researcharticle deformable models for segmentation based ...2 mathematicalproblemsinengineering...

Research ArticleDeformable Models for Segmentation Based on Local Analysis

Jimena Olveres1 Erik Carbajal-Degante2 Boris Escalante-Ramiacuterez1

Enrique Vallejo3 and Carla Mariacutea Garciacutea-Moreno4

1Facultad de Ingenierıa Universidad Nacional Autonoma de Mexico Mexico City Mexico2Posgrado en Ciencia e Ingenierıa de la Computacion Universidad Nacional Autonoma de Mexico Mexico City Mexico3Centro Medico ABC Mexico City Mexico4Hospital Angeles Lomas Huixquilucan MEX Mexico

Correspondence should be addressed to Boris Escalante-Ramırez borisunammx

Received 9 May 2017 Revised 15 August 2017 Accepted 1 October 2017 Published 28 November 2017

Academic Editor David Gonzalez

Copyright copy 2017 Jimena Olveres et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Segmentation tasks in medical imaging represent an exhaustive challenge for scientists since the image acquisition nature yieldsissues that hamper the correct reconstruction and visualization processes Depending on the specific image modality we haveto consider limitations such as the presence of noise vanished edges or high intensity differences known in most cases asinhomogeneities New algorithms in segmentation are required to provide a better performanceThis paper presents a new unifiedapproach to improve traditional segmentation methods as Active Shape Models and Chan-Vese model based on level set Theapproach introduces a combination of local analysis implementations with classic segmentation algorithms that incorporates localtexture information given by the Hermite transform and Local Binary Patterns The mixture of both region-based methods andlocal descriptors highlights relevant regions by considering extra information which is helpful to delimit structures We performedsegmentation experiments on 2D images includingmidbrain inMagnetic Resonance Imaging and heartrsquos left ventricle endocardiumin Computed Tomography Quantitative evaluation was obtained with Dice coefficient and Hausdorff distance measures Resultsdisplay a substantial advantage over the original methods when we include our characterization schemes We propose furtherresearch validation on different organ structures with promising results

1 Introduction

Segmentation tasks are routinely used in a great variety ofapplications being one of the most common task imageanalysis [1] It has become an important tool in the support onclinical diagnosis planning and evaluation of patient treat-ments with noninvasive techniques It is the correct delin-eation of organ boundaries in which signal processing tech-niques have been adapted to recognize anatomical structuresaccording to clinical needs Obtaining high precision andaccurate results withminimal human interaction is becomingmore important for the cliniciansrsquo on their everyday tasks

Among the different existingmedical imagingmodalitiesthe most popular are as follows X-rays Ultrasound Com-puted Tomography (CT) and Magnetic Resonance Imaging(MRI) Each human organ has specific anatomical char-acteristics and each imaging modality has its own imagequality properties and limitations such as sharpness noise

and contrast All the above hampers the segmentation taskThere is a great importance in selecting the correct algorithmthat fits the organ boundary Up to date the efforts focus onlocating the desired structure identifying it in the image anddetermining its contour [2]

Several techniques have been developed in order to coverthe segmentation task Some of the more popular methodsin medical image analysis are those based on deformablemodels [3ndash5] They include algorithms such as Active ShapeModels (ASM) and level sets (LS) which are widely docu-mented in the literature [6] Efficient implementations havebeen designed aiming at faster methods [7 8] Howeverclassic deformable models are associated with a functionalminimization which often leads to oversegmentation [9] Asa part of thesemethods an extra step for validation requires aground truth for comparisonThis is usually obtained by usermanual delineation and prone to user errors

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1646720 13 pageshttpsdoiorg10115520171646720

2 Mathematical Problems in Engineering

In recent years segmentation performance has increasedwith new improvements by combining other techniques andalso having the inclusion of more information cues liketexture or movement Such is the case of Jiang et al whocombined level set framework with seeded region growingalgorithm for segmentation of hippocampus [10] Ma et al[11] used a Haar classifier to detect the heart area and afterthis step the segmentation is performed with an ASM Broxet al [12] introduced simultaneously texture features graylevel and optic flow on a level-based segmentation processAn extension of ASMs has been proposed with the ActiveAppearance Model (AAM) including textural informationwithin the model as a whole [13] but the algorithm results inlow efficiency in real-time systems and high computationalcost besides having inconsistent robustness when appliedunder different circumstances [14]

On the other hand besides combining different tech-niques there exist information features inherent to theimages such as color texture or morphology that canbe included in order to provide especial characteristics toimprove the final segmentation It is known in general thatintensity inhomogeneity complicates many image segmen-tation methods making it difficult to identify the desiredregions based only on pixel intensity Level sets andAAMs areused as based-regionmethods but when dealing with imagesthat contain intensity inhomogeneities they tamper its per-formance [15] For this reason it is important to add featuresto the algorithms that improve the performance in spite ofinhomogeneities presence

Xu et al [16] propose to combine an automatic globaledge and region force field guided method with nonlinearexponential point evolution for lung field segmentationexpanding the ASM performance with this informationKeomany and Marcel [17] proposed to use LBPs also withASMs for the profile search instead of having only gray valuesInstead of gray values it uses LBP information avoiding thelimitations of AAMs

In the case of the Hermite transform (HT) and LocalBinary Patterns (LBP) both methods provide additionalfeatures that improve the segmentation algorithms LBPs arebeneficial when they are combined with other methods suchas ASMs giving a more solid result modeling local structuresin spite of illumination changes [18] In addition LBPs havevery low computational complexity and provide a robustway for describing local texture patterns [19] The Hermitetransform [20 21] is one of the mathematical models thatresembles the human visual system (HVS) These HVS mod-els have increased in popularity because they allow imagesto be expanded into a local decomposition that describesintrinsic attributes related to important cues and highlightstructuresThe information obtained from theHermite trans-form coefficients has previously been used as textural data forclassification tasks [22]

The present study proposes to combine segmentationalgorithms with information given by the HT and LBPs toimprove efficiency of current segmentation techniques suchas ASMs and LS In order to test our approach we used twoimage modalities for different organ studies such as cranialmidbrain MRI and endocardium left ventricle CT

This paper is organized as follows Section 2 introducesthe mathematical background for the proposal based onASMs and a deformablemodel Chan-Vese (CV) algorithm InSection 3 the method is explained in Section 4 the dataset isdescribed and in Section 5we display the experimentrsquos resultsand provide a discussion Finally Section 6 concludes ourwork

2 Theoretical Background

In this section we present the mathematical foundationThe deformable models used here are ASMs a statisticaldeformable model and vector value Chan-Vese model amethod based on level sets

21 Chan-Vese Model with Level Sets The active contourmodel without edges was proposed by [23] and it is inspiredby theMumford-Shahmethod for image segmentationwhichevolve an object by minimization of an energy functional[24] The original definition of the 2-dimensional Chan-Vesemodel (CV) is to letΩ be a bounded open set ofR2 with 120597Ωas the boundary We define an image as 119868 Ω rarr R Nowlet 119862 be an involving curve in Ω as a boundary between tworegions in such a way that Ω = Ω1 cup 119862 cup Ω2 whereΩ1 andΩ2 represent the inner and outer region of 119862 respectivelyFinally the formulation of the energy functional tries toidentify the best partition on 119868 as follows

inf1198881 1198882 119862

119865 (1198881 1198882 119862) (1)

where 1198881 and 1198882 are the two average regions of intensity valuesinside and outside the contour respectively The energy isminimized if the contour 119862 is located at the boundary of theobject of interestThe 2-dimensional energy functional of theCV model in the domain 119909 isin R2 is defined as follows

119865 (1198881 1198882 119862) = 120583int119862119889119904 + 1205821 int

Ω1

(119868 minus 1198881)2 119889119909+ 1205822 int

Ω2

(119868 minus 1198882)2 119889119909(2)

where 1205821 1205822 are fixed nonnegative constants for data fittingand 120583 is introduced to regularize the surface 119862 The minimalpartition problem given by (1) can be solved using the levelset method [25] The contour 119862 can be represented by thenonzero level set of a Lipschitz function called 120601 Ω rarr Rsuch that 119862 = 119909 isin Ω | 120601(119909) = 0 This approach has manyadvantages because it allows awide study of topologies whichis made on a rectangular grid The functional in (2) can bewritten in terms of the level set as follows

119865 (1198881 1198882 120601) = 120583intΩ120575 (120601 (119909)) (nabla120601 (119909)) 119889119909

+ intΩ1205821 (119868 (119909) minus 1198881)2119867(120601 (119909)) 119889119909

+ intΩ1205822 (119868 (119909) minus 1198882)2 (1 minus 119867 (120601 (119909))) 119889119909

(3)

Mathematical Problems in Engineering 3

Considering the approximations of the Heaviside 119867 andDirac 120575 functions [23] as specified in the original paper theminimization is solved with the Euler-Lagrange equation foran artificial time 119905 ge 0

120597120601120597119905 = 120575 (120601) [120583 div( nabla1206011003816100381610038161003816nabla1206011003816100381610038161003816) minus 1205821 (119868 (119909) minus 1198881)2+ 1205822 (119868 (119909) minus 1198882)2]

(4)

Equation (4) is implemented as a discrete model throughfinite differences The vector value model for Chan-Vese(VVCV) proposed in [26] uses119870-extra information in orderto obtain a better segmentation

120597120601120597119905 = 120575 (120601) [120583 div( nabla1206011003816100381610038161003816nabla1206011003816100381610038161003816) minus 1119870119870sum119894=1

120582+119894 (119868119894 (119909) minus 119888+119894 )2

+ 1119870119870sum119894=1

120582minus119894 (119868119894 (119909) minus 119888minus119894 )2] (5)

where 119888+ and 119888minus are constant vectors that represent theaverage value of one of theK layers of 119868119894 inside and outside thecurve 119862 respectively The parameters 120582+119894 and 120582minus119894 are constantweights defined manually for the user and further discussedin the experimental results section We will consider in thisstudy planar images

The solution of (4) and (5) achieves a convergence bychecking whether the solution is stationary Setting a thre-shold is used to verify if the energy is not changing or ifthe contour is not moving too much between two iterationssuch threshold can be found intuitively or experimentallyAnother very used criterion to stop the curve evolution isto manually define a stopping term with a large number ofiterations this assumption considers that the desired contourwill be obtained at the end of the process similar to obtaininga global minima

22 Active Shape Models (ASM) ASMs published by Cooteset al [27] incorporate shape information as part of its modelIt consists of a training phase where the shape informationis modeled statistically and a fitting phase where new shapesare recognized ASMs use a set of points that define the shapewithin the image These points also incorporate featuressuch as the gray-level information that drives the fitting(recognition) phase

In the training phase the points or landmarks definethe prior knowledge and this is accomplished by definingthe set of points correspondent to the images used fortraining These images were aligned one with respect to theother using pose transformations that include translationrotation and scaling In order to avoid variability within thetraining set Cootes developed the point distribution model(PDM) where 119873 shapes 1198841 1198842 119884119873 are defined with 2-dimensional coordinates (1199090 1199100 1199091 1199101 119909119899 minus 1 119910119899 minus 1)We obtain the mean shape from 119884 = (1119873)sum119873minus1119894=0 119884119894 The

variations of themean shape are obtained by computing prin-cipal component analysis [27] Any shape 119884 can be approxi-mated by

= 119884 + 119875b (6)

where 119875 is the matrix of the 119905 first principal componentsb is the weight vector and is the estimated shape And acovariance matrix 119878 is computed as follows

119878 = 1119873 minus 1119873sum119894=1

(119884119894 minus 119884) (119884119894 minus 119884)119879 (7)

The eigenvectors of the largest eigenvalues from thecovariance matrix describe the most significant modes ofvariance [3] and they are used to characterize the variabilityof the training set During the same training phase thealgorithm includes the creation of a gray-level model for eachlandmark that defines the mean shape

In the fitting phase the following steps are executed

(i) Initialize the shape parameters and generate themeanshape 119884 from the training phase

(ii) Convey a gray-level profile search in order to adjustthe mean shape landmarks to the image 119884 shouldbe placed close to the object of interest manuallyEach landmark in119884 is compared with its correspond-ing gray-level profile The landmarks positions areadjusted according to a distance-based optimizationcriterion

(iii) Posemodel is adjusted including translation rotationand scaling aligning the points model to current found in the previous step

(iv) Compute shape adjustment in order to update 119887parameter according to (6)

Except the initialization the steps are repeated iterativelyuntil convergence is achieved

23 Hermite Transform The Hermite analysis functionscalled 119863119899 [20 21] are a set of orthogonal polynomialsobtained from the product of a Gaussian window definedby 119892 and the Hermite polynomials 119867119899 where 119899 in this caseindicates the order of analysis

119863119899 (119909) = (minus1)119899radic2119899119899 1120590radic120587119867119899 (119909120590) exp(minus1199092

1205902) (8)

TheHermite coefficients 119871 can be obtained by convolvingthe main function 119868 with the analysis filters 119863119899 This processinvolves a localized description of the function in termsof the orthogonal basis The 2-dimensional representationcan be obtained since the analysis functions are spatiallyseparable and rotationally symmetric that is119863119899minus119898119898(119909 119910) =119863119899minus119898(119909)119863119898(119910)119871119899minus119898119898 (1199090 1199100)

= int119909int119910119868 (119909 119910)119863119899minus119898119898 (1199090 minus 119909 1199100 minus 119910) 119889119909 119889119910 (9)


with 119863119899minus119898119898(119909 119910) = 119867119899minus119898119898(minus119909 minus119910) sdot 1198922(minus119909 minus119910) where(119899 minus 119898) and119898 denote the analysis order in 119909- and 119910-axes for119899 = 0 infin and 119898 = 0 119899 The associated orthogonalpolynomials are defined as follows

119867119899minus119898119898 (119909 119910)= 1radic2119899119898 (119899 minus 119898)119867119899minus119898 (119909120590)119867119898 (119910120590)

(10)

A steered version of the Hermite transform is based onthe principle of steerable filters [28] and is implemented bylinearly combining the Cartesian Hermite coefficients Thisrepresentation allows adapting the analysis direction to thelocal image orientation content according to a criterion ofmaximum oriented energy Therefore a set of rotated coef-ficients is constructed named the steered Hermite transform(SHT) [29 30]

119871120579119899minus119898119898 (1199090 1199100) = 119899sum119896=0

119871119899minus119896119896 (1199090 1199100) 119877119899minus119896119896 (120579) (11)

119877119899minus119898119898 (120579) = radic(119898119899) cos119899minus119898 (120579) sin119898 (120579) (12)

where 119877119899minus119898119898(120579) are known as the Cartesian angular func-tions that express the directional selectivity of the filter atall window positions by using a simple relation tan(120579) =1198710111987110 which approximates the direction of maximumoriented energy

Since the general theory of the Hermite transform statesthis transformation can be considered as linear and locallyunitary the energy is preserved in terms of its coefficientsThis is expressed by Parsevalrsquos formula

119864infin = infinsum119899=0

119899sum119898=0

[119871119899minus119898119898]2 = infinsum119899=0

119899sum119898=0

[119871120579119899minus119898119898]2 (13)

The SHT version generates energy compaction up toorder 119873 by steering the local coefficients into the directionof maximum energy The more oriented the local structurethe more compact the representation We can distinguish thelocal energy terms 1198641119863119873 and 1198642119863119873 adapted to local orientationcontent given by 1119863 and 2119863 patterns respectively

1198641119863119873 = 119873sum119899=1

[1198711205791198990]2 1198642119863119873 = 119873sum

119899=1

119899sum119898=1

[119871120579119899minus119898119898]2 (14)

Total energy can be written in combination as follows119864119873 = 1198640 + 1198641119863119873 + 1198642119863119873 where 1198640 represents the DC com-ponent In the SHT high compaction of energy is achieved in1198641119863119873 term this is useful for image compression purposes andseveral other applications [29]

24 Local Binary Patterns (LBP) Ojala et al proposed theLocal Binary Pattern descriptor in 1994 [18] It is based onthe idea that textural properties within homogeneous regionscan be mapped into patterns that represent microfeaturesInitially it makes use of a fixed 3 times 3mask which representsa neighborhood around a central pixel The values arecompared with their central pixel and the pixels take 28possible labels that describe intensity differences

In a more general way LBPs use a circular neighborhoodinstead of a fixed rectangular region The sampling coordi-nates of the neighbors are calculated using the expression(119909119888 minus 119877 sin[2120587119901119875] 119910119888 + 119877 cos[2120587119901119875]) [31] If a coordinatedoes not fall on an integer position then the intensity val-ues are bilinearly interpolated Such implementations allowchoosing the spatial resolution (119877) and the number of sampl-ing points (119875) as follows

LBP119875119877 (119892119888) = 119875minus1sum119901=0

119904 (119892119901 minus 119892119888) 2119901 (15)

where 119875 is the number of sampling points 119877 represents theradius of the neighborhood 119892119888 is the central pixel at (119909119888 119910119888)and 119892119901 | 119901 = 0 119875 minus 1 are the values of the neighborswhereas the comparison function 119904(119909) is defined as follows

119904 (119909) = 1 if 119909 ge 00 if 119909 lt 0 (16)

LBPs also possess a variation on this definition calledUni-form Local Binary Patterns (LBPuni

119875119877) [31] It takes advantageof the uniformity of the patterns and over 90 of LBP texturepatterns can be described with few spatial transitions whichare the changes (01) in a pattern chain This uniformity isdefined as follows 119880(LBP119875119877(119892119888)) = |119904(119892119875minus1 minus 119892119888) minus 119904(1198920 minus119892119888)|+sum119875minus1119901=1 |119904(119892119901minus119892119888)minus119904(119892119901minus1minus119892119888)| which corresponds to thenumber of spatial transitions so that LBPuni119875119877 can be obtainedas follows

LBPuni119875119877 (119892119888)= 119875minus1sum119901=0

119904 (119892119901 minus 119892119888) if 119880 (LBP119875119877 (119892119888)) le 2119875 + 1 otherwise

(17)

The pixelwise information is encoded as a histogram119867119894so that it can be interpreted as a fingerprint or a signature ofthe analyzed object

3 The Proposed Method

The aim of the proposed methodology is to compare thesegmentation performance of the original ASM and levelsets methods with improved version of these methods thatinclude information obtained from LBP and SHT The nextsubsections explain different combination procedures usedto improve the chosen algorithms We evaluated differentcombinations of ASM and CV level set algorithms


HT VVCVSHT

Medical image Segmented

image

Figure 1This example shows the Hermite-based procedure with vector-valued Chan-Vese segmentation for midbrain from 119899 = 0 up to 119899 = 3order of expansion

31 Combining VVCV with SHT In this section wedescribe the Hermite transform-based segmentation for pla-nar images This segmentation approach combines the Her-mite coefficients and the vector-valued Chan-Vese algorithmbased on level sets Both image representation and segmen-tation techniques were computed using 2D algorithms

Figure 1 shows the segmentation procedure used as partof this work First the HT is calculated up to order 119899 = 3on every slice of the initial volume 119868(119909 119910 119911) Then the SHT iscomputed by locally rotating the HT coefficients From theseonly the first row coefficients are used that is the coefficientsthat carry most of the 1D energy

A number of 119870 = 4 image coefficients produce the finalsegmentation that is computed iteratively by the formula in(5) in combination with (11)

120597120601120597119905 = 120575 (120601) [120583 div( nabla1206011003816100381610038161003816nabla1206011003816100381610038161003816)minus 14 [120582+0 (11987112057900 (119909 119910) minus 119888+0 )2+ 120582+1 (11987112057910 (119909 119910) minus 119888+1 )2 + 120582+2 (11987112057920 (119909 119910) minus 119888+2 )2+ 120582+3 (11987112057930 (119909 119910) minus 119888+3 )2]+ 14 [120582minus0 (11987112057900 (119909 119910) minus 119888minus0 )2+ 120582minus1 (11987112057910 (119909 119910) minus 119888minus1 )2 + 120582minus2 (11987112057920 (119909 119910) minus 119888minus2 )2+ 120582minus3 (11987112057930 (119909 119910) minus 119888minus3 )2]]

(18)

where 11987112057900 11987112057910 11987112057920 and 11987112057930 are the image coefficients ofSHT for orders 119899 = 0 1 2 3 and119898 = 0 respectively

In the final stage of Figure 1 the VVCV procedure startswith the initialization of a circle with radius 119903 called 120601 andcentered in a specified pixel The initial contour should beplaced close to the object of interest to segment and the

algorithm allows the curve evolution according to the tuningparameters and a convergence criterion

32 Combining LBP with ASM This combination is used aspresented in [32] In the training phase

(1) for every training image a set of points is taken fromthe expert manual annotations and aligned by meansof pose transformations

(2) create a statistical point distribution model (PDM)with point position data from all training images

(3) instead of creating gray-level profiles across the land-marks we generate a model based on LBPs calculatedwithin a neighborhood around the landmarks TheLBPs are calculated over four square regions of 5 times 5pixels around every landmark A histogram is takenfor each square as in the method described by Ojalaet al [31] The four histograms are concatenated asshown in Figure 2

(4) finally for each landmark a mean concatenated his-togram is obtained from all training images Our finalmodel consists of the original ASM model plus anaverage concatenated histogram for each landmark

The LBP concatenated mean histograms take the place ofthe gray-level profile of the original ASM algorithm usingLBPs defined by Ojala et al [18] and the Uniform-LBP (LBP-uni) Figure 3 shows the pipeline process As in the originalASM algorithm this model is used with the shape modelto estimate the optimal position for each landmark in therecognition phase

In the recognition phase a new image is processed by theASM shape and LBP-landmark fitting stage using the modelfrom the previous training phase

(1) Initialize shape parameters and generate the meanshape from the training phase in order to start theprofile search of the points model with the meanshape Before fitting the profile model with the PDM119884must be manually placed close to the object


8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

Figure 2 Combine LBPs with ASMs A quadratic region is used for the LBPrsquos histogram for each contour landmark

77

86 102

101

101

101

101

101

101

101

95 70

15Threshold

Binary code = 11000110

1 1

1 1

0

000

77 83 55

Trainingimages

TrainedASM model

ASM train

Shape model - PDM

LBP model

AverageLBP

histogramExpert manual annotations

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

10

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

9586 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold

Binary code = 11000110Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

Y

Figure 3 Pipeline of LBPs with ASMs training algorithm The training phase incorporates LBPs

(2) For each landmark in 119884 a comparison is donebetween the mean concatenated trained histogramand the corresponding histograms calculated at dif-ferent positions along the search profile Differences

are evaluated with the Chi-square distanceThe land-mark is moved to the position with the smallest Chi-square distance A small distance means more regionsimilarity [33]


ASMrecognition

Segmentedimage

Testimage

Fitting phase

TrainedASM model

Region ofinterest

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

Figure 4 Pipeline of LBPs with ASMs fitting algorithm The recognition phase incorporates LBPs

L0 L1 L2 L3

Figure 5 Construction of the Hermite histograms for the ASM algorithm 1198710 1198711 1198712 and 1198713 are the steered Hermite coefficients

(3) Posemodel is adjusted including translation rotationand scaling aligning the points model to the currentshape found in the previous step

(4) Adjust the statistical model (6) to the new shapeupdating the parameter b within limits defined in theoriginal algorithm

(5) Repeat step (2) and iterate until convergence isreached

Figure 4 displays the pipeline of the recognition phase

33 Combining SHT with ASM This procedure is similar tothe ASMLBP but instead of computing LBP histograms wecompute histograms of quadratic local regions on four imagescorresponding to the first four SHT coefficients 119871120579119899(119909 119910)for 119899 = 0 3 This method resembles a vector-valuedprocessing because at each square region we use four steeredHermite coefficients each one being a channel of the vector-valued ASM

119901 (119903119871119899) = 1119872119873 1198991198711205790 1198991198711205791 1198991198711205792 1198991198711205793 (19)

where 1198991198711205790

1198991198711205791

1198991198711205792

and 1198991198711205793

are the histograms of eachlandmark for each steered Hermite coefficients and 119901 is the

notation for the coefficient histograms concatenated in thequadratic region

The process is the same as in ASMLBP except for theconstruction of the histogram information this is applied inthe training and recognition phases Figure 5 displays how thehistograms are built and finally they are concatenated

4 Materials

The first dataset consists of cranial annotated midbrainstudies from 10 normal subjectsThis studies were labeled andrandomly selected by a highly trained neurology specialist atHospital Interlomas (Mexico) The T2 images were obtainedon a 3 Tesla Scanner generating volumes of size 512 times 448 times176 pixels with a resolution of 044921 times 044921 times 09mmThese data were processed with a normalization of grayintensities We used 10 magnetic resonance brain volumescontaining 10 slices each where the midbrain is located

Our second dataset is 11 annotated tomographic cardiacstudies taken during the cardiac cycle from healthy sub-jects obtained with a CT Siemens dual source scanner (128channels) at Hospital Angeles Pedregal Mexico by a highlytrained cardiology specialist The studies go from a finaldiastolic phase (relaxing) throughout the systolic phase(contraction) and return to the diastolic phase providing10 sets of images at 0 10 20 30 40 50 60


0 3 6 9

06

08

T

Dic

e ind

ex [0

-1]

CVVVCVSHT

(a)

0 3 6 90

20

40

60

80

T

Hau

sdor

ff di

stan

ce

CVVVCVSHT

(b)

Figure 6 Average distances between the segmented volume and expert annotation at each iteration of Δ119905 = 01 up to 119879 = 10 The bestsegmentation results of the midbrain volumes were achieved before119879 = 3 for both CV and VVCVSHT implementations this value indicatesthat the segmentation was obtained with a maximum of 30 iterations

70 80 and 90of the cardiac cycleThe spatial resolutionvalues range from0302734times 0302735times 15 [mm] to 0433593times 0433593 times 1 [mm] In order to remove negative valuesHounsfield values are shifted by an offset The experimentswere evaluated onmid-third slice and the consecutive imagesof the cardiac cycle were retrospectively reconstructed

In both datasets the algorithms were executed with a 2Dsegmentation analysis

5 Experimental Results

Thealgorithmrsquos results were validatedwithmanual segmenta-tions generated by expert cliniciansThe initialization steps ofthe algorithms are similar to the midbrain segmentation andleft ventriclersquos endocardium segmentation For the trainingphase of the ASM algorithms we used leave-one-out cross-validation to reduce bias

51 Midbrain Segmentation We performed a comparisonbetween the texture-basedmethods described previously andthe classic segmentation methods applied directly over eachimage corresponding to midbrain

Approaches based on level set methods profit fromsemiautomatic procedures where the user interaction in theinitialization stage helps to achieve a fast convergence of thecurve evolution The initial circle with radius 119903 was locatedat the central pixel (1199090 1199100 1199110) where the point (1199090 1199100) wasdefined by the user and it is the same for every slice 1199110 =1 10 This value assumes to be close to the central partof the midbrain then the procedure was repeated as an initialstep for each data volume

The stopping term 119879 is the result of applying a certainconvergence criterion for the curve evolution that indicatesthe interval in which the segmentation was achieved Thisterm is crucial and can be experimentally obtained whilethe value of time step Δ119905 can be assigned arbitrarily [34]Since selecting a large value of Δ119905 implies the fast evolutionof the curve process it yields usually unsatisfactory resultsOn the other hand selecting a small value of Δ119905 generates

a slow curve motion and the number of iterations neededto achieve convergence increases substantially Moreoverchoosing a suitable value ofΔ119905 and convergence criteriamightsolve some stability and resolution issues in the developmentFinally the number of iterations can be computed as a ratiobetween the stopping term 119879 and the time step Δ119905

The parameters of 120582+minus119894 can be viewed as weight valuesthat assign priority to the measure of uniformity inside andoutside the curve in terms of intensity Many applications canbe found in the literature where the parameters are set to 1although they can be modified according to a criterion aftercarrying out several experiments This selection is closelyrelated to the object of interest to segment We defined a rela-tion of 120582+119894 gt 120582minus119894 that would allow less variations in the fore-ground compared to the background this rule aims atidentifying an internal homogeneous region compared to theexternal region while the curve is growing [35] Clearly thisrelation improves the delimitation of a single structure amongsome others in the same image because the structure of inter-est was preserved along the first four orders of the chosenexpansion

In the midbrain volume assessing the CV and VVCVSHT parameters in (4) and (18) respectively were definedexperimentally as follows 120582+119894 = 3 120582minus119894 = 15 and 120583 = 05We applied a criterion of time step Δ119905 = 01 to further com-pute the error at each iteration In order to exhibit the sta-bility differences between both methodologies the resultingaverage curves from all volumes segmentation are displayedin Figure 6 using Dice index and Hausdorff distance in theinterval of 0 le 119879 le 10

The Dice index curves can be viewed in Figure 6(a)where the highest value close to 1 corresponds to the bestresult In this case the average distances at each iterationusing VVCVSHT are higher than CV despite the incrementof iterations It is worth highlighting that the curve in CVdecreases suddenly after the highest value is achieved Simi-lar performance of the error curves using Hausdorff metricFigure 6(b) have shown a faster error increment in CVwhile the VVCVSHT values remain lower in the most of


0 3 6 90

2

4

6

TEn

ergy

CVVVCVSHT

middot107

Figure 7 Energy of the contour using Δ119905 = 01 for a single MRI brain image in the interval 0 le 119879 le 10 Midbrain segmentation is reachedwhen the curves exhibit stability

1 2 3 4 5 6 7 8 9 10

06

08

1

Slice

Dic

e ind

ex [0

-1]

ASM

ASMLBPASMLBP-uniCVVVCVSHT

(a)

1 2 3 4 5 6 7 8 9 100

10

20

30

ASM


Slice

Hau

sdor

ff di

stanc

e

(b)

Figure 8 Average distances between the experimental results and the expert contours with Dice coefficient and Hausdorff distance formidbrain segmentation

the iterations The performance of VVCVSHT implemen-tation indicates a good improvement compared to the CValgorithm The error curves show that the evolution canbe stopped easier in VVCVSHT than in CV by defining athresholding criterion or a fixed number of iterations becausethe error remains lower (Dice index close to one Hausdorffdistance close to zero) despite increasing the time step orthe number of iterations The energy of the contour for asingle brain image is displayed in Figure 7 the curve exhibitsstability with VVCVSHT

Moreover in order to quantify the differences among alltechniques both Dice and Hausdorff measures were com-puted according to the expert annotations see Figure 8The best resulting distances in the experiments are displayedas different color lines Notice how the texture descriptorsimprove the classic methods of segmentation In order toillustrate the segmentation results qualitatively we create avolume render shown in Figure 9 trying to highlight the dif-ferences between the best results using CV and VVCVSHTimplementations

52 Left Ventricle Segmentation It is worth pointing out thatthe CV andVVCV parameters can be defined experimentally

as in the case of the midbrain evaluation and the previousconsiderations of Δ119905 119879 and 120582+119894 gt 120582minus119894 terms In this casewe determined the values as follows 120582+119894 = 2 120582minus119894 = 15 and120583 = 05 The performance of the algorithms was comparedusing the Hausdorff distance and Dice coefficient as shownin Figure 10 In general we obtained good performance forthe ASMLBP and VVCVSHT algorithmsThe performanceof the algorithms varies with the phase of the cardiac cycleIn most segmentation algorithms performance decreases insystole and our algorithms are not an exception This couldbe explained because all algorithms are tested under similarconditions such as iteration number and in this phase theLV reaches maximum contraction Also because of organmotion the LV gets a less regular shape of a contractedventricle and a smaller area is obtained It is worth noticingthat in spite of the fact that CT images possess more noisethan MRI images texture-based implementations exhibitbetter quantitative results than the rest of algorithms

Figure 11 shows the segmentation results during the car-diac cycle with changes in timeNotice how the segmentationwith the level set approach detects the endocardial papillarymuscles (yellow line) After this step we applied a convexhull operation (blue line) in order to resemble the clinical


(a) (b) (c) (d)

(e) Az = 0 El = 60 (f) Az = 45 El = 30 (g) Az = 250 El = 40 (h) Az = minus53 El = 38

Figure 9 Render image of the first midbrain segmentation result with VVCVSHT at the top and CV algorithm at the bottom The rendersare displayed in different views in azimuth (Az) and elevation (El) axes

0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM

ASMLBPASMSHTCVVVCVSHT

(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)

Figure 10 Average distances between the experimental and the expert contours with Dice coefficient andHausdorff distance for left ventriclesegmentation of 0 10 90 of cardiac cycle

segmentation and be able to compare it with the traditionaldelineations (red line) through all cardiac cycle percentages

6 Conclusions and Future Work

In this research we first introduced common deformablemodels for image segmentation and we further showedthe improvement provided by texture descriptors for localanalysis steered Hermite transform and Local Binary Pat-terns We assessed their performance segmenting midbrainin Magnetic Resonance Imaging and heartrsquos left ventricle inComputed Tomography

This approach compared the original algorithms withtheir modified versions that include additional local textureinformationWe found that our proposal results in segmenta-tions closer to the expert annotationsThis fact provides clearevidence in the rationale for combiningmethodologies whichconsider either a level of homogeneity in certain zones or agroup of edges that best encloses the structure of interest

In general we obtained the most consistent results whencombining CV level set method with SHT Adding localimage information to the original segmentation algorithmsproved to be advantageous to delimit the shape and contoursbecause of the energy compaction of the SHT Furthermorein the case of ASM the boundaries converge with highaccuracy when we included the information given by theLBPs However it is important to notice that ASM requiresa training phase which takes an extra effort compared tothe CV procedure this fact adds an advantage to level setsmethods regarding computer time and complexityMoreoverthis study is helpful to verify that our VVCVSHT methodcan obtain favorable results when the parameters in the curveevolution such as weights time step and convergence crite-rion are selected intuitively as in most of the segmentationschemes based on active contours

Future work should focus on improving ourmethods andexpanding them to 3D analysis as well as including moreorgan structures in our assessments


(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80

Figure 11 Examples of the left ventricle endocardium segmentation during different phases of the cardiac cycle We display results from 0to 80 90 is not displayed because the image and resulting segmentation is almost the same as in 0 phase The red line is the groundtruth boundary yellow line is the VVCVSHT segmentation and blue line is a convex hull operation that helps remove papillary muscles


Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This work has been sponsored by UNAM Grants PAPIITIN116917 and SECITI 1102015 Jimena Olveres and ErikCarbajal-Degante acknowledge Consejo Nacional de Cienciay Tecnologıa (CONACYT)

References

[1] D L PhamC Xu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 no 2000 pp 315ndash337 2000

[2] M N Bugdol and E Pietka ldquoMathematical model in left ven-tricle segmentationrdquoComputers in Biology andMedicine vol 57pp 187ndash200 2015

[3] C Xu D L Pham and J L Prince ldquoImage segmentation usingdeformable modelsrdquo in Handbook of Medical Imaging Vol 2Medical Image Processing and Analysis vol 2 pp 175ndash272 SPIE2009

[4] C Petitjean and J Dacher ldquoA review of segmentation methodsin short axis cardiac MR imagesrdquo Medical Image Analysis vol15 no 2 pp 169ndash184 2011

[5] J Dolz L Massoptier and M Vermandel ldquoSegmentation algo-rithms of subcortical brain structures on MRI for radiotherapyand radiosurgery A surveyrdquo IRBM vol 36 no 4 pp 200ndash2122015

[6] J S Suri K Liu S Singh S N Laxminarayan X Zeng and LReden ldquoShape recovery algorithms using level sets in 2-D3-Dmedical imagery a state-of-the-art reviewrdquo IEEE Transactionson Information Technology in Biomedicine vol 6 no 1 pp 8ndash28 2002

[7] J Kratky and J Kybic ldquoThree-dimensional segmentation ofbones from CT and MRI using fast level setsrdquo in Proceedingsof the Medical Imaging 2008 Image Processing USA February2008

[8] S Li and Q Zhang ldquoFast image segmentation based on efficientimplementation of the chan-vese model with discrete gray levelsetsrdquoMathematical Problems in Engineering vol 2013 no 2013Article ID 508543 p 16 2013

[9] X Bresson S Esedoglu P Vandergheynst J-P Thiran and SOsher ldquoFast global minimization of the active contoursnakemodelrdquo Journal of Mathematical Imaging and Vision vol 28 no2 pp 151ndash167 2007

[10] X Jiang Z Zhou X Ding X Deng L Zou and B Li ldquoLevel setbased hippocampus segmentation inMR images with improvedinitialization using region growingrdquo Computational and Math-ematical Methods in Medicine vol 2017 no 2017 Article ID5256346 p 11 2017

[11] F Ma J Liu B Wang and H Duan ldquoAutomatic segmentationof the full heart in cardiac computed tomography images usinga Haar classifier and a statistical modelrdquo Journal of MedicalImaging and Health Informatics vol 6 no 5 pp 1298ndash13022016

[12] T Brox M Rousson R Deriche and J Weickert ldquoColour tex-ture and motion in level set based segmentation and trackingrdquoImage and Vision Computing vol 28 no 3 pp 376ndash390 2010

[13] T F Cootes G J Edwards and C J Taylor ldquoActive appearancemodelsrdquo in Proceedings of the 5th European Conference onComputer Vision H Burkhardt and B Neumann Eds vol II ofLecture Notes in Computer Science pp 484ndash498 Springer BerlinHeidelberg 1998

[14] Y C Hum K W Lai N P Utama M I Mohamad Salimand Y M Myint ldquoReview on segmentation of computer-aidedskeletal maturity assessmentrdquo in Advances in Medical Diag-nostic Technology Lecture Notes in Bioengineering pp 23ndash51Springer Singapore Singapore 2014

[15] C Li R Huang Z Ding J C Gatenby D N Metaxas and J CGore ldquoA level set method for image segmentation in the pre-sence of intensity inhomogeneities with application to MRIrdquoIEEE Transactions on Image Processing vol 20 no 7 pp 2007ndash2016 2011

[16] T Xu M Mandal R Long I Cheng and A Basu ldquoAn edge-region force guided active shape approach for automatic lungfield detection in chest radiographsrdquo Computerized MedicalImaging and Graphics vol 36 no 6 pp 452ndash463 2012

[17] J Keomany and S Marcel ldquoActive shape models using localbinary patternsrdquo Tech Rep IDIAP 2006

[18] T Ojala M Pietikainen and D Harwood ldquoPerformance evalu-ation of texture measures with classification based on Kullbackdiscrimination of distributionsrdquo in Proceedings of the 12thInternational Conference on Pattern Recognition - Conference AComputer Vision Image Processing (IAPR) vol 1 pp 582ndash585Jerusalem Israel 1994

[19] P Schelkens R Nava G Cristobal et al ldquoA comprehensivestudy of texture analysis based on local binary patternsrdquo inProceedings of SPIE 8436 Optics Photonics and Digital Techno-logies for Multimedia Applications II Brussels Belgium 2012

[20] J-B Martens ldquoThe Hermite TransformmdashTheoryrdquo IEEE Trans-actions on Signal Processing vol 38 no 9 pp 1595ndash1606 1990

[21] J-B Martens ldquoThe Hermite TransformmdashApplicationsrdquo IEEETransactions on Signal Processing vol 38 no 9 pp 1607ndash16181990

[22] A Estudillo-Romero and B Escalante-Ramirez ldquoRotation-invariant texture features from the steered Hermite transformrdquoPattern Recognition Letters vol 32 no 16 pp 2150ndash2162 2011

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

[24] DMumford and J Shah ldquoOptimal approximations by piecewisesmooth functions and associated variational problemsrdquo Com-munications on Pure and Applied Mathematics vol 42 no 5pp 577ndash685 1989

[25] S Osher and J A Sethian ldquoFronts propagating with curvature-dependent speed algorithms based onHamilton-Jacobi formu-lationsrdquo Journal of Computational Physics vol 79 no 1 pp 12ndash49 1988

[26] T F Chan B Y Sandberg and L A Vese ldquoActive contourswithout edges for vector-valued imagesrdquo Journal of Visual Com-munication and Image Representation vol 11 no 2 pp 130ndash1412000

[27] T F Cootes C J Taylor D H Cooper and J Graham ldquoActiveshapemodelsmdashtheir training and applicationrdquoComputer Visionand Image Understanding vol 61 no 1 pp 38ndash59 1995

[28] W T Freeman and E H Adelson ldquoThe design and use of steer-able filtersrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 13 no 9 pp 891ndash906 1991


[29] A M Van Dijk and J-B Martens ldquoImage representation andcompression with steered Hermite transformsrdquo Signal Process-ing vol 56 no 1 pp 1ndash16 1997

[30] J-B Martens ldquoLocal orientation analysis in images by means ofthe Hermite transformrdquo IEEE Transactions on Image Processingvol 6 no 8 pp 1103ndash1116 1997

[31] T Ojala M Pietikainen and T Maenpaa ldquoMultiresolutiongray-scale and rotation invariant texture classificationwith localbinary patternsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 24 no 7 pp 971ndash987 2002

[32] J Olveres R Nava B Escalante-Ramırez G Cristobal andC M Garcıa-Moreno ldquoMidbrain volume segmentation usingactive shape models and LBPsrdquo in Proceedings of the Appli-cations of Digital Image Processing XXXVI vol 8856 pp88561Fndash1ndash88561Fndash11 USA August 2013

[33] OpenCV ldquoHistogram comparisonrdquo accessed 2017-09-15(2017) URL httpdocsopencvorg24doctutorialsimgprochistogramshistogram comparisonhistogram comparisonhtml

[34] K N Chaudhury and K R Ramakrishnan ldquoStability and con-vergence of the level set method in computer visionrdquo PatternRecognition Letters vol 28 no 7 pp 884ndash893 2007

[35] B Sandberg T Chan and L Vese ldquoA level-set and gabor-basedactive contour algorithm for segmenting textured imagesrdquo TechRep UCLA Department of Mathematics CAM 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In recent years segmentation performance has increasedwith new improvements by combining other techniques andalso having the inclusion of more information cues liketexture or movement Such is the case of Jiang et al whocombined level set framework with seeded region growingalgorithm for segmentation of hippocampus [10] Ma et al[11] used a Haar classifier to detect the heart area and afterthis step the segmentation is performed with an ASM Broxet al [12] introduced simultaneously texture features graylevel and optic flow on a level-based segmentation processAn extension of ASMs has been proposed with the ActiveAppearance Model (AAM) including textural informationwithin the model as a whole [13] but the algorithm results inlow efficiency in real-time systems and high computationalcost besides having inconsistent robustness when appliedunder different circumstances [14]

On the other hand besides combining different tech-niques there exist information features inherent to theimages such as color texture or morphology that canbe included in order to provide especial characteristics toimprove the final segmentation It is known in general thatintensity inhomogeneity complicates many image segmen-tation methods making it difficult to identify the desiredregions based only on pixel intensity Level sets andAAMs areused as based-regionmethods but when dealing with imagesthat contain intensity inhomogeneities they tamper its per-formance [15] For this reason it is important to add featuresto the algorithms that improve the performance in spite ofinhomogeneities presence

Xu et al [16] propose to combine an automatic globaledge and region force field guided method with nonlinearexponential point evolution for lung field segmentationexpanding the ASM performance with this informationKeomany and Marcel [17] proposed to use LBPs also withASMs for the profile search instead of having only gray valuesInstead of gray values it uses LBP information avoiding thelimitations of AAMs

In the case of the Hermite transform (HT) and LocalBinary Patterns (LBP) both methods provide additionalfeatures that improve the segmentation algorithms LBPs arebeneficial when they are combined with other methods suchas ASMs giving a more solid result modeling local structuresin spite of illumination changes [18] In addition LBPs havevery low computational complexity and provide a robustway for describing local texture patterns [19] The Hermitetransform [20 21] is one of the mathematical models thatresembles the human visual system (HVS) These HVS mod-els have increased in popularity because they allow imagesto be expanded into a local decomposition that describesintrinsic attributes related to important cues and highlightstructuresThe information obtained from theHermite trans-form coefficients has previously been used as textural data forclassification tasks [22]

The present study proposes to combine segmentationalgorithms with information given by the HT and LBPs toimprove efficiency of current segmentation techniques suchas ASMs and LS In order to test our approach we used twoimage modalities for different organ studies such as cranialmidbrain MRI and endocardium left ventricle CT

This paper is organized as follows Section 2 introducesthe mathematical background for the proposal based onASMs and a deformablemodel Chan-Vese (CV) algorithm InSection 3 the method is explained in Section 4 the dataset isdescribed and in Section 5we display the experimentrsquos resultsand provide a discussion Finally Section 6 concludes ourwork

2 Theoretical Background

In this section we present the mathematical foundationThe deformable models used here are ASMs a statisticaldeformable model and vector value Chan-Vese model amethod based on level sets

21 Chan-Vese Model with Level Sets The active contourmodel without edges was proposed by [23] and it is inspiredby theMumford-Shahmethod for image segmentationwhichevolve an object by minimization of an energy functional[24] The original definition of the 2-dimensional Chan-Vesemodel (CV) is to letΩ be a bounded open set ofR2 with 120597Ωas the boundary We define an image as 119868 Ω rarr R Nowlet 119862 be an involving curve in Ω as a boundary between tworegions in such a way that Ω = Ω1 cup 119862 cup Ω2 whereΩ1 andΩ2 represent the inner and outer region of 119862 respectivelyFinally the formulation of the energy functional tries toidentify the best partition on 119868 as follows

inf1198881 1198882 119862

119865 (1198881 1198882 119862) (1)

where 1198881 and 1198882 are the two average regions of intensity valuesinside and outside the contour respectively The energy isminimized if the contour 119862 is located at the boundary of theobject of interestThe 2-dimensional energy functional of theCV model in the domain 119909 isin R2 is defined as follows

119865 (1198881 1198882 119862) = 120583int119862119889119904 + 1205821 int

Ω1

(119868 minus 1198881)2 119889119909+ 1205822 int

Ω2

(119868 minus 1198882)2 119889119909(2)

where 1205821 1205822 are fixed nonnegative constants for data fittingand 120583 is introduced to regularize the surface 119862 The minimalpartition problem given by (1) can be solved using the levelset method [25] The contour 119862 can be represented by thenonzero level set of a Lipschitz function called 120601 Ω rarr Rsuch that 119862 = 119909 isin Ω | 120601(119909) = 0 This approach has manyadvantages because it allows awide study of topologies whichis made on a rectangular grid The functional in (2) can bewritten in terms of the level set as follows

119865 (1198881 1198882 120601) = 120583intΩ120575 (120601 (119909)) (nabla120601 (119909)) 119889119909

+ intΩ1205821 (119868 (119909) minus 1198881)2119867(120601 (119909)) 119889119909

+ intΩ1205822 (119868 (119909) minus 1198882)2 (1 minus 119867 (120601 (119909))) 119889119909

(3)




(4)



120582+119894 (119868119894 (119909) minus 119888+119894 )2

+ 1119870119870sum119894=1







= 119884 + 119875b (6)


119878 = 1119873 minus 1119873sum119894=1

(119884119894 minus 119884) (119884119894 minus 119884)119879 (7)










1205902) (8)






(10)


119871120579119899minus119898119898 (1199090 1199100) = 119899sum119896=0

119871119899minus119896119896 (1199090 1199100) 119877119899minus119896119896 (120579) (11)





119899sum119898=0

[119871119899minus119898119898]2 = infinsum119899=0

119899sum119898=0

[119871120579119899minus119898119898]2 (13)


1198641119863119873 = 119873sum119899=1

[1198711205791198990]2 1198642119863119873 = 119873sum

119899=1

119899sum119898=1

[119871120579119899minus119898119898]2 (14)




LBP119875119877 (119892119888) = 119875minus1sum119901=0

119904 (119892119901 minus 119892119888) 2119901 (15)


119904 (119909) = 1 if 119909 ge 00 if 119909 lt 0 (16)



LBPuni119875119877 (119892119888)= 119875minus1sum119901=0


(17)





HT VVCVSHT


image






(18)













8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1


77

86 102

101

101

101

101

101

101

101

95 70

15Threshold


1 1

1 1

0

000

77 83 55

Trainingimages

TrainedASM model

ASM train

Shape model - PDM

LBP model

AverageLBP


86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

10

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

9586 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

Y





ASMrecognition

Segmentedimage

Testimage

Fitting phase

TrainedASM model

Region ofinterest

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56


L0 L1 L2 L3







119901 (119903119871119899) = 1119872119873 1198991198711205790 1198991198711205791 1198991198711205792 1198991198711205793 (19)

where 1198991198711205790

1198991198711205791

1198991198711205792

and 1198991198711205793




4 Materials




0 3 6 9

06

08

T

Dic

e ind

ex [0

-1]

CVVVCVSHT

(a)

0 3 6 90

20

40

60

80

T

Hau

sdor

ff di

stan

ce

CVVVCVSHT

(b)














0 3 6 90

2

4

6

TEn

ergy

CVVVCVSHT

middot107


1 2 3 4 5 6 7 8 9 10

06

08

1

Slice

Dic

e ind

ex [0

-1]

ASM


(a)

1 2 3 4 5 6 7 8 9 100

10

20

30

ASM


Slice

Hau

sdor

ff di

stanc

e

(b)








(a) (b) (c) (d)



0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM


(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)









(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







(4)



120582+119894 (119868119894 (119909) minus 119888+119894 )2

+ 1119870119870sum119894=1







= 119884 + 119875b (6)


119878 = 1119873 minus 1119873sum119894=1

(119884119894 minus 119884) (119884119894 minus 119884)119879 (7)










1205902) (8)






(10)


119871120579119899minus119898119898 (1199090 1199100) = 119899sum119896=0

119871119899minus119896119896 (1199090 1199100) 119877119899minus119896119896 (120579) (11)





119899sum119898=0

[119871119899minus119898119898]2 = infinsum119899=0

119899sum119898=0

[119871120579119899minus119898119898]2 (13)


1198641119863119873 = 119873sum119899=1

[1198711205791198990]2 1198642119863119873 = 119873sum

119899=1

119899sum119898=1

[119871120579119899minus119898119898]2 (14)




LBP119875119877 (119892119888) = 119875minus1sum119901=0

119904 (119892119901 minus 119892119888) 2119901 (15)


119904 (119909) = 1 if 119909 ge 00 if 119909 lt 0 (16)



LBPuni119875119877 (119892119888)= 119875minus1sum119901=0


(17)





HT VVCVSHT


image






(18)













8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1


77

86 102

101

101

101

101

101

101

101

95 70

15Threshold


1 1

1 1

0

000

77 83 55

Trainingimages

TrainedASM model

ASM train

Shape model - PDM

LBP model

AverageLBP


86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

10

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

9586 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

Y





ASMrecognition

Segmentedimage

Testimage

Fitting phase

TrainedASM model

Region ofinterest

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56


L0 L1 L2 L3







119901 (119903119871119899) = 1119872119873 1198991198711205790 1198991198711205791 1198991198711205792 1198991198711205793 (19)

where 1198991198711205790

1198991198711205791

1198991198711205792

and 1198991198711205793




4 Materials




0 3 6 9

06

08

T

Dic

e ind

ex [0

-1]

CVVVCVSHT

(a)

0 3 6 90

20

40

60

80

T

Hau

sdor

ff di

stan

ce

CVVVCVSHT

(b)














0 3 6 90

2

4

6

TEn

ergy

CVVVCVSHT

middot107


1 2 3 4 5 6 7 8 9 10

06

08

1

Slice

Dic

e ind

ex [0

-1]

ASM


(a)

1 2 3 4 5 6 7 8 9 100

10

20

30

ASM


Slice

Hau

sdor

ff di

stanc

e

(b)








(a) (b) (c) (d)



0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM


(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)









(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







(10)


119871120579119899minus119898119898 (1199090 1199100) = 119899sum119896=0

119871119899minus119896119896 (1199090 1199100) 119877119899minus119896119896 (120579) (11)





119899sum119898=0

[119871119899minus119898119898]2 = infinsum119899=0

119899sum119898=0

[119871120579119899minus119898119898]2 (13)


1198641119863119873 = 119873sum119899=1

[1198711205791198990]2 1198642119863119873 = 119873sum

119899=1

119899sum119898=1

[119871120579119899minus119898119898]2 (14)




LBP119875119877 (119892119888) = 119875minus1sum119901=0

119904 (119892119901 minus 119892119888) 2119901 (15)


119904 (119909) = 1 if 119909 ge 00 if 119909 lt 0 (16)



LBPuni119875119877 (119892119888)= 119875minus1sum119901=0


(17)





HT VVCVSHT


image






(18)













8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1


77

86 102

101

101

101

101

101

101

101

95 70

15Threshold


1 1

1 1

0

000

77 83 55

Trainingimages

TrainedASM model

ASM train

Shape model - PDM

LBP model

AverageLBP


86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

10

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

9586 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

Y





ASMrecognition

Segmentedimage

Testimage

Fitting phase

TrainedASM model

Region ofinterest

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56


L0 L1 L2 L3







119901 (119903119871119899) = 1119872119873 1198991198711205790 1198991198711205791 1198991198711205792 1198991198711205793 (19)

where 1198991198711205790

1198991198711205791

1198991198711205792

and 1198991198711205793




4 Materials




0 3 6 9

06

08

T

Dic

e ind

ex [0

-1]

CVVVCVSHT

(a)

0 3 6 90

20

40

60

80

T

Hau

sdor

ff di

stan

ce

CVVVCVSHT

(b)














0 3 6 90

2

4

6

TEn

ergy

CVVVCVSHT

middot107


1 2 3 4 5 6 7 8 9 10

06

08

1

Slice

Dic

e ind

ex [0

-1]

ASM


(a)

1 2 3 4 5 6 7 8 9 100

10

20

30

ASM


Slice

Hau

sdor

ff di

stanc

e

(b)








(a) (b) (c) (d)



0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM


(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)









(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





HT VVCVSHT


image






(18)













8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1


77

86 102

101

101

101

101

101

101

101

95 70

15Threshold


1 1

1 1

0

000

77 83 55

Trainingimages

TrainedASM model

ASM train

Shape model - PDM

LBP model

AverageLBP


86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

10

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

9586 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

Y





ASMrecognition

Segmentedimage

Testimage

Fitting phase

TrainedASM model

Region ofinterest

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56


L0 L1 L2 L3







119901 (119903119871119899) = 1119872119873 1198991198711205790 1198991198711205791 1198991198711205792 1198991198711205793 (19)

where 1198991198711205790

1198991198711205791

1198991198711205792

and 1198991198711205793




4 Materials




0 3 6 9

06

08

T

Dic

e ind

ex [0

-1]

CVVVCVSHT

(a)

0 3 6 90

20

40

60

80

T

Hau

sdor

ff di

stan

ce

CVVVCVSHT

(b)














0 3 6 90

2

4

6

TEn

ergy

CVVVCVSHT

middot107


1 2 3 4 5 6 7 8 9 10

06

08

1

Slice

Dic

e ind

ex [0

-1]

ASM


(a)

1 2 3 4 5 6 7 8 9 100

10

20

30

ASM


Slice

Hau

sdor

ff di

stanc

e

(b)








(a) (b) (c) (d)



0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM


(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)









(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1

8000

6000

4000

2000

0

0 01 02 03 04 05 06 07 08 09 1


77

86 102

101

101

101

101

101

101

101

95 70

15Threshold


1 1

1 1

0

000

77 83 55

Trainingimages

TrainedASM model

ASM train

Shape model - PDM

LBP model

AverageLBP


86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

10

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

9586 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

Y





ASMrecognition

Segmentedimage

Testimage

Fitting phase

TrainedASM model

Region ofinterest

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56


L0 L1 L2 L3







119901 (119903119871119899) = 1119872119873 1198991198711205790 1198991198711205791 1198991198711205792 1198991198711205793 (19)

where 1198991198711205790

1198991198711205791

1198991198711205792

and 1198991198711205793




4 Materials




0 3 6 9

06

08

T

Dic

e ind

ex [0

-1]

CVVVCVSHT

(a)

0 3 6 90

20

40

60

80

T

Hau

sdor

ff di

stan

ce

CVVVCVSHT

(b)














0 3 6 90

2

4

6

TEn

ergy

CVVVCVSHT

middot107


1 2 3 4 5 6 7 8 9 10

06

08

1

Slice

Dic

e ind

ex [0

-1]

ASM


(a)

1 2 3 4 5 6 7 8 9 100

10

20

30

ASM


Slice

Hau

sdor

ff di

stanc

e

(b)








(a) (b) (c) (d)



0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM


(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)









(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





ASMrecognition

Segmentedimage

Testimage

Fitting phase

TrainedASM model

Region ofinterest

86 102

101 95 70

15Threshold


Decimal value = 128 + 64 + 4 + 2 = 198

1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56

86 102

101 95 70

15Threshold


1 1

1 1

0

00

0

77 83 56


L0 L1 L2 L3







119901 (119903119871119899) = 1119872119873 1198991198711205790 1198991198711205791 1198991198711205792 1198991198711205793 (19)

where 1198991198711205790

1198991198711205791

1198991198711205792

and 1198991198711205793




4 Materials




0 3 6 9

06

08

T

Dic

e ind

ex [0

-1]

CVVVCVSHT

(a)

0 3 6 90

20

40

60

80

T

Hau

sdor

ff di

stan

ce

CVVVCVSHT

(b)














0 3 6 90

2

4

6

TEn

ergy

CVVVCVSHT

middot107


1 2 3 4 5 6 7 8 9 10

06

08

1

Slice

Dic

e ind

ex [0

-1]

ASM


(a)

1 2 3 4 5 6 7 8 9 100

10

20

30

ASM


Slice

Hau

sdor

ff di

stanc

e

(b)








(a) (b) (c) (d)



0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM


(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)









(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0 3 6 9

06

08

T

Dic

e ind

ex [0

-1]

CVVVCVSHT

(a)

0 3 6 90

20

40

60

80

T

Hau

sdor

ff di

stan

ce

CVVVCVSHT

(b)














0 3 6 90

2

4

6

TEn

ergy

CVVVCVSHT

middot107


1 2 3 4 5 6 7 8 9 10

06

08

1

Slice

Dic

e ind

ex [0

-1]

ASM


(a)

1 2 3 4 5 6 7 8 9 100

10

20

30

ASM


Slice

Hau

sdor

ff di

stanc

e

(b)








(a) (b) (c) (d)



0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM


(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)









(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0 3 6 90

2

4

6

TEn

ergy

CVVVCVSHT

middot107


1 2 3 4 5 6 7 8 9 10

06

08

1

Slice

Dic

e ind

ex [0

-1]

ASM


(a)

1 2 3 4 5 6 7 8 9 100

10

20

30

ASM


Slice

Hau

sdor

ff di

stanc

e

(b)








(a) (b) (c) (d)



0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM


(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)









(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





(a) (b) (c) (d)



0 10 20 30 40 50 60 70 80 9008

085

09

095

Cardiac cycle ()

Dic

e ind

ex [0

-1]

ASM


(a)

0 10 20 30 40 50 60 70 80 90

10

20

30

ASM


Cardiac cycle ()

Hau

sdor

ff di

stanc

e

(b)









(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





(a) 0 (b) 10 (c) 20

(d) 30 (e) 40 (f) 50

(g) 60 (h) 70 (i) 80





Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




researcharticle deformable models for segmentation based ...2 mathematicalproblemsinengineering...

Documents