-organizingmaps for pattern classificationrdquo in Proceedings of the 1st IEEEInternational Conference onCognitive Informatics (ICCI rsquo02) pp304ndash312 Calgary Canada 2002

[92] I Middleton and R I Damper ldquoSegmentation of magneticresonance images using a combination of neural networks andactive contourmodelsrdquoMedical Engineering andPhysics vol 26no 1 pp 71ndash86 2004

[93] Y V Venkatesh S K Raja and N Ramya ldquoMultiple contourextraction from graylevel images using an artificial neuralnetworkrdquo IEEE Transactions on Image Processing vol 15 no 4pp 892ndash899 2006

[94] G S Vasconcelos C A C M Bastos T I Ren and G D CCavalcanti ldquoBSOM network for pupil segmentationrdquo in Pro-ceedings of the International Joint Conference on Neural Network(IJCNN rsquo11) pp 2704ndash2709 usa August 2011

[95] C A C M Bastos I R Tsang G S Vasconcelos and G D CCavalcanti ldquoPupil segmentation using pulling amp pushing andBSOMneural networkrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo12) pp2359ndash2364 October 2012

[96] M Izadi and R Safabakhsh ldquoAn improved time-adaptive self-organizingmap for high-speed shapemodelingrdquo Pattern Recog-nition vol 42 no 7 pp 1361ndash1370 2009

[97] D Zeng Z Zhou and S Xie ldquoCoarse-to-fine boundary loca-tion with a SOM-like methodrdquo IEEE Transactions on NeuralNetworks vol 21 no 3 pp 481ndash493 2010

[98] F Sadeghi H Izadinia and R Safabakhsh ldquoA new active cont-our model based on the conscience archiving andmean-move-ment mechanisms and the SOMrdquo Pattern Recognition Lettersvol 32 no 12 pp 1622ndash1634 2011

[99] Y Ma X Gu and Y Wang ldquoContour detection based on self-organizing feature clusteringrdquo in Proceedings of the 3rd Interna-tional Conference onNatural Computation (ICNC rsquo07) vol 2 pp221ndash226 August 2007

[100] W-G Teng and P-L Chang ldquoIdentifying regions of interest inmedical images using self-organizing mapsrdquo Journal of MedicalSystems vol 36 no 5 pp 2761ndash2768 2012

[101] Z Yang Z Bai J Wu and Y Chen ldquoTarget region locationbased on texture analysis and active contour modelrdquo Transac-tions of Tianjin University vol 15 no 3 pp 157ndash161 2009

[102] MM Abdelsamea G Gnecco andMM Gaber ldquoA concurrentSOM-based Chan-Vese model for image segmentationrdquo inAdvances in Intelligent Systems andComputing vol 295 pp 199ndash208 Springer Berlin Germany 2014

[103] M M Abdelsamea G Gnecco and M M Gaber ldquoAn efficientSelf-Organizing Active Contour model for image segmenta-tionrdquo Neurocomputing vol 149 pp 820ndash835 2015

[104] M M Abdelsamea G Gnecco and M M Gaber ldquoA SOM-based Chan-VeseModel for unsupervised image segmentationrdquoIMT Technical Report 2014

[105] M M Abdelsamea and G Gnecco ldquoRobust local-global SOM-based ACMrdquo Electronics Letters vol 51 no 2 pp 142ndash143 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



background) However they have not the ability to segmentimages with a unimodal distribution (eg images containingsmall objects with different intensities) and their outputs aresensitive to noiseThus postprocessing operations are usuallyrequired to obtain a final satisfactory segmentation

The second category of methods is called boundary-based segmentation [4] These methods detect boundariesand discontinuities in the image based on the assumptionthat the intensity values of the pixels linking the foregroundand the background are distinct The firstsecond orderderivatives of the image intensity are usually used to highlightthose pixels (eg this is the case of Sobel and Prewitt edgedetectors [4] as first-order methods and the Laplace edgedetector [1] as a second-order method resp) The differencebetween first and second order methods is that the latter canalso localize the local displacement and orientation of theboundary By far the most accurate technique of detectingboundaries and discontinuities in an image is the Cannyedge detector [5] The Canny edge detector is less sensitiveto noise than other edge detectors as it convolves the inputimage with a Gaussian filter The result is a slightly blurredversion of the input image This method is also very easyto be implemented However it is still sensitive to noiseand leads often to a segmentation result characterized by adiscontinuous detection of the object boundaries

The third category of methods is called region-basedsegmentation [6] Region-based segmentation techniquesdivide an image into subsets based on the assumption thatall neighboring pixels within one subset have a similar valueof some feature for example the image intensity Regiongrowing [7] is the most popular region-based segmentationtechnique In region growing one has to identify at first aset of seeds as initial representatives of the subsets Thenthe features of each pixel are compared to the features ofits neighbor(s) If a suitable predefined criterion is satisfiedthen the pixel is classified as belonging to the same subsetassociated with its ldquomost similarrdquo seed Accordingly regiongrowing relies on the prior information given by the seedsand the predefined classification criterion A second popularregion-based segmentation method is region ldquosplitting andmergingrdquo In suchmethod the input image is first divided intoseveral small regionsThen on the regions a series of splittingand merging operations are performed and controlled by asuitable predefined criterion As region-based segmentationis an intensity-based method the segmentation result ingeneral leads to a nonsmooth and badly shaped boundary forthe segmented object

The fourth category of methods is learning-based seg-mentation [8]There are two general strategies for developinglearning-based segmentation algorithms namely generativelearning and discriminative learning Generative learning[9] utilizes a data set of examples to build a probabilisticmodel by finding the best estimate of its parameters for someprespecified parametric form of a probability distributionOne problem with these methods is that the best estimate ofthe parametersmay not provide a satisfactorymodel becausethe parametricmodel itselfmaynot be correct Another prob-lem is that the classificationclustering framework associatedwith a parametric probabilistic model may not provide an

accurate description of the data due to the limited numberof parameters in the model even in the case in whichits training is well performed Techniques following thegenerative approach include K-means [10] the Expectation-Maximization algorithm [11] and Gaussian Mixture Models[12] Discriminative learning [13 14] ignores probability andattempts to construct a good decision boundary directly Suchan approach is often extremely successful especially when noreasonable parametric probabilistic model of the data existsDiscriminative learning assumes that the decision boundarycomes from another class of nonparametric solutions andchooses the best element of that class according to a suitableoptimality criterion Techniques following the discriminativeapproach include Linear Discriminative Analysis [15] NeuralNetworks [16] and Support Vector Machines [17] The mainproblems with the application of these methods to image seg-mentation are their sensitivity to noise and the discontinuityof the resulting object boundaries

The last category of methods is energy-based segmenta-tion [18 19]This class ofmethods is based on an energy func-tional and deals with the segmentation problem as a func-tional optimization problem whose goal is to partition theimage into regions based on the maximizationminimizationof the energy functional (Loosely speaking a functional isdefined as a function of a function that is a functional takesa function as its input argument and returns a scalar) Themost well-known energy-based segmentation techniques arecalled ldquoactive contoursrdquo or Active Contour Models (ACMs)The main idea of active contours is to choose an initialcontour inside the image domain to be segmented and thenmake such a contour evolve by using a series of shrinking andexpanding operations Some advantages of the active con-tours over the aforementioned methods are that topologicalchanges of the objects to be segmented can be often handledimplicitlyMore importantly complex shapes can bemodeledwithout the need of prior knowledge about the image Finallyrich information can be inserted into the energy functionalitself (eg boundary-based and region-based information)

More specifically ACMs usually deal with the segmen-tation problem as an optimization problem formulated interms of a suitable ldquoenergyrdquo functional constructed in sucha way that its minimum is achieved in correspondence witha contour that is a close approximation of the actual objectboundary Starting from an initial contour the optimizationis performed iteratively evolving the current contour withthe aim of approximating better and better the actual objectboundary (hence the denomination ldquoactive contourrdquo modelswhich are used also formodels that evolve the contour but arenot based on the explicit minimization of a functional [20])In order to guide efficiently the evolution of the current con-tour ACMs allow to integrate various kinds of informationinside the energy functional such as local information (egfeatures based on spatial dependencies among pixels) globalinformation (eg features which are not influenced by suchspatial dependencies) shape information prior informationand a-posteriori information learned from examples (Dueto the possible lack of a precise prior information on theshape of the objects to be segmented in this respect mostACMs make only the assumption that it is preferable to have









s = 0

s = 02

s = 1



120601(x) lt 0



119862 = 119909 isin Ω 119909 = (1199091 (119904) 1199092 (119904)) 0 le 119904 le 1 (1)

where 1199091(119904) and 119909





119862 = 119909 isin Ω 120601 (119909) = 0 (2)


120601 (119909) =

+120588 for119909 isin inside (119862)

0 for119909 isin 119862


(3)




Local Global




CV [26] No Yes



SBGFRLS [68] No Yes



LBF [71] Yes No











SISOM [38] No No





TASOM [39] No No





BSOM [93] No Yes











Table 1 Continued


Local Global

CAM-SOM [98] No Yes






SOAC [103] Yes No









119864 (120601) = 119864in (120601) + 119864out (120601) + 119864119862 (120601) (4)




119864in (120601) = int120601(119909)gt0

119890 (119909) 119889119909 = intΩ

119867(120601 (119909)) sdot 119890 (119909) 119889119909

119864out (120601) = int120601(119909)lt0

119890 (119909) 119889119909 = intΩ

(1 minus 119867 (120601 (119909))) sdot 119890 (119909) 119889119909

119864119862(120601) = int

Ω

1003817100381710038171003817nabla119867 (120601 (119909))1003817100381710038171003817 119889119909 = int

Ω

120575 (120601 (119909)) sdot1003817100381710038171003817nabla120601 (119909)

1003817100381710038171003817 119889119909

(5)


119867(119911) =

1 if 119911 ge 0

0 if 119911 lt 0

120575 (119911) =119889

119889119911119867 (119911)

(6)




120597120601

120597119905= minus

120597119864 (120601)

120597120601 (7)







120597120601

120597119905= 119892

1003817100381710038171003817nabla1206011003817100381710038171003817 (nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

) + 120572) + nabla119892 sdot nabla120601 (8)


119892 =1

1 +1003817100381710038171003817nabla119892120590 lowast 119868

1003817100381710038171003817

2=

1

1 +1003817100381710038171003817119892120590 lowast nabla119868

1003817100381710038171003817

2 (9)









+ 120582+intin(119862)

(119868 (119909) minus 119888+(119862))2

119889119909


(119868 (119909) minus 119888minus(119862))2

119889119909

(10)


119888+(119862) = mean (119868 (119909) | 119909 isin in (119862))


(11)


+ 120582minus


+(119862))2119889119909 and intout(119862)(119868(119909) minus 119888

minus(119862))2119889119909 respec-



119862 = 119909 isin Ω 120601 (119909) = 0 (12)


120601 (119909) =

119889 (119909 119862) 119909 isin in (119862)

0 119909 isin 119862

minus119889 (119909 119862) 119909 isin out (119862)

(13)





120597120601

120597119905= 120575 (120601) [120583nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817


+ 120582minus(119868 minus 119888

minus(120601))2

]

(14)



+(120601) and 119888







120597120601

120597119905= spf (119868 (119909)) sdot 120572 1003817100381710038171003817nabla120601

1003817100381710038171003817 (15)


spf (119868 (119909)) =119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2)

max119909isinΩ (119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2))

(16)



that is

120601 larr997888 119892120590lowast 120601 (17)









120597120601

120597119905= minus 120575

120576(120601) (120582

11198901minus 12058221198902) + V120575

120576(120601) nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

)


nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

))

(18)



follows

1198901 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

1 (119909))2119889119910

1198902 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

2 (119909))2119889119910

(19)

where 1198911and 119891





120575120576 (119909) =

1

120587

120576

1205762 + 1199092 (20)


2are defined as

1198911 (119909) =

119892120590 (119909) [119867 (120601 (119909)) 119868 (119909)]

119892120590 (119909)119867 (120601 (119909))

1198912 (119909) =

119892120590 (119909) [(1 minus 119867 (120601 (119909))) 119868 (119909)]

119892120590 (119909) (1 minus 119867 (120601 (119909)))

(21)



120597120601

120597119905= (119868 minus 119868

LFI) (1198981minus 1198982) 120575120576(120601) (22)


119868LFI

= 1198981119867120576(120601) + 119898

2(1 minus 119867

120576(120601)) (23)

where 1198981and 119898







119864LRCV (119862) = 120582+intin(119862)

(119868 (119909) minus 119888+(119909 119862))

2

119889119909


(119868 (119909) minus 119888minus(119909 119862))

2

119889119909

(24)



119888+(119909 119862) =

intin(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intin(119862) 119892120590 (119909 minus 119910) 119889119910

119888minus(119909 119862) =

intout(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intout(119862) 119892120590 (119909 minus 119910) 119889119910

(25)


R2119892120590(119909)119889119909 = 1



120597120601

120597119905= 120575 (120601) [minus120582

+(119868 minus 119888

+(119909 120601))

2

+ 120582minus(119868 minus 119888

minus(119909 120601))

2

]

(26)






120597120601

120597119905= (1198892minus 1198891) 120575 (120601) (27)

where 1198891and 119889





119890119894 (119909) = minus log (119901

119894 (119868 (119909))) forall119894 isin I (28)






119894(119868(119909))) quantifies






119890119894 (119909) = minus log (119901

119894 (I (119909))) forall119894 isin I (29)

where 119901119894(I(119909)) = 119901(I(119909) | 119909 isin Ω

119894)



120597120601




119901in (I (119909)) 119901out (I (119909)) =119870

sum

119896=1

120572119896N (120583119896 Σ119896 I (119909)) (31)







119901in (119868 (119909)) =1

|119871+|

|119871+|

sum

119894=1

K(119868 (119909) minus 119868 (119909

+

119894)

120590KDE)

119901out (119868 (119909)) =1

|119871minus|

|119871minus|

sum

119894=1

K(119868 (119909) minus 119868 (119909

minus

119894)

120590KDE)

(32)


minus


sets 119871+ and 119871


K (119906) =1


2

2) (33)








3 SOM-Based ACMs









119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)






functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)



constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887


























SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in











s = 0

s = 02

s = 1



120601(x) lt 0



119862 = 119909 isin Ω 119909 = (1199091 (119904) 1199092 (119904)) 0 le 119904 le 1 (1)

where 1199091(119904) and 119909





119862 = 119909 isin Ω 120601 (119909) = 0 (2)


120601 (119909) =

+120588 for119909 isin inside (119862)

0 for119909 isin 119862


(3)




Local Global




CV [26] No Yes



SBGFRLS [68] No Yes



LBF [71] Yes No











SISOM [38] No No





TASOM [39] No No





BSOM [93] No Yes











Table 1 Continued


Local Global

CAM-SOM [98] No Yes






SOAC [103] Yes No









119864 (120601) = 119864in (120601) + 119864out (120601) + 119864119862 (120601) (4)




119864in (120601) = int120601(119909)gt0

119890 (119909) 119889119909 = intΩ

119867(120601 (119909)) sdot 119890 (119909) 119889119909

119864out (120601) = int120601(119909)lt0

119890 (119909) 119889119909 = intΩ

(1 minus 119867 (120601 (119909))) sdot 119890 (119909) 119889119909

119864119862(120601) = int

Ω

1003817100381710038171003817nabla119867 (120601 (119909))1003817100381710038171003817 119889119909 = int

Ω

120575 (120601 (119909)) sdot1003817100381710038171003817nabla120601 (119909)

1003817100381710038171003817 119889119909

(5)


119867(119911) =

1 if 119911 ge 0

0 if 119911 lt 0

120575 (119911) =119889

119889119911119867 (119911)

(6)




120597120601

120597119905= minus

120597119864 (120601)

120597120601 (7)







120597120601

120597119905= 119892

1003817100381710038171003817nabla1206011003817100381710038171003817 (nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

) + 120572) + nabla119892 sdot nabla120601 (8)


119892 =1

1 +1003817100381710038171003817nabla119892120590 lowast 119868

1003817100381710038171003817

2=

1

1 +1003817100381710038171003817119892120590 lowast nabla119868

1003817100381710038171003817

2 (9)









+ 120582+intin(119862)

(119868 (119909) minus 119888+(119862))2

119889119909


(119868 (119909) minus 119888minus(119862))2

119889119909

(10)


119888+(119862) = mean (119868 (119909) | 119909 isin in (119862))


(11)


+ 120582minus


+(119862))2119889119909 and intout(119862)(119868(119909) minus 119888

minus(119862))2119889119909 respec-



119862 = 119909 isin Ω 120601 (119909) = 0 (12)


120601 (119909) =

119889 (119909 119862) 119909 isin in (119862)

0 119909 isin 119862

minus119889 (119909 119862) 119909 isin out (119862)

(13)





120597120601

120597119905= 120575 (120601) [120583nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817


+ 120582minus(119868 minus 119888

minus(120601))2

]

(14)



+(120601) and 119888







120597120601

120597119905= spf (119868 (119909)) sdot 120572 1003817100381710038171003817nabla120601

1003817100381710038171003817 (15)


spf (119868 (119909)) =119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2)

max119909isinΩ (119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2))

(16)



that is

120601 larr997888 119892120590lowast 120601 (17)









120597120601

120597119905= minus 120575

120576(120601) (120582

11198901minus 12058221198902) + V120575

120576(120601) nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

)


nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

))

(18)



follows

1198901 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

1 (119909))2119889119910

1198902 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

2 (119909))2119889119910

(19)

where 1198911and 119891





120575120576 (119909) =

1

120587

120576

1205762 + 1199092 (20)


2are defined as

1198911 (119909) =

119892120590 (119909) [119867 (120601 (119909)) 119868 (119909)]

119892120590 (119909)119867 (120601 (119909))

1198912 (119909) =

119892120590 (119909) [(1 minus 119867 (120601 (119909))) 119868 (119909)]

119892120590 (119909) (1 minus 119867 (120601 (119909)))

(21)



120597120601

120597119905= (119868 minus 119868

LFI) (1198981minus 1198982) 120575120576(120601) (22)


119868LFI

= 1198981119867120576(120601) + 119898

2(1 minus 119867

120576(120601)) (23)

where 1198981and 119898







119864LRCV (119862) = 120582+intin(119862)

(119868 (119909) minus 119888+(119909 119862))

2

119889119909


(119868 (119909) minus 119888minus(119909 119862))

2

119889119909

(24)



119888+(119909 119862) =

intin(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intin(119862) 119892120590 (119909 minus 119910) 119889119910

119888minus(119909 119862) =

intout(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intout(119862) 119892120590 (119909 minus 119910) 119889119910

(25)


R2119892120590(119909)119889119909 = 1



120597120601

120597119905= 120575 (120601) [minus120582

+(119868 minus 119888

+(119909 120601))

2

+ 120582minus(119868 minus 119888

minus(119909 120601))

2

]

(26)






120597120601

120597119905= (1198892minus 1198891) 120575 (120601) (27)

where 1198891and 119889





119890119894 (119909) = minus log (119901

119894 (119868 (119909))) forall119894 isin I (28)






119894(119868(119909))) quantifies






119890119894 (119909) = minus log (119901

119894 (I (119909))) forall119894 isin I (29)

where 119901119894(I(119909)) = 119901(I(119909) | 119909 isin Ω

119894)



120597120601




119901in (I (119909)) 119901out (I (119909)) =119870

sum

119896=1

120572119896N (120583119896 Σ119896 I (119909)) (31)







119901in (119868 (119909)) =1

|119871+|

|119871+|

sum

119894=1

K(119868 (119909) minus 119868 (119909

+

119894)

120590KDE)

119901out (119868 (119909)) =1

|119871minus|

|119871minus|

sum

119894=1

K(119868 (119909) minus 119868 (119909

minus

119894)

120590KDE)

(32)


minus


sets 119871+ and 119871


K (119906) =1


2

2) (33)








3 SOM-Based ACMs









119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)






functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)



constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887


























SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Local Global




CV [26] No Yes



SBGFRLS [68] No Yes



LBF [71] Yes No











SISOM [38] No No





TASOM [39] No No





BSOM [93] No Yes











Table 1 Continued


Local Global

CAM-SOM [98] No Yes






SOAC [103] Yes No









119864 (120601) = 119864in (120601) + 119864out (120601) + 119864119862 (120601) (4)




119864in (120601) = int120601(119909)gt0

119890 (119909) 119889119909 = intΩ

119867(120601 (119909)) sdot 119890 (119909) 119889119909

119864out (120601) = int120601(119909)lt0

119890 (119909) 119889119909 = intΩ

(1 minus 119867 (120601 (119909))) sdot 119890 (119909) 119889119909

119864119862(120601) = int

Ω

1003817100381710038171003817nabla119867 (120601 (119909))1003817100381710038171003817 119889119909 = int

Ω

120575 (120601 (119909)) sdot1003817100381710038171003817nabla120601 (119909)

1003817100381710038171003817 119889119909

(5)


119867(119911) =

1 if 119911 ge 0

0 if 119911 lt 0

120575 (119911) =119889

119889119911119867 (119911)

(6)




120597120601

120597119905= minus

120597119864 (120601)

120597120601 (7)







120597120601

120597119905= 119892

1003817100381710038171003817nabla1206011003817100381710038171003817 (nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

) + 120572) + nabla119892 sdot nabla120601 (8)


119892 =1

1 +1003817100381710038171003817nabla119892120590 lowast 119868

1003817100381710038171003817

2=

1

1 +1003817100381710038171003817119892120590 lowast nabla119868

1003817100381710038171003817

2 (9)









+ 120582+intin(119862)

(119868 (119909) minus 119888+(119862))2

119889119909


(119868 (119909) minus 119888minus(119862))2

119889119909

(10)


119888+(119862) = mean (119868 (119909) | 119909 isin in (119862))


(11)


+ 120582minus


+(119862))2119889119909 and intout(119862)(119868(119909) minus 119888

minus(119862))2119889119909 respec-



119862 = 119909 isin Ω 120601 (119909) = 0 (12)


120601 (119909) =

119889 (119909 119862) 119909 isin in (119862)

0 119909 isin 119862

minus119889 (119909 119862) 119909 isin out (119862)

(13)





120597120601

120597119905= 120575 (120601) [120583nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817


+ 120582minus(119868 minus 119888

minus(120601))2

]

(14)



+(120601) and 119888







120597120601

120597119905= spf (119868 (119909)) sdot 120572 1003817100381710038171003817nabla120601

1003817100381710038171003817 (15)


spf (119868 (119909)) =119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2)

max119909isinΩ (119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2))

(16)



that is

120601 larr997888 119892120590lowast 120601 (17)









120597120601

120597119905= minus 120575

120576(120601) (120582

11198901minus 12058221198902) + V120575

120576(120601) nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

)


nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

))

(18)



follows

1198901 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

1 (119909))2119889119910

1198902 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

2 (119909))2119889119910

(19)

where 1198911and 119891





120575120576 (119909) =

1

120587

120576

1205762 + 1199092 (20)


2are defined as

1198911 (119909) =

119892120590 (119909) [119867 (120601 (119909)) 119868 (119909)]

119892120590 (119909)119867 (120601 (119909))

1198912 (119909) =

119892120590 (119909) [(1 minus 119867 (120601 (119909))) 119868 (119909)]

119892120590 (119909) (1 minus 119867 (120601 (119909)))

(21)



120597120601

120597119905= (119868 minus 119868

LFI) (1198981minus 1198982) 120575120576(120601) (22)


119868LFI

= 1198981119867120576(120601) + 119898

2(1 minus 119867

120576(120601)) (23)

where 1198981and 119898







119864LRCV (119862) = 120582+intin(119862)

(119868 (119909) minus 119888+(119909 119862))

2

119889119909


(119868 (119909) minus 119888minus(119909 119862))

2

119889119909

(24)



119888+(119909 119862) =

intin(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intin(119862) 119892120590 (119909 minus 119910) 119889119910

119888minus(119909 119862) =

intout(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intout(119862) 119892120590 (119909 minus 119910) 119889119910

(25)


R2119892120590(119909)119889119909 = 1



120597120601

120597119905= 120575 (120601) [minus120582

+(119868 minus 119888

+(119909 120601))

2

+ 120582minus(119868 minus 119888

minus(119909 120601))

2

]

(26)






120597120601

120597119905= (1198892minus 1198891) 120575 (120601) (27)

where 1198891and 119889





119890119894 (119909) = minus log (119901

119894 (119868 (119909))) forall119894 isin I (28)






119894(119868(119909))) quantifies






119890119894 (119909) = minus log (119901

119894 (I (119909))) forall119894 isin I (29)

where 119901119894(I(119909)) = 119901(I(119909) | 119909 isin Ω

119894)



120597120601




119901in (I (119909)) 119901out (I (119909)) =119870

sum

119896=1

120572119896N (120583119896 Σ119896 I (119909)) (31)







119901in (119868 (119909)) =1

|119871+|

|119871+|

sum

119894=1

K(119868 (119909) minus 119868 (119909

+

119894)

120590KDE)

119901out (119868 (119909)) =1

|119871minus|

|119871minus|

sum

119894=1

K(119868 (119909) minus 119868 (119909

minus

119894)

120590KDE)

(32)


minus


sets 119871+ and 119871


K (119906) =1


2

2) (33)








3 SOM-Based ACMs









119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)






functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)



constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887


























SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table 1 Continued


Local Global

CAM-SOM [98] No Yes






SOAC [103] Yes No









119864 (120601) = 119864in (120601) + 119864out (120601) + 119864119862 (120601) (4)




119864in (120601) = int120601(119909)gt0

119890 (119909) 119889119909 = intΩ

119867(120601 (119909)) sdot 119890 (119909) 119889119909

119864out (120601) = int120601(119909)lt0

119890 (119909) 119889119909 = intΩ

(1 minus 119867 (120601 (119909))) sdot 119890 (119909) 119889119909

119864119862(120601) = int

Ω

1003817100381710038171003817nabla119867 (120601 (119909))1003817100381710038171003817 119889119909 = int

Ω

120575 (120601 (119909)) sdot1003817100381710038171003817nabla120601 (119909)

1003817100381710038171003817 119889119909

(5)


119867(119911) =

1 if 119911 ge 0

0 if 119911 lt 0

120575 (119911) =119889

119889119911119867 (119911)

(6)




120597120601

120597119905= minus

120597119864 (120601)

120597120601 (7)







120597120601

120597119905= 119892

1003817100381710038171003817nabla1206011003817100381710038171003817 (nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

) + 120572) + nabla119892 sdot nabla120601 (8)


119892 =1

1 +1003817100381710038171003817nabla119892120590 lowast 119868

1003817100381710038171003817

2=

1

1 +1003817100381710038171003817119892120590 lowast nabla119868

1003817100381710038171003817

2 (9)









+ 120582+intin(119862)

(119868 (119909) minus 119888+(119862))2

119889119909


(119868 (119909) minus 119888minus(119862))2

119889119909

(10)


119888+(119862) = mean (119868 (119909) | 119909 isin in (119862))


(11)


+ 120582minus


+(119862))2119889119909 and intout(119862)(119868(119909) minus 119888

minus(119862))2119889119909 respec-



119862 = 119909 isin Ω 120601 (119909) = 0 (12)


120601 (119909) =

119889 (119909 119862) 119909 isin in (119862)

0 119909 isin 119862

minus119889 (119909 119862) 119909 isin out (119862)

(13)





120597120601

120597119905= 120575 (120601) [120583nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817


+ 120582minus(119868 minus 119888

minus(120601))2

]

(14)



+(120601) and 119888







120597120601

120597119905= spf (119868 (119909)) sdot 120572 1003817100381710038171003817nabla120601

1003817100381710038171003817 (15)


spf (119868 (119909)) =119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2)

max119909isinΩ (119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2))

(16)



that is

120601 larr997888 119892120590lowast 120601 (17)









120597120601

120597119905= minus 120575

120576(120601) (120582

11198901minus 12058221198902) + V120575

120576(120601) nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

)


nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

))

(18)



follows

1198901 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

1 (119909))2119889119910

1198902 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

2 (119909))2119889119910

(19)

where 1198911and 119891





120575120576 (119909) =

1

120587

120576

1205762 + 1199092 (20)


2are defined as

1198911 (119909) =

119892120590 (119909) [119867 (120601 (119909)) 119868 (119909)]

119892120590 (119909)119867 (120601 (119909))

1198912 (119909) =

119892120590 (119909) [(1 minus 119867 (120601 (119909))) 119868 (119909)]

119892120590 (119909) (1 minus 119867 (120601 (119909)))

(21)



120597120601

120597119905= (119868 minus 119868

LFI) (1198981minus 1198982) 120575120576(120601) (22)


119868LFI

= 1198981119867120576(120601) + 119898

2(1 minus 119867

120576(120601)) (23)

where 1198981and 119898







119864LRCV (119862) = 120582+intin(119862)

(119868 (119909) minus 119888+(119909 119862))

2

119889119909


(119868 (119909) minus 119888minus(119909 119862))

2

119889119909

(24)



119888+(119909 119862) =

intin(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intin(119862) 119892120590 (119909 minus 119910) 119889119910

119888minus(119909 119862) =

intout(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intout(119862) 119892120590 (119909 minus 119910) 119889119910

(25)


R2119892120590(119909)119889119909 = 1



120597120601

120597119905= 120575 (120601) [minus120582

+(119868 minus 119888

+(119909 120601))

2

+ 120582minus(119868 minus 119888

minus(119909 120601))

2

]

(26)






120597120601

120597119905= (1198892minus 1198891) 120575 (120601) (27)

where 1198891and 119889





119890119894 (119909) = minus log (119901

119894 (119868 (119909))) forall119894 isin I (28)






119894(119868(119909))) quantifies






119890119894 (119909) = minus log (119901

119894 (I (119909))) forall119894 isin I (29)

where 119901119894(I(119909)) = 119901(I(119909) | 119909 isin Ω

119894)



120597120601




119901in (I (119909)) 119901out (I (119909)) =119870

sum

119896=1

120572119896N (120583119896 Σ119896 I (119909)) (31)







119901in (119868 (119909)) =1

|119871+|

|119871+|

sum

119894=1

K(119868 (119909) minus 119868 (119909

+

119894)

120590KDE)

119901out (119868 (119909)) =1

|119871minus|

|119871minus|

sum

119894=1

K(119868 (119909) minus 119868 (119909

minus

119894)

120590KDE)

(32)


minus


sets 119871+ and 119871


K (119906) =1


2

2) (33)








3 SOM-Based ACMs









119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)






functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)



constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887


























SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







120597120601

120597119905= 119892

1003817100381710038171003817nabla1206011003817100381710038171003817 (nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

) + 120572) + nabla119892 sdot nabla120601 (8)


119892 =1

1 +1003817100381710038171003817nabla119892120590 lowast 119868

1003817100381710038171003817

2=

1

1 +1003817100381710038171003817119892120590 lowast nabla119868

1003817100381710038171003817

2 (9)









+ 120582+intin(119862)

(119868 (119909) minus 119888+(119862))2

119889119909


(119868 (119909) minus 119888minus(119862))2

119889119909

(10)


119888+(119862) = mean (119868 (119909) | 119909 isin in (119862))


(11)


+ 120582minus


+(119862))2119889119909 and intout(119862)(119868(119909) minus 119888

minus(119862))2119889119909 respec-



119862 = 119909 isin Ω 120601 (119909) = 0 (12)


120601 (119909) =

119889 (119909 119862) 119909 isin in (119862)

0 119909 isin 119862

minus119889 (119909 119862) 119909 isin out (119862)

(13)





120597120601

120597119905= 120575 (120601) [120583nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817


+ 120582minus(119868 minus 119888

minus(120601))2

]

(14)



+(120601) and 119888







120597120601

120597119905= spf (119868 (119909)) sdot 120572 1003817100381710038171003817nabla120601

1003817100381710038171003817 (15)


spf (119868 (119909)) =119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2)

max119909isinΩ (119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2))

(16)



that is

120601 larr997888 119892120590lowast 120601 (17)









120597120601

120597119905= minus 120575

120576(120601) (120582

11198901minus 12058221198902) + V120575

120576(120601) nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

)


nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

))

(18)



follows

1198901 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

1 (119909))2119889119910

1198902 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

2 (119909))2119889119910

(19)

where 1198911and 119891





120575120576 (119909) =

1

120587

120576

1205762 + 1199092 (20)


2are defined as

1198911 (119909) =

119892120590 (119909) [119867 (120601 (119909)) 119868 (119909)]

119892120590 (119909)119867 (120601 (119909))

1198912 (119909) =

119892120590 (119909) [(1 minus 119867 (120601 (119909))) 119868 (119909)]

119892120590 (119909) (1 minus 119867 (120601 (119909)))

(21)



120597120601

120597119905= (119868 minus 119868

LFI) (1198981minus 1198982) 120575120576(120601) (22)


119868LFI

= 1198981119867120576(120601) + 119898

2(1 minus 119867

120576(120601)) (23)

where 1198981and 119898







119864LRCV (119862) = 120582+intin(119862)

(119868 (119909) minus 119888+(119909 119862))

2

119889119909


(119868 (119909) minus 119888minus(119909 119862))

2

119889119909

(24)



119888+(119909 119862) =

intin(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intin(119862) 119892120590 (119909 minus 119910) 119889119910

119888minus(119909 119862) =

intout(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intout(119862) 119892120590 (119909 minus 119910) 119889119910

(25)


R2119892120590(119909)119889119909 = 1



120597120601

120597119905= 120575 (120601) [minus120582

+(119868 minus 119888

+(119909 120601))

2

+ 120582minus(119868 minus 119888

minus(119909 120601))

2

]

(26)






120597120601

120597119905= (1198892minus 1198891) 120575 (120601) (27)

where 1198891and 119889





119890119894 (119909) = minus log (119901

119894 (119868 (119909))) forall119894 isin I (28)






119894(119868(119909))) quantifies






119890119894 (119909) = minus log (119901

119894 (I (119909))) forall119894 isin I (29)

where 119901119894(I(119909)) = 119901(I(119909) | 119909 isin Ω

119894)



120597120601




119901in (I (119909)) 119901out (I (119909)) =119870

sum

119896=1

120572119896N (120583119896 Σ119896 I (119909)) (31)







119901in (119868 (119909)) =1

|119871+|

|119871+|

sum

119894=1

K(119868 (119909) minus 119868 (119909

+

119894)

120590KDE)

119901out (119868 (119909)) =1

|119871minus|

|119871minus|

sum

119894=1

K(119868 (119909) minus 119868 (119909

minus

119894)

120590KDE)

(32)


minus


sets 119871+ and 119871


K (119906) =1


2

2) (33)








3 SOM-Based ACMs









119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)






functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)



constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887


























SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

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SOM-based priorinfo





References





















































































































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Volume 2014












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Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120597120601

120597119905= 120575 (120601) [120583nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817


+ 120582minus(119868 minus 119888

minus(120601))2

]

(14)



+(120601) and 119888







120597120601

120597119905= spf (119868 (119909)) sdot 120572 1003817100381710038171003817nabla120601

1003817100381710038171003817 (15)


spf (119868 (119909)) =119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2)

max119909isinΩ (119868 (119909) minus ((119888

+(119862) + 119888

minus(119862)) 2))

(16)



that is

120601 larr997888 119892120590lowast 120601 (17)









120597120601

120597119905= minus 120575

120576(120601) (120582

11198901minus 12058221198902) + V120575

120576(120601) nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

)


nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

))

(18)



follows

1198901 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

1 (119909))2119889119910

1198902 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

2 (119909))2119889119910

(19)

where 1198911and 119891





120575120576 (119909) =

1

120587

120576

1205762 + 1199092 (20)


2are defined as

1198911 (119909) =

119892120590 (119909) [119867 (120601 (119909)) 119868 (119909)]

119892120590 (119909)119867 (120601 (119909))

1198912 (119909) =

119892120590 (119909) [(1 minus 119867 (120601 (119909))) 119868 (119909)]

119892120590 (119909) (1 minus 119867 (120601 (119909)))

(21)



120597120601

120597119905= (119868 minus 119868

LFI) (1198981minus 1198982) 120575120576(120601) (22)


119868LFI

= 1198981119867120576(120601) + 119898

2(1 minus 119867

120576(120601)) (23)

where 1198981and 119898







119864LRCV (119862) = 120582+intin(119862)

(119868 (119909) minus 119888+(119909 119862))

2

119889119909


(119868 (119909) minus 119888minus(119909 119862))

2

119889119909

(24)



119888+(119909 119862) =

intin(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intin(119862) 119892120590 (119909 minus 119910) 119889119910

119888minus(119909 119862) =

intout(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intout(119862) 119892120590 (119909 minus 119910) 119889119910

(25)


R2119892120590(119909)119889119909 = 1



120597120601

120597119905= 120575 (120601) [minus120582

+(119868 minus 119888

+(119909 120601))

2

+ 120582minus(119868 minus 119888

minus(119909 120601))

2

]

(26)






120597120601

120597119905= (1198892minus 1198891) 120575 (120601) (27)

where 1198891and 119889





119890119894 (119909) = minus log (119901

119894 (119868 (119909))) forall119894 isin I (28)






119894(119868(119909))) quantifies






119890119894 (119909) = minus log (119901

119894 (I (119909))) forall119894 isin I (29)

where 119901119894(I(119909)) = 119901(I(119909) | 119909 isin Ω

119894)



120597120601




119901in (I (119909)) 119901out (I (119909)) =119870

sum

119896=1

120572119896N (120583119896 Σ119896 I (119909)) (31)







119901in (119868 (119909)) =1

|119871+|

|119871+|

sum

119894=1

K(119868 (119909) minus 119868 (119909

+

119894)

120590KDE)

119901out (119868 (119909)) =1

|119871minus|

|119871minus|

sum

119894=1

K(119868 (119909) minus 119868 (119909

minus

119894)

120590KDE)

(32)


minus


sets 119871+ and 119871


K (119906) =1


2

2) (33)








3 SOM-Based ACMs









119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)






functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)



constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887


























SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






120597120601

120597119905= minus 120575

120576(120601) (120582

11198901minus 12058221198902) + V120575

120576(120601) nabla sdot (

nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

)


nabla120601

1003817100381710038171003817nabla1206011003817100381710038171003817

))

(18)



follows

1198901 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

1 (119909))2119889119910

1198902 (119909) = int

Ω

119892120590(119909 minus 119910) (119868 (119910) minus 119891

2 (119909))2119889119910

(19)

where 1198911and 119891





120575120576 (119909) =

1

120587

120576

1205762 + 1199092 (20)


2are defined as

1198911 (119909) =

119892120590 (119909) [119867 (120601 (119909)) 119868 (119909)]

119892120590 (119909)119867 (120601 (119909))

1198912 (119909) =

119892120590 (119909) [(1 minus 119867 (120601 (119909))) 119868 (119909)]

119892120590 (119909) (1 minus 119867 (120601 (119909)))

(21)



120597120601

120597119905= (119868 minus 119868

LFI) (1198981minus 1198982) 120575120576(120601) (22)


119868LFI

= 1198981119867120576(120601) + 119898

2(1 minus 119867

120576(120601)) (23)

where 1198981and 119898







119864LRCV (119862) = 120582+intin(119862)

(119868 (119909) minus 119888+(119909 119862))

2

119889119909


(119868 (119909) minus 119888minus(119909 119862))

2

119889119909

(24)



119888+(119909 119862) =

intin(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intin(119862) 119892120590 (119909 minus 119910) 119889119910

119888minus(119909 119862) =

intout(119862) 119892120590 (119909 minus 119910) 119868 (119910) 119889119910

intout(119862) 119892120590 (119909 minus 119910) 119889119910

(25)


R2119892120590(119909)119889119909 = 1



120597120601

120597119905= 120575 (120601) [minus120582

+(119868 minus 119888

+(119909 120601))

2

+ 120582minus(119868 minus 119888

minus(119909 120601))

2

]

(26)






120597120601

120597119905= (1198892minus 1198891) 120575 (120601) (27)

where 1198891and 119889





119890119894 (119909) = minus log (119901

119894 (119868 (119909))) forall119894 isin I (28)






119894(119868(119909))) quantifies






119890119894 (119909) = minus log (119901

119894 (I (119909))) forall119894 isin I (29)

where 119901119894(I(119909)) = 119901(I(119909) | 119909 isin Ω

119894)



120597120601




119901in (I (119909)) 119901out (I (119909)) =119870

sum

119896=1

120572119896N (120583119896 Σ119896 I (119909)) (31)







119901in (119868 (119909)) =1

|119871+|

|119871+|

sum

119894=1

K(119868 (119909) minus 119868 (119909

+

119894)

120590KDE)

119901out (119868 (119909)) =1

|119871minus|

|119871minus|

sum

119894=1

K(119868 (119909) minus 119868 (119909

minus

119894)

120590KDE)

(32)


minus


sets 119871+ and 119871


K (119906) =1


2

2) (33)








3 SOM-Based ACMs









119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)






functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)



constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887


























SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







120597120601

120597119905= (1198892minus 1198891) 120575 (120601) (27)

where 1198891and 119889





119890119894 (119909) = minus log (119901

119894 (119868 (119909))) forall119894 isin I (28)






119894(119868(119909))) quantifies






119890119894 (119909) = minus log (119901

119894 (I (119909))) forall119894 isin I (29)

where 119901119894(I(119909)) = 119901(I(119909) | 119909 isin Ω

119894)



120597120601




119901in (I (119909)) 119901out (I (119909)) =119870

sum

119896=1

120572119896N (120583119896 Σ119896 I (119909)) (31)







119901in (119868 (119909)) =1

|119871+|

|119871+|

sum

119894=1

K(119868 (119909) minus 119868 (119909

+

119894)

120590KDE)

119901out (119868 (119909)) =1

|119871minus|

|119871minus|

sum

119894=1

K(119868 (119909) minus 119868 (119909

minus

119894)

120590KDE)

(32)


minus


sets 119871+ and 119871


K (119906) =1


2

2) (33)








3 SOM-Based ACMs









119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)






functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)



constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887


























SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

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3 SOM-Based ACMs









119908119895 (119905 + 1) = 119908

119895 (119905) + 120578 (119905) ℎ119895119887 (119905) (119909(119905)minus 119908119895 (119905)) (34)






functions above are

120578 (119905) = 1205780exp(minus 119905

120591120578

) (35)



constant and

ℎ119895119887 (119905) = exp(minus(

1198892

119895119887

1205902

119887(119905)

)) (36)

where 119889119895119887


























SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






















SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




SOM






119895 119908119895+1)


119895+1is smaller






as follows

120578119895 (119905 + 1) = (1 minus 120572) 120578119895 (119905) + 120572119891(

10038171003817100381710038171003817119909(119905)minus 119908119895 (119905)

10038171003817100381710038171003817

119904119891sdot 119904119897 (119905)

) (37)






119906 (119895) =

119873119903

sum

119896=1

sgn (119868 (119901119886119896

) minus 119868 (119901119887119896

)) (38)



119886119896

119901119887119896


119886119896

and 119901119887119896




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




SOM















120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









120597120601

120597119905= 120575 (120601) [minus120582

+119890++ 120582minus119890minus] (39)


119890+(119909 119862) = (119868(119909) minus 119908

+

119887(119862))2

(40)

119890minus(119909 119862) = (119868(119909) minus 119908

minus

119887(119862))2

(41)

where 119908+








119887(119909 119862) and 119908minus

119887(119909 119862) (which are used


minus

119887(119862) in (4) and


119887(119862) and 119908

minus

119887(119862) which are











prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in












prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




prior info

Regional info

Boundary info

Shape info

Imageshape







SOM-based priorinfo





References





















































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







































































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in


































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Documents

Review Article On the Relationship between Variational ...eprints.imtlucca.it/3125/1/109029.pdf · Review Article On the Relationship between Variational Level Set-Based and SOM-Based