Vessel Extraction in Medical Images by Wave Propagation and Traceback

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 2, FEBRUARY 2001 117

Vessel Extraction in Medical Images byWave-Propagation and Traceback

Francis K. H. Quek*, Member, IEEEand Cemil Kirbas

Abstract—This paper presents an approach for the extractionof vasculature from angiography images by using a wave propaga-tion and traceback mechanism. We discuss both the theory and theimplementation of the approach. Using a dual-sigmoidal filter, welabel each pixel in an angiogram with the likelihood that it is withina vessel. Representing the reciprocal of this likelihood image as anarray of refractive indexes, we propagate a digital wave throughthe image from the base of the vascular tree. This wave “washes”over the vasculature, ignoring local noise perturbations. The ex-traction of the vasculature becomes that of tracing the wave alongthe local normals to the waveform. While the approach is inher-ently single instruction stream multiple data stream (SIMD), wepresent an efficient sequential algorithm for the wave propagationand discuss the traceback algorithm. We demonstrate the effec-tiveness of our integer image neighborhood-based algorithm andits robustness to image noise.

Index Terms—Traceback, vascular extraction, wave propaga-tion.

I. INTRODUCTION

A VARIETY of medical imaging techniques such as X-rayangiography, computed tomography (CT) and magnetic

resonance imaging (MR) imaging /angiography are capable ofobtaining data on vasculature. As with the processing of mostmedical images, signal noise, drift in image intensity and lackof image contrast pose significant challenges to the extractionof blood vessels. In X-ray angiography, for example, the con-sistency of the pixel intensity is dependent on a number of fac-tors. Angiograms are obtained by injecting the patient with aradio-opaque dye before imaging. The consistency of this dyein the vessel, the depth of the vessel, and noise in the imagingprocess result in images that pose a challenge, even for a humanexpert, to interpret. Furthermore, because X-ray angiograms areprojections of the three-dimensional (3-D) reality into a two-di-mensional (2-D) representation, there is a fair amount of selfocclusion among the vessels. Hence, one cannot extract thesevessels directly using the image intensities or gradients (edges)alone. This paper describes a wave propagation-based approach

Manuscript received November 15, 1999; revised October 25, 2000. Thiswork was supported by the Whitaker Foundation under Biomedical EngineeringResearch Grant 96-0458. The Associate Editor responsible for coordinating thereview of this paper and recommending its publication was M. Sonka.Asteriskindicates corresponding author.

*F. K. H. Quek is with Vision Interfaces and Systems Laboratory (VISLab),Department of Computer Science and Engineering, Wright State University,Dayton, OH 45435-0001 USA (e-mail: [email protected])

C. Kirbas is with Vision Interfaces and Systems Laboratory (VISLab),Department of Computer Science and Engineering, Wright State University,Dayton, OH 45435-0001 USA.

Publisher Item Identifier S 0278-0062(01)01233-2.

to the extraction of this vasculature from the image data to ob-tain its connection morphology and the structure of each vessel.

II. V ASCULAR EXTRACTION FROM IMAGE DATA

Researchers have taken different approaches to this problem.We shall briefly review these under the following headings.

• Traditional pattern recognition techniques.• Tracking-based approaches.• Model-based approaches.• Artificial intelligence approaches.• Neural Network-based approaches.• Miscellaneous tube-like object detection approaches.• Front Propagation methods.

Representative of traditional pattern recognition approachesare Ritchings and Colchester [1] who apply a syntactic patternrecognition scheme, and Thackray and Nelson [2] who use mor-phological and adaptive thresholding operations. The formerprocesses X-ray angiograms by applying an edge detector andpairing the resulting edge segments that may be parallel op-posing edges of a vessel segment. These may be thought of as“ribbon segments” which may be grouped to obtain extendedvessel tracts. Each of these segments are labeled asnormal,widening, unsure, andabnormaldepending on the shape of theopposing edge segments using a syntactic pattern recognitionsystem. The system does not attempt to determine the structureof the arterial system. The goal was to obtain these labels for thediagnosis of vascular abnormality. Thakray and Nelson [2] de-scribe an approach which extracts vascular segments using a setof eight morphological operators, each of which represents anoriented vessel segment (in eight orientations). The system alsoapplies an adaptive thresholding scheme to extract the vascularsegments from the intensity image. The system was used to ex-tract vessel segments in a capillary angiogram of mice, and doesnot extract the vascular interconnection structure. It appears thatthe range of vessel widths the system handles is limited by thesetting of the eight morphological operators.

The second set of approaches are vascular tracking ap-proaches. Liu and Sun [3] present an approach that extractsextended tracts of vasculature in X-ray angiograms by anadaptive tracking algorithm. Given an initial point within avessel, they apply an “extrapolation update” scheme [4] thatinvolves the estimation of local vessel trajectories. Once asegment has been tracked, it is deleted in the angiogram imageby growing the “deletion intensity value” over the grey levelsrepresenting the vessel. This procedure is performed recursivelyto extract the vascular tree. This algorithm requires the user to

0278–0062/01$10.00 © 2001 IEEE

118 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 2, FEBRUARY 2001

specify vessel starting points, and does not appear extensibleto 3-D extraction. Aylwardet al. [5] describe an approach bywhich the medial axes tubular objects such as vessels in anangiogram are approximated as directed “intensity ridges.”As with [3], these ridges are tracked by estimating the localvessel directions. The authors show results of a vascular treeextracted from a MR angiogram. This required a fair amountof user intervention (105 mouse clicks in all). Stevensonetal. [6] propose a system by which the user locates vesselbifurcations and vessels are tracked between these bifurcations.The uniqueness of their work is that they use these vesselsegments extracted from two different X-ray viewpoints toestimate the 3-D structure of the vasculature. Parvinet al. [7]describe their work in extracting the structure of DNA chainsfrom epi-fluorescence microscopy images. These structuresare tubular objects and their technique might have applicationto the extraction of vasculature. They apply a hierarchicalfeature detection approach in which low-level edge segmentsare grouped as opposing sides of a tube by examining the localsymmetries. They extract these contours by applying high-levelconstraints to guide the extraction. These constraints areformulated as a cost function that determines which competingedge segments are recruited to the contour. They also use thehigh-level information of the tubes, thus obtained, to guide theextraction of the medial axes of these tubes.

The third class of approaches are model-based in that theyapply explicit vessel models to extract the vasculature. The workof Thackray and Nelson [2] described earlier may be thought ofas model-based in that the eight morphological operators are es-sentially explicit oriented vessel models. Kleinet al.[8] describean approach to extract vessels from X-ray angiograms using de-formable spline models or snakes. In their approach, the userprovides an initial estimate of the location of the vascular en-tity, and the system refines the estimate by deforming a snake tominimize some energy function. They use a B-spline model intheir snake implementation. The energy function defines suchconstraints as the smoothness or coherence of the contour, thecloseness the contour is to image edge pixels, and the compact-ness of the boundary. They use a Gabor filter to determine theimage (or edge) energy term to attract the snake. The approachis most suitable for the accurate extraction of vascular segments.The amount of user interaction and computation required makesit impractical for extracting entire vascular structures.

The fourth class of approaches may be described as artifi-cial-intelligence-based. Stansfield [9] describes a rule-basedexpert system to segment coronary vessels from digital sub-tracted angiograms. Her system extracts vessel segments astrapezoidal units using an OPS-5 production system. The rule setis used to determine which edge segments may participate theformation of these trapezoidal strips and which segments arisefrom image noise. The system does not combine these units toform an extended vascular structure. Smetset al. [10] present aknowledge-based system for the delineation of blood vessels onsubtracted angiograms. The system encodes general knowledgeabout appearance of blood vessels in these images in the formof 11 rules (e.g., that vessels have high-intensity center lines,comprise high-intensity regions bordered by parallel edges etc.).These rules facilitate the formulation of a four-level hierarchy

(pixels, center lines, bars, and segments) each of which is derivedfrom the preceding level by a subset of the 11 rules. They showresults of their system that indicate that the system is successfulwhere the image contains high contrast between the vessel andthe background, and that the system has considerable problemsat vessel bifurcations and self-occlusions.

The fifth approach in the research literature employs neuralnetworks in image segmentation and vessel detection. Nekoveiand Sun [11] describe their back-propagation network for thedetection of blood vessels in X-ray angiography. This systemapplies the neural network directly to the angiogram pixelswithout prior feature detection. Since angiograms are typicallyvery large, the network is applied to a small subwindow whichslides across the angiogram. The pixels of this subwindow aredirectly fed as input to the network. Prelabeled angiograms areused as the training set to set the network’s weights. A modifiedversion the common delta-rule is to obtain these weights. Thissystem does not extract the vascular structure. Its purpose is tolabel the pixels as vessel or nonvessel.

The sixth class of research approaches deals with the ex-traction of tubular structures from images. This is actually a“miscellaneous” class of approaches that may be applicableto vascular extraction in that vessels are tubular entities, butthese approaches were not designed for vessel extractionperse. Huang and Stockman [12] describe a system that usesgeneralized cylindersto extract tubular structures in 2-D inten-sity images. These cylinders are defined by a cross-sectionalelement that is swept along the axis of the tube using somesweep rule. [12] shows results of this algorithm applied tothe extraction of tree-roots. Grimsonet al. [13] observe thatcylindrical objects in 3-D range images appear as conic profilesalong the scan lines. They use a conic detector to detect suchprofiles and extract tubular objects from such range images.

The last class is front propagation approaches. Caselleset al.[14] and Malladiet al. [15] use propagating interfaces undera curvature-dependent speed function to model anatomicalshapes. They used the level set method approach developedby Osher and Sethian [16] and adapted it to shape recognitionprocess. The main idea behind the level set method is torepresent propagating curves as the zero level set of a higherdimensional function which is given in the Eulerian coordinatesystem. Hence, a moving front is captured implicitly by thelevel set function. This approach has some advantages thatmake it attractive. First, it can handle complex interfaces whichdevelop sharp corners and change its topology during thedevelopment. Second, intrinsic properties of the propagatingfront such as the curvature of and normal to the curve can beeasily extracted from the level set function. Third, since thelevel set function is given in the Eulerian coordinate system,discrete grids can be used together with finite differencesmethods to obtain a numerical approximation to the solution.Finally, it is easily extendible to higher dimensions. Sethiandeveloped another method, called the fast marching method[17], which uses a wave propagation approach for specializedfront problems. fast marching methods are used in the problemswhere the front advances monotonically with a speed that doesnot change its sign. The fast marching method’s advantage overthe level set methods is that it is more computationally efficient.

QUEK AND KIRBAS: VESSEL EXTRACTION IN MEDICAL IMAGES BY WAVE-PROPAGATION AND TRACEBACK 119

Fig. 1. A vascular angiogram.

III. T HE PROBLEM REFRAMED

We shall reframe the problem of vascular detection before wediscuss our wave propagation and traceback approach, outlineour algorithm and results in 2-D vessel extraction in X-ray an-giography, and describe the extension of this algorithm to 3-Dvascular extraction in MR and CT images.

Fig. 1 shows an X-ray angiogram. Notice the variation inintensity within the darker vessel pixels, and the considerablenoise and self-occlusion. The challenge is to ignore the local in-tensity variations while extracting the entire vasculature. Thereis no global threshold that one can impose on the data that willcleanly separate the vessels from the background. While onecannot easily determine a clear set of thresholds to determinewhich pixels represent vessels, one can ascertain that certainpixels are more likely to belong to vessels.

If we assigned each pixel in an image with the likelihood thatthe pixel is a vessel-pixel, we obtain a vessel-likelihood image.We can think of the reciprocal of this image as acost functionarray . Given a point of origin at the base of the vasculartree, the problem of detecting the vascular path to any extremalpoint in the tree may be thought of as that of minimizing thepath integral

(1)

where is some path through the cost space, for all pos-sible paths. Unfortunately, this is an intractable NP-completeproblem.

If instead, we model the cost function as a set of refractiveindexes, and the image as a medium through which a wave maybe propagated, we could propagate a wave from the base of thevascular tree. The wave would travel faster through low refrac-tive index (likely to be vessel) pixels, and slower through highrefractive index pixels. The problem of finding that min-imizes (1) from any extremal point in the vasculature becomesone of tracing the wave back along the direction of local normals

Fig. 2. The dual-sigmoid function to set vessel likelihood.

to the wavefront. To do this, we make pixels that are definitelynot vessel-pixels “barriers” to the wave, and we assign lower re-fractive indexes to more likely vessel-pixels.

One may think of the approach as modeling the image as apond where the lower indexes are shallower than the larger in-dexes. If a stone were dropped into the pond, the wave wouldpropagate faster over the shallower regions and slower over thedeeper regions. A trace following the local wavefronts from anypoint in the pond over which the wave propagates, will invari-ably take us back to the origin of the wave. This approach is ableto find the vessels through the intensity variations because thewave is able to “wash” over these variations while maintainingits general path through the vasculature.

The problem of vascular extraction, then, becomes that ofcreating the refractive index image. Pixels that are definitely notvessels are assigned an index of infinity (they are barriers to thewave), and vessel-pixels are assigned values in the interval [0,1], where zero corresponds to the maximal likelihood that thepixel is a vessel pixel, and one is the least likely.

We can achieve this by applying the dual-sigmoidal filtershown in Fig. 2 to the original image. The positive going sig-moid function is described by

(2)

where the constant controls the steepness of the curve andsets the 50% point (the pixel value corresponding to a 0.5

certainty that it is a vessel pixel). The reciprocal of this imageconstitutes the refractive index image.

IV. DIGITAL WAVE PROPAGATION AND TRACEBACK

We apply a general wave propagation technique that producesa digital saw-tooth waveform over a medium of any dimension.For simplicity of discussion, we shall begin with one-dimen-sional (1-D) wave propagation, and extend it through the thirddimension. We shall also describe the algorithm to trace thewave back to the source.

A. Digital Wave Propagation

Our wave propagation approach is to apply local integerneighborhood operations in such a way as to make the resultsglobally robust (i.e., insensitive to local noise variations).We define two kinds of scalar quantities,medium states,

, andwave states, . Wedefine amedium pointas any -dimensionaldata point thathas amedium statevalue, and awave pointas a data point that


Fig. 3. One-dimensional wave-propagation example with fourmedium statesand eightwave states.

has awave state. The general wave propagation algorithm maythen be described as follows:

Set Current Wave State toTag the desired wave origin withwhile medium points that are neighbors

of wave pointspropagate the wave by one step

Decrement Current Wave Stateif Current Wave State is less than

Reset Current Wave State tomedium point that is neighbor of wave

pointsDecrement once within one wave step

even if it is has several wave pointneighbors

Decrement the medium point state byone

if the medium point state isSet the medium point to a wave point

with the current wave state

Fig. 3 shows a 1-D wave propagation sequence with fourmedium states and eight wave states . Thefirst row of the figure shows 11 points where the middle point iswave origin. The wave origin was set to (the startingcurrentwave state). Each successive line in the figure shows the state ofthe data after a successive processing step. After each step, thecurrent wave stateis decremented. The diagram shows 11 pro-cessing steps. The resulting waveform after the eleventh step isshown at the bottom of the figure.

At the first wave step, the wave origin has twomedium stateneighbors, both with value . After the first wave step, the


(a)

(b)

Fig. 4. (a) Four-neighborhood and (b) eight-neighborhood propagation andtheir saw-tooth form over a homogeneous medium.

first neighbors were set to which is less than . Hence, thedata points are replaced with thecurrent wave statewhich was

. At the second step, themedium statepoints bordering thewave front are and , respectively. These points are decre-mented to and , respectively. This process was repeatedfor 11 bprocessing steps until the wave had propagated over theentire medium. The waveform displayed at the bottom of thefigure shows the characteristic saw-tooth waveform producedby the algorithm.

B. Two-Dimensional Wave Propagation

Since the propagation algorithm is based entirely on localneighborhood operations, in extending the wave propagationto 2-D, the kind ofneighborhoodin the pixelated representa-tion needs to be defined. Obviously, the kind of neighborhoodused affects the shape of the waveform generated. Fig. 4 showsa comparison between four- and eight-neighborhood propaga-tion, respectively. In this example, the original medium stateswere all set to and the number of wave states was 16. Neitherpropagation morphology adequately approximates isotropic ra-dial propagation. We show a profile of each of the propagatedwaves to show the 1-D sawtooth in Fig. 4. In our implemen-tation, we alternated between four and eight neighborhoods onsuccessive process cycles, yielding the result shown in Fig. 5.Since the algorithm is essentially SIMD in nature, it is actuallyeasier to express the 2-D algorithm as a pixelwise parallel op-eration. We shall begin with the SIMD expression and show themodifications necessary for efficient implementation in a gen-eral purpose single CPU.

1) Parallel Algorithm: Assuming a separate processor maybe assigned to each pixel per cycle, the pixelwise wave propa-gation algorithm may be expressed as follows:

medium point that is neighbor of wavepoints

Decrement once within one wave stepeven if it is has several wave pointneighbors

Decrement the medium point state byone

if the medium point state is less than

Set the medium point to a wave pointwith the current wave state

Parallel Wave Propagation Algorithm (3)

Obviously, one would have to alternate the neighborhood def-inition on alternate cycles. This algorithm may be efficientlyimplemented in any parallel image processing architecture. Ex-amples of true array processors capable of integer neighbor-hood operations are the Lockheed Martin’s CISP architecture[18], [19] the Maspar computer architecture [20], Thinking Ma-chines’ CM-2 [21], and the SILT [22]. Alternatively, one mightimplement the algorithm in a pipeline architecture such as thereconfigurable Datacube MV20 [23] pipeline process. Fig. 6 il-lustrates how a scalable pipeline machine may be configuredto implement our parallel algorithm 3. Each pipeline stage inFig. 6 performs a single process cycle through the entire image.It’s output is fed to the next stage. As can be seen in the figure,the first stage processor may begin its neighborhood operationonce data points have been read into its buffer, where

is the width of the image. As each successive point is read,the pipeline buffer simply functions as a pixel-shift queue. Aslong the pixel-queue is wide, the neighborhood relationshipof the surrounding pixels to the focus pixel (marked with an)


Fig. 5. Alternating four- and eight-neighborhood propagation and wavefront list.

Fig. 6. Pipelined execution of wave-propagation algorithm.

is maintained. The output stream from the first process wouldbe the result of one cycle of algorithm 3.

As each pixel is processed, it is output to the pixel queueof the next processor. Once pixels have been pushedinto the next processor queue, the second processor can beginprocessing the second image cycle. Hence, each processor re-

quires a pixel-shift queue, and each stage has a “la-tency” of operations before it can begin functioning.Pixels shifting “off the queue” are simply discarded. Obviously,the neighborhood definition of each successive processor needsto be alternated between four and eight neighborhood.

Our algorithm is entirely scalable in that the number of pro-cessors assigned simply determines what number of process cy-cles of algorithm 3 is completed each time an image is streamedcompletely through the processors. The only requirement is thatthe number of processors is even so that each processor cankeep a constant neighborhood definition. The output of thethprocessor is placed in the image buffer to be fed to the first pro-cessor once the original input buffer is exhausted. Hence, thesystem needs only an image memory buffer that func-tions as a “circular” queue. The latency occursonly once, and only determines the number of cycles of algo-rithm 3 is executed each pass through the image. The wave prop-agation is complete when no pixels change on any processingstage.

Defining the algorithm this way poses a challenge at the imageboundaries, as it assumes that the first pixel of each image lineis a neighbor of the th pixel of the preceding image line, andthat the first row of the th input (where is a constantdescribing the number of image passes) is adjacent to the last rowof the th input. This may be easily solved by creating a one-pixel border around the image of “barrier” pixels. Since wavesdo not propagate through barrier pixels, the boundary conditionswill not affect the result.

2) Single CPU Algorithm:In a general purpose CPU, how-ever, testing every medium pixel to see if it has a wave stateneighbor for every wave step would be prohibitively costly. Inour implementation, we invert the testing focus from wavefront-neighbor pixels (algorithm 3) to the wavefront pixels. We em-ploy awavefront list(shown in Fig. 5) whose neighbors are in-spected to see if they need to be decremented or propagated overin each process cycle. For each wave step, we test and update themedium pixelneighbors of each wavefront point in this list ac-cording to our wave propagation rules. If a wavefront point hasno more remainingmedium pointneighbors, it is removed fromthe wavefront list. Also, new wave points generated during anywave stateiteration are added to thewavefront listin prepara-tion for the nextwave stateiteration. Our algorithm performsthe wave propagation “in place,” obviating the need to generate


Fig. 7. Two-dimensional neighborhood index.

a separate output image for each process cycle without intro-ducing propagating errors by the way we update thewavefrontlist. We prepend new wavefront pixels to the head of thewave-front list so that new wavefront pixels will not affect the currentprocess cycle. We perform the wave propagation until thewave-front list is empty.

Using awavefront listin this way poses a problem becausea medium pixel may be a neighbor to more than one wave-front pixels. We do not want to decrement the medium pixelmore than once per wave state iteration. Hence, we include apair of first-visitedand iteration-toggletag bits to each pixel.The first-visitedflag is initially set to zero before propagation.When a medium pixel is visited for the first time as a wave pixelneighbor, this flag is set to one. In each wave step, thecurrentwave statemay either be odd or even. We correspond this to theiteration-toggle. If the current wave state is odd, and theiter-ation-toggleof a medium pixel is zero, we know the mediumpixel is being visited for the first time in this wave step, and wedecrement its medium state and set theiteration toggleto one.The converse is applied when thecurrent wave stateis even. Theinclusion of these two bits is critical to the sequential algorithm.Notice that there is no need to reset any flagafter each cyclesince theiteration-toggleinforms the system if a wave-neighborpixel has been seen on alternating odd and even iterations.

To implement the alternating four- and eight-neighborhoodpropagation shown in Fig. 5, we add anotherweak neighbortagbit to each pixel. For the central point C show in Fig. 7, points[1, 2, 3, 4, 5, 6, 7, 8] are all strong neighbors in eight neighbor-hood, while in four neighborhood, points [2, 4, 6, 8] are strongneighbors and points [1, 3, 5, 7] are weak neighbors. We wantto perform four- and eight-neighborhood propagations on oddand evencurrent wave stateiterations, respectively. Theweakneighbortag bit is initially set to zero. On oddcurrent wavestate iterations, when a medium pixel neighbor to the wave-front is inspected for the first time, this bit is set to one if it is aweak neighbor (i.e., a diagonal neighbor), and its medium stateis left unchanged while thefirst-visitedanditeration-toggletagbits are set the same way as before. If this medium pixel is vis-ited again in the samewave stateiteration because it is a strongneighbor of another wave pixel, we know that it had been visited,but not yet decremented in the current iteration if both theiter-ation toggleand theweak neighbor tagare one. In this case, weclear theweak neighbor tagand decrement the medium state.Hence, we know that a medium pixel had been decremented

(a)

(b)

Fig. 8. (a) Biased traceback and (b) improved traceback in homogeneousmedium.

in an oddwave stateiteration if theiteration toggleandweakneighbor tagare one and zero, respectively. On evenwave stateiterations, all neighbors need to be decremented. Therefore, wesimply set allweak neighbor tagsto zero on even iterations.

C. Traceback Algorithm

As with the wave propagation, our traceback algorithm usesonly a local neighborhood process. As described in Section III,the traceback is tracing the wave back along the direction oflocal normals to the wavefront. Let the current traceback pointbe designated as. The normal to the wavefront at would bein the direction of the steepest ascent in the wave from. How-ever, more than one pixel having the same wave state may sat-isfy this steepest ascent criterion. Hence, a good scanning rulefor finding the steepest ascent path is necessary. Consider thelabeled 2-D neighborhood of aroundin Fig. 7. If we simplytook the sweep order [1, 2, 3, 4, 5, 6, 7, 8], we would get a biasedtraceback direction toward thenorthwestdirection as shown inFig. 8(a). This is undesirable.

In fact, any fixed scanning sequence will result in a direc-tional bias. In our implementation, we implement an alternating


Fig. 9. One-dimensional wave with five medium and eight wave states.Aliasing results in ambiguity in traceback direction.

sweep scheme during traceback. Using the notation of Fig. 7,we apply the sweep sequence of [1, 2, 3, 4, 5, 6, 7, 8] alternatingwith the reverse sweep of [8, 7, 6, 5, 4, 3, 2, 1]. This removesthe directional bias since any direction will not be permitted todominate. Fig. 8(b) shows the result of our the traceback usingthis alternating sweep sequence.

A key characteristic in our wave propagation paradigm is ev-idenced in our traceback algorithm. Since the waveform is acyclical saw-tooth, one would not know what the steepest de-scent wave pixel is if there are moremedium statesthan halfthe number ofwave states. Take the situation with four mediumstates and eight wave states. If a wave pixel has wave state,the steepest ascent pixel must be within the set . Anywave neighbor in the set would actually belongto the previous wave cycle. If, however, we had five mediumstates, there would be a directional ambiguity during traceback.Consider the two examples shown in Fig. 9. In both cases, thenumber of wave states is eight. In the first case the medium statesare followed by . In the second case, the medium statesare followed by . The first and second rows of Fig. 9show the before- and after-propagation states of both cases, re-spectively. In both cases, the wave was propagated from left toright, and the correct traceback direction is rightward. As canbe seen in our examples, an ambiguity arises as to whether aneighbor to the current pixel resulted from a propagation from

down to through a medium or from a to athrough a medium. This is an example of aliasing that occurswhen the wave sampling frequency is less than twice the max-imum frequency of change in the medium. Hence, our systemconforms the classical Nyquist sampling theorem.

V. WAVE PROPAGATION IN X-RAY ANGIOGRAMS

Fig. 10 shows our 2-D wave propagation system interface.The user is able to set the parameters of the dual-sigmoid func-tion [see (2)], set the number of medium (or “ground”) andwave states, select a wave origin, propagate the wave, selectvarious traceback start points, and activate the traceback algo-rithm. We get the gray level range most likely to be vessels usinga sigmoid function and then convert this range into bins for theground states specified. Fig. 11 shows the histogram and its cor-responding ground states bins after using likelihood functionfor the image used in Fig. 23. The parameters of the sigmoidfunction are estimated using the histogram-based p-tile method.

This method is typically used to find a hard threshold to ex-tract objects from the backgrounds. In noisy images such as an-giograms, it is difficult to find a good threshold to extract vesselsclearly. We use the p-tile generated values as parameters in oursigmoid function shown in Fig. 2. Values that are definitely notin the vessel range are assigned the value of “Barrier.” Valuesthat are strongly in the vessel range are assigned the lowest costvalues. The sigmoid function is a“rule-of-thumb” heuristic thatassigns less likely pixel values to an increasingly higher costs.The output of this sigmoid function is quantized as shown inFig. 11 to produce the corresponding ground states.

To show the results of the wave propagation and tracebackmore clearly, we reproduce a small vessel segment in Fig. 12.The figure shows our 2-D wave propagation in an angiogramsegment. The wave origin is marked by the cross and the trace-back can be seen as the light colored line through the wave. No-tice that the traceback does indeed approximate the lowest costpath rather than the medial axis. It snakes through the vessel,cutting the corners to find the shortest path through the vessel.

A. Noise Resistance

A key advantage of our wave propagation and trace approachis its robustness to image noise. Since the wave propagates overthe medium, local high-frequency image variations do not af-fect the general direction of the waveform. Since the tracebackdepends only on the propagated wave, it is able to ignore imagenoise and always find the wave origin.

In Fig. 13, we added 50% Gaussian noise to the image seg-ment in Fig. 12. The wave propagation and traceback points areidentical in the two examples. The robustness of the algorithmis evident as the traceback is almost identical to that in Fig. 12.

B. Wave Count Segments

If we count the number of wave cycles of the propagatedwave, and label an image with these“wave counts,” the re-sulting image yields a “medium-weighted distance measure”from the wave origin. Fig. 14(a) shows a 32-wave stateprop-agation through a homogeneous medium. Fig. 14(b) is labeledwith thewave countthat tracks the number of the wave state cy-cles across the propagated wave. Thiswave count imageyieldsanother side effect of our wave propagation approach that is ben-eficial to the extraction of the vascular network structure.

Fig. 15 shows the concomitant wave propagation andwavecountimages for our example angiogram segment (this time weuse 32 wave states). Thewave countimage effectively segmentsthe vasculature into axial vessel segments. The connectivity ofsegments reflect the connectivity and self-occlusion of the vas-cular tree in the angiogram. This is especially useful in 3-D wavepropagation through MR or CT images since these do not ex-hibit self-occlusion. Hence, all wave count segments with threeor more adjacent segments are necessarily bifurcations in such3-D wave propagations.

Fig. 16 illustrates how the wave segment adjacency may beused to obtain the structure of the vessel tree. Fig. 16(b) showsan exploded illustration of wave count segments emphasizingthe connectivity of each segment throughout the network. Theconnectivity information may be used to construct the branchesof the vessel tree both for the visualization and the analysis.


Fig. 10. Our prototype wave propagation system interface.

Fig. 11. Histogram and its ground states bins after applying likelihoodfunction.

C. Computational Complexity

Our algorithm requires wave cycles to propagate throughsome pixel with cost . The total cost would be

(4)

In Section IV-B1, we discussed all the inherent parallel natureof our approach. Fig. 17 shows the worst case scenario whereparallelization is hindered. If the image comprises a one pixelwide corridor winding through the image separated by barriers,the algorithm is forced into sequential operations. In this case,while the number of wave cycle is high, the computation time ineach wave cycle is low since the wave boundary comprises onlyone pixel. There would be ground pixels which has to betraversed sequentially at cost and the overall computationtime from (4) becomes

(5)

where image size is and is the number of the pixelsin the image.


Fig. 12. Wave propagation through an angiogram segment.

Fig. 13. Wave propagation through an angiogram segment with added noise.

In the case of a homogeneous medium as seen in Fig. 14(a),the worst case occurs when all pixels are at the maximum cost.In this case, however, parallelism is of greater advantage. Thenumber of operations, from (4), would be equal to .The number of wave cycles and the length of the boundary atany point is obviously dependent upon the complexity of thecostscape through which the wave propagates. If, however, asingle CPU is used, applying the algorithm of Section IV-B-2represents the worst case cost complexity.

(b)

(b)

Fig. 14. (a) Wave propagation and (b) wave count images through anangiogram segment.

(a) (b)

Fig. 15. (a) Wave propagation and (b) wave count images through anangiogram segment.

In either case, the complexity of our algorithm is directly pro-portional to the number of ground pixels and the cost of eachpixel. Since the cost of each pixel varies, the average complexity,from (4), is .


(a) (b)

Fig. 16. (a) Wave propagated angiogram image and (b) Exploded illustrationof wave count segments showing the connectivity derived from consecutiveadjacent “wave-count regions.”

D. Comparison to Other Methods—Advantages andDisadvantages

The approaches that are the most similar to our wave-prop-agation method are the level set method approach developedby Osher and Sethian [16] and the fast marching method de-veloped by Sethian [17]. We give a brief information aboutthese methods in Section II. There are some advantages of ourwave-propagation method that makes it attractive. First, it iseasy to implement. The algorithm is strictly integer operationwhich can be implemented in an image pipeline architecture.Second, it is very fast. After obtaining good cost functions forthe image pixels, the wave-propagation process is spontaneous.Third, it can propagate through any arbitrary shapes, there is noneed to havea priori assumption about the object’s topology.Fourth, it extracts connectivity and morphological informationof the vessel tree which can be used both in image visualizationand analysis. Finally, like both methods mentioned above, ourmethod can easily be extendible to 3-D.

The disadvantage of our algorithm is that it relies on findinggood cost functions for different images. We developed an inter-active system to assign good cost functions to each pixel in animage. In a case where image is fuzzy and a global cost functioncan not be found, the user can assign costs locally by opening awindow in the corresponding area of the image. Fig. 18 showsan image where pixel costs are assigned by global cost func-tion. Notice that the vessels in the window on the left side of theimage are not well segmented. Fig. 19 shows the same imagewith better pixel costs assigned locally. Notice the differencebetween the two windows.

(a)

(b)

Fig. 17. (a) Propagated wave and (b) corresponding wave counts illustrationof the maximum computation time scenario.

E. Results

We run our wave-propagation algorithm on a set of neurovas-cular angiogram images. The results we have obtained so farhas been very promising. Figs. 20–23 show a set of originaland wave-propagated angiogram images with measured vesselsegments. We have used attentionally based interaction model(AIM) system [24] developed in our lab to measure the ves-sels’ widths. We have taken 110 measurement over the vesselsegments with widths between three pixels (0.53 mm) and 35pixels (6.24 mm), from a set of six angiogram images, using theAIM system. Then we run our wave-propagation algorithm onthese images and checked all these measurement points on thepropagated vessels. The system could not propagate and extract


Fig. 18. Image converted to ground states using global likelihood function.

Fig. 19. A section of the image converted to ground states using local likelihood function.

vessel segments only on four of these 110 points. The rest of thevessel segments were successfully propagated and segmented

by the system. The image sections in which four unsegmentedvessel points lay are very fuzzy.


(a) (b)

Fig. 20. (a) Original and (b) wave-propagated angiograms with measured vessel segments.

(a) (b)



(a) (b)


(a) (b)

Fig. 23. (a) Original and (b) wave-propagated angiograms with measuredvessel segments.

VI. CONCLUSION AND FUTURE WORK

This paper describes a new approach for extracting vascula-ture from 2-D medical images. Wave propagation and tracebackallows us to extract not only the individual vessels, but the vas-cular connection morphology as well. Our system is robust tonoise and is able to obtain the global network effectively. Ourwave-propagation and traceback algorithms can be easily ex-

tended to operate in 3-D data. As with 2-D wave propagation,3-D propagation will deal with weak and strong neighborhoods,with eight and four neighborhoods corresponding to 26 and sixneighborhoods, respectively. Extension to traceback sweepingsequence to 3-D operations will require greater care to preventa directional bias, but all the basic operations and ideas are thesame with the 2-D traceback. Our future work will focus on the3-D wave propagation and automated 3-D traceback to extractthe entire vascular tree in 3-D data set.

ACKNOWLEDGMENT

Some earlier experimental version of the code was writtenby X. Gong while she was associated with the VISLab at theUniversity of Illinois at Chicago.

REFERENCES

[1] R. T. Ritchings and A. C. F. Colchester, “Detection of abnormalities oncarotid angiograms,”Pattern Recogn. Lett., vol. 4, pp. 367–374, Oct.1986.

[2] B. D. Thackray and A. C. Nelson, “Semi-automatic segmentation of vas-cular network images using a rotating structuring element (ROSE) withmathematical morphology and dual feature thresholding,”IEEE Trans.Med. Imag., vol. 12, pp. 385–392, Sept. 1993.

[3] I. Liu and Y. Sun, “Recursive tracking of vascular networks in an-giograms based on the detection-deletion scheme,”IEEE Trans. Med.Imag., vol. 12, pp. 334–341, June 1993.

[4] Y. Sun, “Automated identification of vessel contours in coronary arte-riograms by an adaptive tracking algorithm,”IEEE Trans. Med. Imag.,vol. 8, pp. 78–88, Mar. 1989.

[5] S. Aylward, S. Pizer, E. Bullitt, and D. Eberl, “Intensity ridge andwidths for tabular object segmentation and description,” inProc.Workshop Mathematical Methods in Biomedical. Image Analysis, 1996,pp. 131–138.

[6] D. J. Stevenson, L. D. R. Smith, and G. Robinson, “Working towardthe automatic detection of blood vessels in X-ray angiograms,”PatternRecog. Lett., vol. 6, pp. 107–112, July 1987.


[7] B. A. Parvin, C. Penf, W. Johnston, and F. M. Maestre, “Tracking oftubular objects for scientific applications,” inProc. IEEE Conf. CVPR,1994, pp. 295–301.

[8] A. K. Klein, F. Lee, and A. Amini, “Quantitive coronary angiographywith deformable spline models,”IEEE Trans. Med. Imag., vol. 16, pp.468–482, Oct. 1997.

[9] S. A. Stansfiled, “ANGY: A rule-based expert system for automaticsegmentation of coronary vessels from digital subtracted angiograms,”IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 188–199,Mar. 1986.

[10] C. Smets, G. Verbeeck, P. Suetens, and A. Oosterlinck, “A knowledge-based system for the delineation of blood vessels on subtraction an-giograms,”Pattern Recog. Lett., vol. 8, pp. 113–121, September 1988.

[11] R. Nekovei and Y. Sun, “Back-propagation network and its configura-tion for blood vessel detection in angiograms,”IEEE Trans. Neural Net-works, vol. 6, pp. 64–72, Jan. 1995.

[12] Q. Huang and G. C. Stockman, “Generalized tube model: Recognizing3-D elongated objects from 2-D intensity images,” inProc. IEEE Conf.CVPR, 1993, pp. 104–109.

[13] W. E. L. Grimson, T. Lozano-Perez, N. Nobel, and S. J. White, “Anautomatic tube inspection system that finds cylinders in range data,” inProc. IEEE Conf. CVPR, 1993, pp. 446–452.

[14] V. Caselles, F. Catte, T. Coll, and F. Dibos, “A geometric model foractive contours in image processing,”Numericshe Mathematik, vol. 66,pp. 1–32, 1993.

[15] R. Malladi, J. A. Sethian, and B. C. Vemuri, “Shape modeling with frontpropagation: A level set approach,”IEEE Trans. Pattern Anal. MachineIntell., vol. PAMI-17, pp. 158–175, Feb. 1995.

[16] S. Osher and J. A. Sethian, “Fronts propagating with curvature depen-dent speed: Algorithms based on Hamilton–Jacobi formulation,”J.Computational Phys., vol. 79, pp. 12–49, 1988.

[17] J. A. Sethian, “A fast marching level set method for monotonically ad-vancing fronts,” inProc. Nat. Acad. Sci., vol. 93, 1996, pp. 1591–1595.

[18] E. L. Cloud,Geometric Arithmetic Parallel Processor: Architecture andImplementation. New York: Academic, 1991.

[19] M. S. Tomassi and R. D. Jackson, “An evolving SIMD architecture ap-proach for a changing image processing environment,”DSP MultimediaTechnol., vol. 10, pp. 1–7, Oct. 1994.

[20] T. Blank, “The maspar mp-1 architecture,” inProc. Int. Conf. ComputerArchitecture, Feb. 1990, pp. 20–24.

[21] D. Hillis, The Connection Machines Distinguished Disserta-tions. Cambridge, MA: MIT Press, 1985.

[22] M. Bolotski, R. Barman, J. J. Little, and D. Camporese, “Silt: A dis-tributed bit-parallel architecture for early vision,”Int. J. Comput. Vis.,vol. 11, pp. 63–74, 1993.

[23] “MaxVideo 20 hardware reference manual,” Datacube Inc., Danvers,MA, 1991.

[24] F. Quek, C. Kirbas, and F. Charbel, “Aim: Attentionally-based interac-tion model for the interpretation of vascular angiograph,”IEEE Trans.Inform. Technol. Biomed., vol. 3, pp. 139–150, June 1999.

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Vessel Extraction in Medical Images by Wave Propagation and Traceback