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A self-learning segmentation frameworkthe Taguchi approachDing-Horng Chen, Yung-Nien Sun *

Department of Computer Science and Information Engineering, National Cheng-Kung University, No 1 University Road, Tainan, Taiwan, ROC

Received 4 November 1999; accepted 26 January 2000

Abstract

The detection of object boundary is an interesting and challenging task in computer vision and medical image processing. The activecontour model (snake model) has attracted much attention for object boundary detection in the past decade. However, due to the lack of understanding on the effect of different energy terms to the behavior of related objective functions for an image, the assignment of weights fordifferent energy terms in this model is usually fullled empirically. Few discussions have been brought out specically for assigning theseweights automatically. In this paper, a novel self-learning segmentation framework, based on the snake model is proposed and applied to thedetection of cardiac boundaries from ultrasonic images. The framework consists of a learning section and a detection section, and provides atraining mechanism to obtain the weights from a desired object contour given manually. This mechanism rst employs Taguchi's method todetermine the weight ratios among distinct energy terms, followed by a weight renement step with a genetic algorithm. The rened weightscan be treated as the a priori knowledge embedded in the manually dened contour and be used for subsequent contour detection.Experiments with both synthetic and real echocardiac images were conducted with satisfactory outcomes. Results also show that the presentmethod can be used to analyze successive images of the same object with only one training contour. Finally, the validity of the weightdetermining process was veried by the analysis of variance method (ANOVA). q 2000 Elsevier Science Ltd. All rights reserved.

Keywords : Snake model; Taguchi's method; Genetic algorithm; ANOVA

1. Introduction

Object segmentation is a critical task in the early stage of vision and medical image processing applications. In mostcases, such a task serves as a primary pre-processing step todistinguish the foreground object from its background.Object segmentation is generally classied into two maincategories, i.e. region-based methods [1,2], and boundary-based methods [3,4].

In region-based methods, they testify the homogeneity of the target object and generate the object shape, using regiongrowing and merging techniques. The major drawback of this type of method is that irregular boundaries and cavitiesinside an object are often generated. On the contrary, bound-ary-based methods, which only deal with the informationalong the object boundary, can give much better boundarysmoothness without cavitation problems. The usual toolsbeing employed in boundary-based methods include, localltering approaches such as Canny edge detector [5], Sobeledge operator, or energy minimization mechanisms like the

active contour model (i.e. snake model) [6] and balloonmodels [7].

In this paper, we focus on the active contour model,which has been the most frequently used approach oncontour segmentation, due to its exibility and practicality[3,816]. This model formulates the boundary detectionproblem as an energy minimization process for an energyfunction, and simulates the contour behavior by the manip-ulation of different energy terms. The conventional snakemodel makes an assumption that the denition of the energyterms can describe the dynamic behavior of the deformedcontour. Therefore, the minimization of the energy functioncan achieve an optimal contour, as dened by the preferredimage properties. In general, an energy function consists of three major feature forces, namely, the internal force, theexternal force (also known as image force ), and theconstraint force. The internal force limits the contour tobe a smooth curve. The image force is a function that corre-sponds to certain characteristic functions of a contour point.The constraint force can be included to restrict the contourto a specic shape expectation.

One of the major problems in the snake model is thatit is hard to determine the appropriate ratio (i.e. weights)between different forces in the total energy function of the

Computerized Medical Imaging and Graphics 24 (2000) 283296PERGAMON

Computerized Medical Imaging

and Graphics

0895-6111/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved.PII: S0895-6111(00) 00023-9

www.elsevier.com/locate/compmedimag

* Corresponding author. Tel.: 1 886-6-2757575, ext. 62520; fax:1 886-6-2747076.

E-mail address: [email protected] (Y.-N. Sun).

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model, due to the lack of understanding on the effects of each force on the energy function. Therefore, few convin-cing or systematic ways to acquire reasonable weights thatreect the contribution of the corresponding energy termsare available, and such weights are generally assignedempirically or an arbitrary global-weight (i.e. a same weightfor all feature forces) is used. However, in cases where thereare non-homogenous image properties around the vicinityof the deformed contour, empirical or global-weight assign-ment could render the previously mentioned basic assump-tion of the snake model invalid, and the desired contour

usually cannot be achieved. Although satisfactory weightscan be obtained eventually, through tedious and time-consuming manual trial-and-error processes, there is nostandard procedure to prove that the weights thus obtainedare optimum ones.

In this paper, we propose a novel algorithm, which can beused to automatically detect the contour of a series of dynamic images from a single object, or the contour for aset of similar images. The present proposed method containstwo sections, i.e. the training section and the detectionsection, as illustrated in Fig. 1. The training section aimsto extract the embedded internal knowledge (as representedby the weights of feature forces) of an image contour,through an image outline provided by an expert, usingTaguchi's method and the Genetic Algorithms (GA). Thedetection section then utilizes these weights to detect thecontour of desired objects.

The present method rst trains the computer to learn fromknowledge embedded in a training contour of an imageprovided by an expert who is familiar with such images(for instance, a cardiac surgeon for a ultrasonic echocardiacimage). It rst adopts the expert's sketch as the initialcontour, then use this contour as a template in the snakemodel, and attempts to analyze the embedded informationof the sketch, by assigning proper weights for different

feature forces. Subsequently, image features forces (includ-ing the internal force, external force, and constraint force)are calculated, and a set of weights is assigned to each force,to reect its signicance. Taguchi's method is used in thepresent method to reduce the variable complexity in theexperiment design, and act as a standard procedure to deter-mine the initial weights, which are then rened by the GA.The rened weights can be regarded as the a priori exper-tise, viz. the high level knowledge, acquired from thehuman sketched contour. In the detection section, thegreedy method was adopted to obtain the contour, using

the a priori knowledge. By iterative minimization of theweighted energy function along different assigned search-ing lines, the optimally t contour positions can belocated.

In the following parts of this paper, Section 2 introducesthe theoretical background and the detailed algorithm,including a demonstration on the extraction of desiredcontours for ultrasonic cardiac images. The experimentalresults and their discussions are presented in Section 3,with the conclusions being given in Section 4.

2. The self-learning snake model

Details of the present method are described in thissection, which include the basic snake model, the learningsection and detection section. The active contour model (orsnake model ) which can locate the position of a smoothcontour in a 2D image was rst introduced by Kass et al.[6]. However, its simplicity and exibility in implementa-tion found wide applications in both 2D and 3D imagery[17,18]. In this paper, we demonstrate with 2D illustration, anew self-learning mechanism for the snake model. This newdevelopment teaches the snake model to carry out borderdetection tasks according to the given training.

D.-H. Chen, Y.-N. Sun / Computerized Medical Imaging and Graphics 24 (2000) 283296 284

Fig. 1. The system diagram of the proposed framework.

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2.1. The snake model

The snake model is dened with a function E to representthe energy of a contour, as dened by:

E snake v s V

E int v s 1 E ext v s 1 E con v s ds 1

where v(s) is a point on the curve with C , E int , E ext , and E conrepresenting the previously mentioned internal, external andconstraint forces, respectively, and V is the supportingdomain.

In the energy minimization process of the snake model,the internal, external and constraint forces play differentroles in the contour detection process. The internal forceprovides the a priori knowledge about the contour behaviorsuch as smoothness and continuity. The external (image)force represents the local features of the image functionsuch as edges, lines, regions or textures. The constraintforce restricts the deformation of the contour to specicgeometrical properties, e.g. degree of curvature, of theimage plane.

To clarify the concept in more detail, the energy functionE can be rewritten as:

v U3 E v 1

0E v s ds

1

0w1ivH s i

2 1 w2ivHHs i 1 P v s ds

2

where vHand vHHdenote the rst and the second derivatives of v, respectively. The derivatives retain the curve smoothnessand P is the potential function associated with the externalforces, with P usually being designed according to thedesired properties in contour extraction. For example, if we want the contour to approach the edge points, the edgegradient can be included as one of the potential elds.Sometimes, the constraint force is also employed to limitthe evolution of the contour to certain geometric shapeconditions.

The contour deformation is achieved by using an energy-minimizing approach and begins with an initial contour that

is given either by a human sketch or by pre-processing. Asearch is made on a set of searching lines (or curves) thatextend from the initial contour, to locate the optimal contourthat has the minimum energy according to the given energydenition. These searching lines (curves) are usually ortho-gonal to the initial contour and locate uniformly along thearc length.

2.2. Training section

Let C be the training contour given by an expert on animage. In the present proposed framework, the trainingcontour provides the bases to extract human knowledge.The contour is generally sampled with some control points,on which the search of the optimal contour position isperformed. Let P i, 1 # i # n be the ith control point of the training contour. For each control point, m featuressuch as intensity, gradient magnitude, or curvature, aredened to represent the local image properties. Thus, the

m feature values (i.e. u ij, 1 # i # n ; 1 # j # m for eachcontrol point P i in the training contour C could be written ina matrix form:

Q

Q 1

Q 2

.

.

.

Q n

Hfffffffd

Iggggggge

u 11 u 12 u 1m

u 21 u 22 u 2m

.

.

.] ..

.

u n1 u n2 u nm

Hfffffffd

Iggggggge

3

The feature value u ij represents the local image properties of the control point P i on the training contour C . Fig. 2 showsthe illustrative sketch of control points on the trainingcontour.

2.2.1. The energy functionFor a training contour C , its energy function can be

dened as:

E AQ n

i 1

m

j 1a iju ij 4

where

A

A1

A2

.

.

.

An

Hfffffffd

Iggggggge

a 11 a 12 a 1m

a 21 a 22 a 2m

.

.

.] ..

.

a n1 a n2 a nm

Hfffffffd

Iggggggge

and m j 1 a ij 1 ; 1 # i # n ; () is the scalar product opera-tor. The weight a ij reects the signicance of the featureforce u ij. The values of u ij such as intensity, gradient magni-tude, curvature or other image properties, are normalized tothe same scale. Therefore, feature values with higherweights have a more signicant inuence in the totalcontour energy. In practice, we usually dene the preferred


Fig. 2. The illustrative sketch of the training contour C.

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image properties with higher feature values. The energyterms we used in this study include:

1. Intensity the intensity of the object was higher than the

background, higher intensity values will increase thecorresponding feature values.2. First-order derivative resists the contour from

stretching.3. Second-order derivative resists the contour from

bending.4. Curvature is used to preserve smoothness in object

shape.5. Gradient denotes the edginess of the boundary. High

gradient implies high possibility of being boundarypoints. In this study, the Sobel edge operator is used tocompute the gradient magnitude.

6. Template energy the term used to dene the templateenergy is usually problem-dependent [1014]. It denotesa prototype of the object shape present in the image andcan be viewed as the constraint force in the snake model.In this study, the ratio of the radius on the control point tothe overall average radius is calculated as the templateenergy.

In fact, there are still many other alternatives to deneimage properties for the desired object contour. Forinstance, texture, intensity contrast, or some specic lter-ing output may also be good image features in the energycomputation. The following section presents a systematic

approach to select the most useful features from a givenset of image features.

2.2.2. Initial estimation of weights by Taguchi's method As the feature values u ij are xed for a given training

contour C , the initial population of GA is used as the startingpoint to approximate the exact weights, according to theproposed paradigm. Since we are aiming at obtaininglocal adaptive weight assignment (i.e. different weights forthe same feature force at different control points), a largenumber of experimental factor levels will result, which willpose a demanding challenge for computation. For instance,if there are n control points, each with m feature values, thenmn combinations in selecting the initial population will haveto be tested. Such a large number of experiments will notonly involve too much computation time, but also makes theoptimal selection of model parameters infeasible. Thus, aproper estimation of initial weight is necessary before GAtreatment. As Taguchi's method provides a simplied,

robust, and reproducible means to handle multiple factorselection problems, it is thus used for the initial estimationof the weights a ij.

Taguchi's method was rst developed for quality engi-neering purposes [19,20]. However, the robust optimizationscheme makes it possible to be applied to many other appli-cations. During the design stage of an experimental scheme,the most important task is to control the variability of differ-ent experimental factors. The technique of laying out condi-tions of experiments involving multiple factors, is alsoknown as factorial design of experiment, and was rst intro-duced by Fisher in the 1920s [21]. Although a full factorial

experimental design can describe all possible conditions fora given set of factors, this will result in a large number of experiments. To make an experimental design easier toimplement, a compact design that involves a manageablenumber of experiments is needed. Taguchi's methodsuccessfully resolves two difculties in compacting experi-mental design. First, orthogonal arrays (OAs) [22] thatrepresent the possible experimental conditions were intro-duced. Secondly, a standard procedure to analyze the resultsperformed by the adopted experimental design is provided.Taguchi's method also compares the main effects of eachexperimental factor and analyzes the performance of eachfactor to acquire the best experimental conguration.Finally, the ANVOA technique can then be used to evaluatethe outcome of experimental results to obtain a betterexperimental conguration.

The rst step of designing an experiment with knownnumber of factors in Taguchi's method is to select a mostsuitable OA, which is designed to cover all the possibleexperiment conditions and the factor combination. In thepresent study, since there are six factors, represented by A, B, C , D , E and F ; with ve levels (a , b, c, d and e) for eachfactor, the orthogonal array L 25 (5 6) that involves 25 trialruns is selected to perform the initial weights estimationof the segmentation framework. Table 1 shows an instance


Table 1An instance of the orthogonal array L 25 (5 6)

A B C D E F

1 a a a a a a2 a b b b b b3 a c c c c c

4 a d d d d d 5 a e e e e e6 b a b c d e7 b b c d e a8 b c d e a b9 b d e a b c10 b e a b c d 11 c a c e b d 12 c b d a c e13 c c e b d a14 c d a c e b15 c e b d a c16 d a d b e c17 d b e c a d 18 d c a d b e

19 d d b e c a20 d e c a d b21 e a e d c b22 e b a e d c23 e c b a e d 24 e d c b a e25 e e d c b a

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of the L 25 (5 6 ) OA, with each row representing a trial con-dition with factor levels indicated by the letters in the row.The vertical columns of OA correspond to the factorsspecied in the study.

The factor levels a , b, c, d and e are the physical experi-mental values of each factor, which are being used toperform the trial runs. However in the present study, sincethere are no physical or pre-dened conditions that can be

utilized, all feature values are normalized to the same scale,and the factor levels around 1, i.e. a 0 : 25 ; b 0 : 5 ; c1 : 0; d 1 : 25 and e 1 : 5 ; are assigned as the given experi-ment conditions. The factor levels are then substituted intoa ij in the energy optimization scheme according to Eq. (4).Taguchi's method is then used to calculate the main effect of u ij at different factor levels a ij (i.e. factor levels a , b, c, d ande as dened above), and the combination with the highestenergy is chosen. However, the outcome of Taguchi'smethod only provides a single value for each factor level.In the present study that involves multiple control points,such single values (equivalent to the weight of the respectivefeature force) will have to be assigned to all control points,i.e. global-weight assignment. Therefore, such outcomes canonly be used as the initial states of the GA treatment atsubsequent stages. The detailed implementation of Taguchi'smethod can be found in related references [19,20,22].

2.2.3. Rened weights by GAIn this stage, GA is carried out to ne-tune the global-

weights obtained by Taguchi's method. GA was inspired bythe natural genetics and rst proposed by Holland [23]. Themost powerful feature of GA is its highly parallel and adap-tive property compared with other optimization techniques.These features make GA particularly useful in handling

simultaneous optimizations of a whole population, whileavoiding local minimum solutions.

Generally speaking, GA employs an evolutionary strat-egy that incorporates a selection and search mechanism toachieve a near optimal solution of a given objective func-tion. The objective function is being approached iteratively,with each iteration during the optimization process knownas a generation . In GA treatment, populations of binarystrings are rst chosen as the initial solutions with respectto the objective function. These binary strings represent thecandidate solutions by encoding its numerical values to 0sand 1s. Given a pre-dened integer range, the candidatesolutions are mapped linearly to the range with a xednumber of binary bits, which are used to construct the binarystrings that carry the solution of the desired objective func-tion, and are regarded as genes as in genetics. To evaluatethe tness of the genes, an objective function, called tness function is used to evaluate each binary string (actuallyrepresenting a candidate solution), and a tness value that

reects the performance of the solution is assigned to eachstring. Before the tness value reaches a pre-decided termi-nation criterion, the looping is repeated. The terminationcriterion generally varies with the given problems.

The GA contains three types of operations, namely, selec-tion, crossover and mutation. The selection operation simu-lates nature's survival-of-the-ttest phenomenon, and can becarried out by setting appropriate criteria for selecting indi-vidual genes. Crossover simulates the gene interchange ingenetics. Suppose the binary string length is l, an integerranging from 1 to l 2 1 is randomly selected as a crossoverpoint. Further, a probability value is used as the crossover

rate to indicate the necessity of performing the crossover.When the randomly selected integer is greater than thecrossover rate, crossover operations will take place. Givena pair of binary strings, the crossover operation exchangesthe bit content beyond the crossover point, to generate a newbinary string.

Mutation operations can also proceed after crossover.This operation ips the bit content of the binary stringfrom 1 to 0, and vice versa, thereby providing a mechanismto keep the solution away from the local minimum. A prob-ability value called mutation rate is given to determine thenecessity of performing mutation. Similar to crossover,mutation will proceed only if the value of a random pickedinteger is greater than the mutation rate.

The owchart of GA is sketched as in Fig. 3. Details of GA can be found in a related reference [23].

At the beginning of the GA process, the weights acquiredby Taguchi's method are used as the initial population of GA, and the tness function is dened as:

t 1

1 1 e E 5

where E ni 1m j 1 a ij u ij ;

m j 1 a ij 1; 1 # i # n :

GA evaluates the weights that are bound in a giveninterval around the initial population. After the previously


Fig. 3. The owchart of GA.

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Fig. 4. The conceptual sketch for the contour detection.

Fig. 5. The synthetic image for verifying the effectiveness of the proposed method. (a) The original image. (b) The training contour with manual drawn circles.(c) The normalized feature values on the searching lines.

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mentioned training process is completed, a set of weightsthat represent the signicance of the feature force values isobtained, i.e.

A

A1

A2

.

.

.

An

Hfffffffd

Iggggggge

a 11 a 12 a 1m

a 21 a 22 a 2m

.

.

.] ..

.

a n1 a n2 a nm

Hfffffffd

Iggggggge

andm

j 1a ij 1 ; 1

# i # n (6)

It is worth noting that the acquired weight set is only apossible solution in the solution space, with respect to thetness function. The solution space spanned by the tnessfunction is an n-dimensional hyperspace and each weight setacquired by the GA mechanism approaches to a local mini-mum over the hyperspace. Thus, each weight set is actuallya testing point to nd the real minimum of the tness func-tion. By the central limit theorem, a large number of non-degenerate identically distributed random variables willapproach to a normal distribution. Therefore, the meanand variance will converge after being tested severaltimes. In the present method, the average of the resulting


Fig. 6. The detection section: (a) the initial contour and the result contour (innerthe initial contour, outerthe result); (b) the energy prole of the resultcontour after 9 iterations; (c), (d), (e) and (f) the deformation process of the rst, second, third and fth iteration, respectively.

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weights obtained from each test is chosen to be the repre-sentative weight for the features. For each control point inthe training contour, a distinct weight is obtained and

assigned to the corresponding feature. These weights arethe indicators for the inuence of the features in energyoptimization.

2.3. Detection section

In this section, we use the trained weights obtained in thetraining section to detect the object contour. The weightingparameters, which represent the knowledge embedded in thesketched contour provided by an expert, indicate the relativesignicance of feature forces in the energy function system.In this study, the conventional greedy method is used tominimize the value of the resulting energy function.

Given a target image T , let C 0 be the initial contour in T .C 0 is equally sampled with n control points P i, 1 # i # n :For a specic control point P i, let xi be any one of thecandidate points lying on the searching line L i passingthrough the control point and normal to C 0 . Let a ij, 1 # j #m be the trained weights for the i-th control point and j-thfeature, where m is the number of features. Fig. 4 shows thesearching line on the object contour.

The intermediate contour is located by looking for thepoint with the highest energy along the searching lines, asdened by Eq. (4). With such points being detected on all

searching lines, a new intermediate contour can beconstructed. After a number of iterations, the intermediatecontour will converge to the desired contour.

3. Results and discussions

Experiments were carried out with both synthetic andmulti-plane transesophageal echocardiac images to testifythe present method. The ANOVA analysis is also used toevaluate the successfulness and validity of these experimen-tal results. Further, application of the present method todetect the boundaries of a series of ultrasonic echocardiac

images, without repetitive training, was also demonstrated,to show the suitability of the present method for treatingsequential images that retain similar image properties orfor treating similar images. Re-training is only necessarywhen a sudden change in sequential images occurs.

3.1. Experiments with synthetic image

A synthetic image with one oval shape object in the


Fig. 7. The comparative studies. (a) The result of the weight set obtained by the proposed method. (b) The result of the equally-weighted method.

Fig. 8. The other synthetic image experiments. (a) and (b): the different initial contour settings do not affect the resulting contours.

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center surrounded by a varying background is created andtested as shown in Fig. 5. The object and its surroundingbackground both have gradually changing intensity, but thedirections of their intensity change are opposite. The imageis blurred with a smoothing lter and Gaussian noise added.Therefore, the intensity changes between the object and itsbackground are not homogenous along the boundary. Undersuch circumstances, it is difcult to dene energy terms inthe traditional snake model. Further, manual assignment of weights with respect to different energy terms is compli-cated and tricky. Thus, a simple weight assignment willbe hardly suitable to all changes in local properties alongthe boundary.

The training contour in Fig. 5(a) is sketched manually andsampled with 20 equally spaced control points as shown inFig. 5(b). The features being considered for the image

include intensity value, gradient magnitude, rst derivativevalue, second derivative value, curvature, and templateenergy. All these feature values are normalized to thesame scale, and Fig. 5(c) shows plots of the normalizedirregular and complicate feature prole along the searchinglines. The goal of the present method is to nd suitableweights among different energy terms and blend thesefeature proles into an integrated one, in order to fulllthe task of contour detection in complex environments.The OA dened in Table 1 is used to compute the initialweights by Taguchi's method. Once the initial weights areobtained, GA is applied to rene these weights of the givenenergy functions. There are two conditions to end the train-ing section, one is to simply stop GA iteration after a xednumber of generations; the other is to measure the differenceof the tness values between two successive generations,


Fig. 9. The real image experiment. (a) The original image. (b) The training contour drawn by the expert. (c) The resulting contour detected by the proposedmethod.

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and stop the iteration when their difference is smaller than agiven threshold value. In the synthetic image experiment, theweight training operation was stopped after 100 iterations.

In the synthetic image experiment, after training with theexpert's sketched contour on an image, the same image wasused to verify the detection performance of the presentmethod, using the greedy method to detect the targetcontour. Fig. 6(a) shows both the irregular initial contourand the nal oval shape contour. Fig. 6(b) shows the energyproles after 9 iterations. It is obvious that the resulting

energy prole becomes more regular by adopting theweights acquired in the training section, thus showing thatwith the present method, contour detection can be easier and

more precise. The deformation process is also demon-strated, with the rst, second, third and the fth iterationresults shown in Fig. 6(c)(f), respectively. Despite theimage properties of the boundary prole not being homo-geneous, the present method can still successfully convergeto the desired contour.

To demonstrate the effectiveness of the present method, acomparative study on two sets of weights was conducted onthe same image using the same initial contour, as depicted inFig. 7. The rst set of weights was obtained by the present

method, and the second set simply employed arbitraryglobal-weights. Results of these two experiments after 9iterations (Fig. 7), show that the global-weight approach


Fig. 10. The real image experiment (part two). (a) The third original image of the cardiac sequence. (b) Without re-training, the same initial contour stillproduces a reliable resulting contour using the trained weight acquired by Fig. 9.

Table 2ANOVA results on validity of differences observed. Yes: F

p. F a ; No: F

p# F a

Point F p

F 0.05 F 0.025 a 0: 05 a 0: 025

1 3.971884098 2.244703978 2.609809258 Yes Yes2 2.266573918 2.244703978 2.609809258 Yes No3 2.611998377 2.244703978 2.609809258 Yes Yes4 4.496893489 2.244703978 2.609809258 Yes Yes5 4.729724441 2.244703978 2.609809258 Yes Yes6 3.145555259 2.244703978 2.609809258 Yes Yes

7 1.22628374 2.244703978 2.609809258 No No8 3.830804254 2.244703978 2.609809258 Yes Yes9 2.355824833 2.244703978 2.609809258 Yes No10 3.996787785 2.244703978 2.609809258 Yes Yes11 3.793322368 2.244703978 2.609809258 Yes Yes12 3.703114097 2.244703978 2.609809258 Yes Yes13 2.761463569 2.244703978 2.609809258 Yes Yes14 4.918862555 2.244703978 2.609809258 Yes Yes15 3.41773355 2.244703978 2.609809258 Yes Yes16 3.290474207 2.244703978 2.609809258 Yes Yes17 3.445763895 2.244703978 2.609809258 Yes Yes18 3.009224704 2.244703978 2.609809258 Yes Yes19 3.682369806 2.244703978 2.609809258 Yes Yes20 2.165461 2.244703978 2.609809258 No No

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could not reach the desired contour. Further, it was foundthat for the global-weight approach, increasing the number

of iterations still could not lead to a nal converging solu-tion for the detected contour, due to the presence of largeamount of local minima when approaching the nal contour.In contrast, the present method successfully achieves thedesired contour only after nine iterations. Therefore, theresults show that the assignment of global-weights willnot provide a satisfactory solution. It was also shown inlater experiments, that the present method is also applicablefor detecting contours of real cardiac images. Fig. 8 showsanother two synthetic image experiments with differentinitial contours using the present method. The resultingcontours are not affected by different initial contours, show-ing the robustness of the training paradigm.

3.2. Experiment with real images

A series of multi-plane transesophageal echocardiacimages are used in this part of the experiment. The trainingprocess was carried out after a manually drawn standardtraining contour was prepared. The weights are derivedrst by Taguchi's method and then rened by GA optimiza-tion. Again, the features used in this case are intensity,gradient magnitude, rst and second derivative values,curvature, and template energies. The rened weightswere obtained by GA with 100 iterations. Fig. 9(a) is the

rst image in the cardiac image sequence. The non-homo-geneous image properties along the cardiac image contour,

makes human decision for proper weights in the conventionsnake model very difcult. Fig. 9(b) is the heart shape train-ing contour drawn by an expert, with the centroid indicatedby the small inner circle. To prove that the characteristics of the human sketch contour could be learned during the train-ing section, the present method was applied to the sameimage, using an arbitrary initial contour. From the resultingdetected contour Fig. 9(c), it can be found that the detectedcontour is almost identical with the training contour. Thisdemonstrates that by proper weight assignment, the presentmethod can enable the traditional snake model to adopt toimage characteristics along the cardiac contour. Fig. 10(a) isthe third image of the given echocardiac image sequence.Using the same initial contour, and trained with the sketchon the rst image in Fig. 9(b), the resulting contour of Fig.10(a) is shown in Fig. 10(b), which exhibits a high degree of agreement with the image. Therefore, results in Fig. 10 alsoshow that, the present segmentation framework is both self-adaptive and re-usable, even though the test image isnot the original one. It was also found that the trainedweights could be reused for several continuous framesin the same sequence of echocardiac motion images.Around two or three training processes were requiredin the boundary detection of a complete cardiacsequence of 12 images.


Table 3The Taguchi experiment using the OA dened in Table 1. The asterisked rows indicate the equally-weighted case (experiment 1), the best case (experiment 6),and the worst case (experiment 13), respectively

Exp # Gradient Intensity Curvature First Second Template Fitness

1p

0.1667 0.1667 0.1667 0.1667 0.1667 0.1667 7.9913252 0.0909 0.1818 0.1818 0.1818 0.1818 0.1818 8.356503

3 0.0625 0.1875 0.1875 0.1875 0.1875 0.1875 8.4934444 0.0476 0.1905 0.1905 0.1905 0.1905 0.1905 8.5651765 0.0385 0.1923 0.1923 0.1923 0.1923 0.1923 8.609318

6p

0.1176 0.0588 0.1176 0.1765 0.2353 0.2941 10.198367 0.1176 0.1176 0.1765 0.2353 0.2941 0.0588 6.9026868 0.1176 0.1765 0.2353 0.2941 0.0588 0.1176 8.0415939 0.1176 0.2353 0.2941 0.0588 0.1176 0.1765 7.10338810 0.1176 0.2941 0.0588 0.1176 0.1765 0.2353 8.89205711 0.1667 0.0556 0.1667 0.2778 0.1111 0.2222 9.67689212 0.1667 0.1111 0.2222 0.0556 0.1667 0.2778 8.790809

13p

0.1667 0.1667 0.2778 0.1111 0.2222 0.0556 5.97841714 0.1667 0.2222 0.0556 0.1667 0.2778 0.1111 7.41230115 0.1667 0.2778 0.1111 0.2222 0.0556 0.1667 8.44316416 0.2105 0.0526 0.2105 0.1053 0.2632 0.1579 7.351849

17 0.2105 0.1053 0.2632 0.1579 0.0526 0.2105 8.37087118 0.2105 0.1579 0.0526 0.2105 0.1053 0.2632 9.97125919 0.2105 0.2105 0.1053 0.2632 0.1579 0.0526 7.31413220 0.2105 0.2632 0.1579 0.0526 0.2105 0.1053 6.52488121 0.2500 0.0500 0.2500 0.2000 0.1500 0.1000 7.02485522 0.2500 0.1000 0.0500 0.2500 0.2000 0.1500 8.54522423 0.2500 0.1500 0.1000 0.0500 0.2500 0.2000 7.7477524 0.2500 0.2000 0.1500 0.1000 0.0500 0.2500 8.7158225 0.2500 0.2500 0.2000 0.1500 0.1000 0.0500 6.475992

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3.3. Verication of the experiments

As mentioned previously, the present method assumedthat the weights within the snake model represent theprior knowledge of the boundary behavior and thus shouldbe distinct according to the feature computation. In thissection, both theoretical proof and visual inspections wereapplied to verify such assumptions.

3.3.1. The ANOVA modelThe results so far agree with the assumption that the

weight of each feature force at different control pointsshould be different. To prove that such an agreement isnot due to noise or statistical error, the ANOVA techniquewas used to analyze some of the results. ANOVA is a statis-tical technique, which is used to test the differences in mean

values between two or more categories of independentvariables.

In this part of the study, a test image and a trainingcontour sampled with 20 control points was rst chosen(Fig. 9(b)). The feature values of the control points on theboundary were then calculated and the initial weightscomputed by Taguchi's method. These initial weightswere applied to GA for 50 iteration cycles, and the averageweights thus obtained were chosen as the rened trainedweights. These rened trained weights represent the signi-cance of different energy features. To prove the validity of the rened trained weights, two hypothesis H 0 and H 1 wereput forward as follows:

H 0 : m11 m12 m1m mnm ; H 1 : not all mij ; 1 # i # n ; 1 # j # m are equal ;


Fig. 11. The comparison test. (a) The best case (experiment 6 in Table 3). (b) The worst case (experiment 13 in Table 3). (c) The equally-weighted case(experiment 1 in Table 3).

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where u ij is the rened weight obtained by GA. The appro-priate decision rule in the single-factor ANOVA model tocontrol the condence level ( a ) is as follows:

If F p

# F 1 2 a ; conclude H 0;If F

p. F 1 2 a ; conclude H 1 , where F

p

is calculated inthe ANOVA model.

In testing H 0 versus H 1 , there are two possible types of errors: H 0 can be falsely classied as H 1 or H 1 can be falselyclassied as H 0 . The rst error type is called Type I error ora false alarm and the second error type is called Type II error or a miss . The ANVOA analysis is performed with50 weight sets. To avoid the Type I error, we applied the F statistics to assert our assumption, with a 0 : 05 : Table 2shows the ANOVA table with 20 control points. We can ndthat only 2 control points disagree with the assumption H 1when a 0 : 05 : This result implies the assumption beingmade, was valid and reasonable.

3.3.2. Visual inspectionIn this section, we like to show the visual difference

between the weight set obtained by the proposed methodand by setting global-weights. We perform the vericationtest with the initial weights acquired by Taguchi's method.

Table 3 shows the experiment performance using Tagu-chi's OA dened in Table 1. The best tness occurs inexperiment 6 and the worst case occurs in experiment 13.The global-weight experiment was conducted in experiment1 according to the OA L 25 (5 6). Given the same initialcontour, the outcome of the best and worst tness experi-ments, together with the global-weight experiment were

compared, to assert the present assumption. Fig. 11(a)shows the result by using the best weight set obtained byTaguchi's method. The detected contour is almost the sameas the desired one provided by experts. On the contrary, theresults of the worst case and the global-weight experimentshown in Fig. 11(b) and (c), respectively, show that neitherexperiment gave satisfactory results. These strongly supportthe assumption in the present proposed method, i.e. the apriori knowledge about the shape can be expressed by thedistinct weight assignment. Further, the present proposedmethod can also improve the segmentation result more ef-ciently and robustly.

4. Conclusions

A novel self-learning segmentation framework based onthe snake model is proposed, and details of the related algo-rithm described. The present method dealt with the mostimportant and difcult problem of the conventional snakemodel, namely the weight assignment of different featureforces. Experiments using both synthetic and real echocar-diac images were made to evaluate the performance of thisproposed method. It was shown that the present methodcould extract the embedded knowledge of a contour

provided by an expert, and successfully assign the weightsfor different energy terms within the snake model, asevidenced by the correct determination of boundarycontours for images with inhomogeneous background inten-sity change. Further, it was also demonstrated that aftertraining with one contour, the present method could beapplied to determine the boundaries of successive imagesof the same object, thus showing its learning capability toadapt for local image properties by adjusting the weights of feature forces. The self-learning mechanism signicantlyimproved contour detection in comparison with the tradi-tional snake model and had obviously enabled the snakemodel to learn to trace an object contour as humans do.All these results, together with the verications by theANOVA method shows that, the basic assumptions for thepresent method are valid and provide a reliable and robustalternative for weights assignment in a snake model forimage boundary determination. It is also envisaged that thepresent method can be extended to other applications, such as

deformable template matching and object recognition.

Acknowledgements

This work was supported in part by grant NSC 88-2213-E006-035 from the National Science Council of Taiwan,ROC.

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Ding-Horng Chen received his BS degree in Mathematics in 1989 andhis MS degree in Information Engineering in 1993, all from NationalCheng-Kung University, Taiwan. Currently he is a PhD candidate at theDepartment of Computer Science and Information Engineering,National Cheng-Kung University. His research interests focus oncomputer graphics, medical image processing, image processing andcomputer vision. He is a member of IEEE, SPIEthe International

Society of Optical Engineering and the Chinese Association of ImageProcessing and Pattern Recognition.

Yung-Nien Sun received his BS degree from National Chiao-TungUniversity, Hsin-chu, Taiwan, Republic of China, in 1978 and hisMS and PhD degrees from University of Pittsburgh, Pittsburgh, Penn-sylvania, in 1983 and 1987, respectively. He was an Assistant Scientistwith the Brookhaven National Laboratory, New York from 1987 to1989, and he is currently Professor at the Department of ComputerScience and Information Engineering, National Cheng-Kung Univer-sity, where he joined in 1989 as an associate professor. He has beenworking on image processing and computer vision since 1982 and haspublished more than 100 papers, half of them in refereed journals. Hiscurrent research interests are in medical image analysis, computergraphics, and virtual reality. He is a member of IEEE, Sigma-Xi, theChinese Association of Image Processing and Pattern Recognition, andthe Chinese Association of Biomedical Engineering.

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