Fuzzy shape based motion evaluation of left ventricle

using genetic algorithm

A. Mishra a,*, P.K. Dutta b, M.K. Ghosh b

a Vanderbilt University, Nashville, TN, USAb Electrical Engineering Department, IIT Kharagpur, West Bengal, India

Received 14 May 2004; received in revised form 11 October 2005; accepted 31 January 2006

Abstract

A shape based non-rigid cardiac motion study is presented using simple fuzzy shape descriptors. The objective of this work is to evaluate the

detail point wise motion trajectories from sequential contours. The shape correspondence on endocardial contour has been performed in multiple

stages with well-defined, level specific curvature information. We incorporate non-uniform expansion and contraction of shape matched templates

to optimize the correspondence in each level. However, final flow field evaluation is a constrained optimization problem, which results into a

smooth mapping of contours. Constrained non-linear optimization with genetic algorithm has shown considerable promise in solving this

problem. The results are quite consistent when correlated with the movement of implanted markers in an experimental set-up. Even though

tracking contours in the reverse direction is irrelevant from a practical standpoint a good correlation between motions in either direction is

observed. The algorithm has been tested over sets of 2D images to quantify the motion of left ventricle (LV) using two different imaging

modalities.

q 2006 Elsevier B.V. All rights reserved.

Keywords: Fuzzy shape properties; Genetic algorithm; Left ventricular motion; MRI and echocardiography images

1. Introduction

Study of complex shape deformity of left ventricle (LV) is

an area of focus since the beginning of 1990s. A good volume

of work has been reported to study the non-rigid non-uniform

motion of endocardial and epicardial wall using 2D image

sequences. Quantitative motion evaluation, to study the cavity

morphology, degree of infracted muscles and other coronary

artery diseases is normally performed on left ventricle images.

Hence, an accurate assessment and interpretation of cardiac

motion is important to clinically diagnose cardiac pathology

related to coronary circulation deficiency and myocardial

artifacts.

1.1. Contour extraction

Number of issues concerns the development of algorithms

for computerized automation of the contour extraction using

0262-8856/$ - see front matter q 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.imavis.2006.01.010

* Corresponding author.

E-mail address: arimishra@rediffmail.com (A. Mishra).

verity of imaging modalities. Some of the early methods [1–3]

require the gray value information to identify the cardiac

boundary, which may not be sufficient to extract the contours

except images with better contrast and SNR. Suh et al. [4]

presented a technique using uncertainty reasoning within the

Dempster–Shafer framework, combining the low-level image

features. The interactivity with contour initialization was

minimal and it could propagate the contour over the entire

study. Work of Fleagle et al. [5] suggests a graph search

method to minimize a cost function based on image gradients.

However, Staib et al. [6] implemented a probabilistic

deformable model considering the boundary to be a 2D

deformable object using maximum posteriori estimate.

Snakes or active contour models have become primary

focus for segmentation and tracking of non-rigid bodies, e.g.

cell mobility from microscopic images [7]. Automated contour

extraction algorithm using an active contour model has been

proposed by Ranganath [8,9] applicable to spin and gradient

echo MRI image sequences with specific intermediate

preprocessing. However, work in [9] is associated with a

contour propagation technique to track the boundary reliably in

a sequence despite its poor temporal resolution. Similar work

of Chalana et al. [10] reports an interesting approach to detect

epicardial and endocardial boundary of short axis

Image and Vision Computing 24 (2006) 436–446

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A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446 437

echocardiography images with a multiple active contour

model. Active surface model where the surface is represented

as a sequence of planner contours. The proposed method

requires user defined initial approximation for epicardial

boundary that detect contours by computing image gradients.

Recent work on segmentation of medical images has been

reported using geometric active deformable models where the

contour propagates with a velocity profile as a function of

curvature [11]. Contour detection on gradient echo MRI

images using micro-genetic algorithm has been reported by our

early work [12]. The proposed algorithm involves less

computational time in comparison to conventional GA,

where as a more generalized version, is reported in [13].

Some of the above techniques are quite robust for contour

extraction and propagation [9,10]. However, it is difficult to

estimate detail point wise motion with snake models because

the optimization is performed globally.

1.2. Shape correspondence based motion estimation

Shape correspondence based point matching is preferred for

its benefits and simplicity of using point features in image

registration. Correspondence between samples on two sequen-

tial contours are established by matching the shape properties

of contour segments surrounding each of the points [14]. Shape

correspondence techniques based on curvature information to

optimize a cost functional in the contour space has been

presented by Cohen et al. [15]. Minimization in this case being

highly non-linear, it is difficult to get a smooth shape based

correspondence in the Euclidean space. A possible 3D

expansion is suggested taking similar considerations in to

account [16]. However, similar work of Demi et al. [17]

implements regularization of the flow vectors using an

interpolation scheme to estimate the shape correspondence

between subsampled shape features. Experimental methods

using physically implanted markers to quantify the motion in

animal heart are presented in [18] where the markers were

corresponded and tracked in a stereo static external co-ordinate

system. This method possesses the disadvantage of invasive-

ness and limitation on implanting markers to track motion at all

locations of interest. The MR spin tagging in the other hand

creates virtual markers to track the motion but the tagging does

not last over the entire cycle and applications are limited to

MRI images only [19].

A significant contribution on the point wise motion

quantification has been reported in the work of McEachen

et al. [20] based on bending energy model. Correspondences

between samples on two sequential contours are established by

matching the shape properties of contour segments surrounding

each of the points [21]. The local curvature difference between

the contours under consideration and the mean normal contour

is found out at a number of equidistant sample points. The

weighted square of these differences added over a set of points

is found to be the regional bending energy. The flexibility and

success of the method depends on meaningful curvature

information. Hence the discrete contour data requires

preprocessing, low pass filtration or smoothing to reduce the

effect of noise and inter observer anomalies in tracing the

boundaries. The final flow field evaluation is an optimization

problem adding a smoothness constraint to the flow vectors,

weighted by two uniquely defined functions (i.e. closeness and

uniqueness) in both Euclidean and contour space. A 3D surface

mapping of the myocardial wall motion has been recently

reported [22] to estimate dense motion field.

Even though the field has strongly moved towards surface

correspondence we feel the necessity of shape based

mapping of 2D contours taking their non-uniform expansion

and contraction into consideration. This has not been given

enough attention while establishing the shape correspon-

dence in most of the non-rigid correspondence problems.

We have implemented recursive maximization of closeness

of the fuzzy shape descriptors assigned to the non-uniform

samples. It results into a vector flow filed, which is then

optimized using genetic algorithm resulting in to a smooth

point matched trajectory of flow vectors. Results are

compared with a bending energy model. A similar approach

with considerable promise has been reported by Mishra et

al. [23,24] to evaluate the 2D motion field using wavelet

based curvature information.

Section 2 presents problem formulation. Section 3 describes

the fuzzy shape based mapping of contours where the non-

uniform variation of contours is taken in to account. In Sections

4 and 5 optimization procedures are presented for the point

tracked motion field using genetic algorithm. Section 6

presents the results and its comparison with wavelet model.

Section 7 provides a general conclusion and future scope of the

work.

2. Problem formulation

2.1. Boundary extraction

Contour extraction of sequential images in a cardiac cycle is

an essential requirement for motion estimation. Heart being a

3D spatially deforming body the motion is often predicted from

2D images at different axial locations. Our algorithm is based

on the assumption that there is minimum out of plane motion

while estimating it from 2D temporal sequences acquired in a

particular plane. A number of methods have been proposed in

literatures to isolate the blood pull in the LV including [1–14].

However, contour extraction procedure is not the prime issue

of this work and in the present context we implement two

different methods to extract the endocardial boundary of left

ventricle i.e. (1) Outlining it manually with the help of trained

observers, radiologists, (2) By implementing the procedures in

[12] and [13] for ultrasound and MRI images.

2.2. Model considerations

Normal procedure to establish correspondence requires

finding out some shape similarity between consecutive

boundaries. The initial shape matching based on bending

energy model assumes the contour length and samples to be

equal in the consecutive frames [20–22]. This issue needs to be

Fig. 1. Initial shape matching.

A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446438

addressed carefully when the temporal resolution is poor,

particularly while analyzing MRI sequences. The maximum

numbers of frames that can be acquired in a cardiac cycle vary

from 16 to 20. The maximum temporal resolution in this case is

approximately 60 ms, which is sufficient for the cardiac muscle

to undergo significant deformation. In addition the cardiac

muscle undergoes non-uniform expansion and contraction at

different locations. We present this work taking the above

considerations in to account with less number of assumptions

and restrictions. The corresponding samples may be of unequal

length when shape similarity is computed by maximizing the

closeness of the fuzzy membership functions.

The point matching method for motion estimation is based

on simple pixel distribution statistics of the perceived contours.

Shape matching is performed by varying the sample dimension

in a definite manner to incorporate the non-uniform shape

variation (contraction/expansion). The point correspondence in

the analysis leads to a monotonic mapping of samples in the

successive contours, which may not necessarily be of equal

length. An initial correspondence for a set of consecutive

contours has been established based on their shape properties

and subsequent matching is performed in a similar manner like

in our previous approach [24].

The allowed variation of the sample length is decided based

on the average variation of the contour length between

consecutive contours, because it is difficult to predict the

piece wise variation of the contour dimension due to non-

uniform expansion/contraction of the LV wall. The samples

having optimum difference between fuzzy shape functions in a

definite class are indexed as the shape matched pair. Coarser

level matching of samples is performed with less number of

samples and once the shape matched vectors are obtained at a

particular level similar matching procedure is repeated in the

corresponding sample space. The contour mapping preserves

monotonicity and provides a non-uniform set of matched

samples. The final flow field evaluation is a non-linear

optimization problem associated with cost the functions

regulating the smoothness and closeness of the shape matched

samples.

3. Fuzzy shape based correspondence

3.1. Fuzzy shape descriptors

Shape description of the samples depends on the pixel

distribution statistics of the perceived contours in a 2D image.

A sample on the contour is either circular or straight if its

boarder satisfies the equation of a circle or straight line. In most

cases the contours appear nearly circular or straight where a

continuous gradation of circularity or straightness can be done

with a fuzzy concept. For example in a set of objects U a fuzzy

subset may defined in terms of some properties of objects in U.

To quantify fuzzy shape membership of a sample on a curve,

which may belong to an elementary class circular, we assign a

membership function mcir between 0 and 1. For an object R2U,

mcir(R) is the degree of membership of R in the set of circular

objects (samples). A simple definition of fuzzy degree of

circularity mcir is defined below

mcirðRÞZ1

exp 1K srLCi

� �2� �b(1)

here, s is the standard deviation of all the radius vectors

connecting the center of the fitting circle to all the border points

on LCi, b is assumed to be one here unless some more specific

membership function is to be defined (Fig. 1).

Similarly to express the fuzzy degree of straightness, the

offset and the slope of the straight line yZa0Ca1x has to be

estimated. The data vector contains the data points (y co-

ordinates) and the model matrix relates the data points and

model parameters (x co-ordinates Eq. (2)).

1 x1

1 x2. .

1 xn

26664

37775

a0

a1

� �

G m

Z

y1

y2.

yn

2664

3775

d

(2)

The least square solution to the Eq. (2) is

mest Z ðGTGÞK1GTd (3)

As the method is based on statistics, the solution includes an

estimate of the errors. The mean deviation between the

measured and predicted data points is directly related to the

norm of the error vector. After mest is evaluated from the whole

set of data, s2 for the individual samples can be computed with

the matrix d and G for the individual samples with a global mest

on the segment Lc. The fuzzy shape membership function ms for

the samples are computed in the similar manner (Eq. (4)). A

suitable threshold value of the radius of the fitting circle

classifies the shape in to two elementary classes.

ms Z 1=expðs2Þ (4)

A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446 439

3.2. Initial shape correspondence

The contours Ci and CiC1 are considered to be subset of the

ordered set of N contours in a 2D image space i.e. Ci3R2ci2[1,.N]. Initial correspondence is established between these

two consecutive and deformable contours at time instants ti and

tiC1. The contours are then divided in to n number of samples

of equal length, which are indexed asmip; p½1;.n�0mi

p2Ci.

A window W is defined in such a way that it includes two

sections of contours comprising of three samples each on Ci

and CiC1 to establish the initial match (Fig. 1).

The starting point (mid-point) of the section LCiis chosen

arbitrarily on the contour Ci. It is assumed that the motion is

concentrated around the center of gravity (c.g. of area enclosed by

Ci). The intersection point on the contour CiC1 by the line

connecting the c.g. to the starting point on Ci is considered as the

midpoint of the section LCiC1to be included on CiC1. The middle

sample on LCiin the window is indexed as the initial sample mi

1.

Segments LCiand LCiC1

are classified in to two simple geometric

shapes i.e. either a straight line or a circle based on their curvature.

This classification is based on radius of curvature of the circles

considering the twoendpoints and themidpoint ofLCito beon the

circumference of a circle. LCiis considered to be a straight line if

the radius of curvature rLCiis more than a threshold value. The

shape matching is performed by finding out the closest fuzzy

membership function associated with the sample on LCiC1, as we

go on shifting it pixel wise on the contour from the left most

sample position (Fig. 1). The process is repeated for all the

contours to provide an initial match.

3.3. Non-linear mapping of subsequent samples

After the initial correspondence is established, the subsequent

samples are matched in a different manner i.e. by incorporating

the non-uniformvariation of the sample dimensions. Thewindow

W is shifted one sample in a particular direction and the shape

membership of the next sample mi2 is calculated. Initially, it is

assumed that sample miC12 is linearly mapped version of mi

2 on

CiC1 as we map it adjacent to the previous sample. Subsequent

shape matching is done by varying the length of the sample in an

Fig. 2. Subsequent sample matching Illustration.

allowed region (m%Km%%). Initially center ofm%Km%% is located

at the end of miC12 .

The fuzzy shape membership function of miC12 is computed

by shifting m% in unit steps (pixel wise) i.e. by successive

inclusion of points on CiC1 in the region m%Km%% (Fig. 2). It is

based on the assumption that the search region is the range of

probable expansion/contraction of the sample. However, one

may encounter following ambiguities in establishing the shape

correspondence for detail motion estimation.

(i) Possibility of getting a sample pair belonging to the same

class cannot be guaranteed. Even though it occurs less

frequently the optimal matching becomes difficult to

correlate two different classes. In this situation the

threshold radius of the fitting circle is increased to allow

both samples to belong to the same class.

(ii) Samples on consecutive contours may attain closest

membership function at more than one location. In that

case we choose the smaller sample as the closest match

in the contraction (EDKES) phase, else the larger one in

the expansion phase (ESKED).

3.4. Multiple level mapping of contours

A multiple level mapping of samples is implemented to

overcome the difficulties with implementation of filters to

preprocess the contour before point matching [21]. The contour

is divided into less number of samples (varies from 8 to 16

depending on the contour size under study) for a coarser level

matching. In this level the contours are optimized with internal

energy alone with a large bending force in a relatively small-

constrained region, motivated by the work in [12,13]. The

objective is to make the contour smooth so that the coarser

level matching can be done with reduced computational

burden. The next stage of matching is confined to the

previously matched sample region where the actual pixel

distribution statistics is considered to compute shape member-

ship functions. It makes the method attractive as it takes the

detail curvature information in to account in the finer level. The

number of divisions in the corresponding sub sample region is

kept constant to maintain uniformity in the flow field. The

process is repeated for all contours from ES to ED, and

similarly from ED to ES.

The two-step shape correspondence is performed to study

the motion field followed by an optimization of the final flow

field. A constrained optimization solves the final flow field

evaluation taking the smoothness and closeness of point

matching into account. The indexed shape matched sample

pair and their respective fuzzy shape membership functions in

the allowed search region are stored for optimization of the

motion field using GA.

4. Flow field optimization

The displacement flow vector dip in the image plane registers

each discrete sample on the contour Ci on to the next temporal

A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446440

sequence CiC1, defined by a mapping function j(pi). A

smoothness term is defined as the squared magnitude of the

first derivative of the flow field (Eq. (5)).

KðpÞZvdðpÞ

vp

��������2

(5)

Smoothening of the vector flow field with the above

criterion alone may not preserve the shape similarity. Hence,

the closeness of the fuzzy shape descriptors between the

matched samples is incorporated as another constraint. The

closeness term for a pair of consecutive contours is defined as

the sum of square of the difference of membership functions

(Eq. (6)). However, both smoothness and closeness parameters

are subject to choice of admissible space (Section 5.2).

MðpÞZvmðpÞ

vp

��������2

(6)

Putting Eqs. (5) and (6) together an optimization functional

is defined to solve a set of smooth mapping �jðpiÞ.

�jðpiÞZ argminj#½a!KðpÞCb!MðpÞ�dpi (7)

Here, a and b are two constants (0–1) regulating the

smoothness and closeness of point matching. Eq. (5) is

descretized with the normal and tangential component of the

flow vectors.

KsðmÞZ diNm KdiNmK1

� �2C diTm KdiTmK1

� �2h i(8)

A similar discretization of the closeness term along with the

smoothness functional may be used to transform the Eq. (7) in

to an energy minimization function in Eq. (9).

EZa!XnmZ1

diNm KdiNmK1

� �2C diTm KdiTmK1

� �2h iCb

!XnmZ1

mimKmiC1

m

� �2(9)

Minimization of the energy functional E (Eq. (9)) is

subjected to two constraints i.e. the length of the matched

sample should not be out side Gl% (depends on mean inter-

frame change in contour dimension) of the sample length and

the deviation in fuzzy shape membership functions should not

be more than a prespecified limit. The weighting function a and

b is chosen between 0 and 1 to acceptably quantify the motion

in the study. It is difficult to solve this non-linear and

constrained optimization problem using conventional optim-

ization methods, hence we solved it using genetic algorithm.

5. Solution approach

5.1. Genetic algorithm

Genetic algorithm (GA) is a heuristic optimization

technique, which is distinguished by its parallel investigation

of several areas of search space, simultaneously by

manipulating a population of candidate solutions [25–27].

GA starts with a fixed population of candidate solutions and

each of the candidates are evaluated with a fitness function that

is a measure of the candidate’s potential as a solution to the

problem. Genetic operators like selection, crossover and

mutations are implemented to simulate the natural evolution.

A population, usually presented by a binary string is modified

by the probabilistic application of the genetic operators from

one generation to the next. It has a potential of multi-

dimensional optimization because it starts searching the

optimum from multiple points rather than a single initial

approximation. Applications in computer vision include edge

detection [28], adoptive image segmentation [29] and

application on rigid body correspondence [30].

5.2. Chromosome coding

Input to the optimization algorithm is a set of corresponding

points PZ Xpjpj pZ1;.n & jpZn1;.n2

j k, where para-

meter vector XpjpZ �mi

pðxÞ; �mipðyÞ; ujp ; vjp

j krepresents the

corresponding point sets. Here, �mipðxÞ and �mi

pðyÞ represents

the Cartesian coordinate of the sample point �mip on the contour

Ci and ujp , vjp represents the points on the shape matched

sample on CiC1 based on the first criteria of allowed variation

of sample length. Parameter ujp and vjp represents the point sets

in the region confined to m%Km%%, which are the candidate

solutions to the GA. The limit of variation n1 and n2 are

converted in to its nearest whole number to pick up the indexed

points on the solution contour. Traditionally genetic algorithm

use bit string to represent chromosomes with each gene either

being zero or one. The parameter vector XpjpðrÞ, r21,.R (R

number of candidate solutions in the population) is used as the

genotypes, where ujp and vjp are generated randomly in the

constrained space (m%Km%%).

5.3. Objective function evaluation

The fitness function in genetic algorithm maps each

candidate solution as a scalar representing the candidate’s

fitness in the population. The displacement vector field (DVF)

dipðpZ1;.nÞ is evaluated based on the genotypes along with

their fuzzy membership functions. The chromosomes ujp and

vjp that do not satisfy the fuzzy shape constraints is disregarded

for genetic operations in the present generation but are

subjected to conditional mutation. The fitness functions are

evaluated for all candidates taking the smoothness and

closeness terms in to account (Eq. (9)) with a shape constraint

for an optimal match in Eq. (10).

VT Z mimKmiC1

m =mim

�� ��; cm2ð1.1Þ (10)

The candidate solutions for which PT, is above a

predefined value is treated as a solution with poor fitness. We

implemented steady state GA, which is suitable for optimiz-

ation problems with large number of variables in the solution

vector [27]. A fixed portion of the candidates is directly

selected based on their fitness value for the next generation.

Fig. 4. Optimized displacement vector field (DVF) of six sequential contours of

the phantom image using fuzzy shape descriptor model.

A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446 441

This subset of the population goes through regular selection for

mating but is not altered going into the next generation. In

addition to the conventional mutation the candidates, not

satisfying the closeness criteria are also subjected to a

conditional mutation. A faster minimization of the energy

function (Eq. (12)) is observed using this mutation scheme. The

optimization of the flow field is performed after the finer level

shape matched vectors are obtained and the process is repeated

for the next contour in the sequence.

6. Experimental results

6.1. Phantom image analyses

To experimentally verify the non-rigid motion we have

studied the deformation that occurs on an elastic balloon. A

CCD camera (resolution-644!480 pixels) interfaced with

Silicon Graphics machine acquired 12 frames from the top of

an air filled balloon as it collapses releasing air [24,31]. The

image contrast is kept deliberately low to make the image

quality comparable to the MRI (spin echo) cardiac images

where blood pull appears dark in comparison to the

surrounding muscles (Fig. 3). A silver foil is glued on the

surface of the balloon as a marker. The trajectory of the marker

is treated as ground truth to validate the algorithm. The point

wise motion has been tracked after extracting the contour by an

automated approach [13].

The flow field optimized with GA is shown in Fig. 4. Fig. 5

shows the contour mapping function where x-axis indexes the

number of samples, with the contour length measured from the

starting point of the second contour in the y-axis. The two

weighting functions a and b, regulating the smoothness and

closeness of the shape-matched vectors are fixed at 0.5 [31]

with a contour expansion constraint PTZ0.2 (Eq. (10)). The

shape correspondence is established in two stages. Initial

matching of samples is based on the smooth boundaries

optimizing the internal energy alone (with bending and

stretching coefficients 0.7 and 0.4, respectively [12]). The

Fig. 3. A sequence of six images acquired by CCD camera (alternate frames

from a 12 frames series) to track the motion of implanted marker (a small silver

foil).

final shape correspondence based flow field evaluation is

performed on the actual perceived contour to incorporate the

true shape characteristic of each contour. The movement of the

implanted markers on the initial contour is tracked and the

results are compared with our previous wavelet approach [24]

and bending energy (BE) model [20].

The mean absolute deviation (MAD) between the algor-

ithm-based trajectory (traced by interpolating the exact marker

position on initial frame in corresponding shape matched

samples) and the true trajectory is used as an index for

performance analysis. The maximum allowed variation of the

sample length is G10%, normally higher than the true mean

variation of contour length. The search space provides

flexibility in either direction to choose befitting shape match

in the analysis (Fig. 2). The candidate solutions having at least

a pair of corresponding samples where the shape similarity

PT(0.2 (Eq. (10)) are subjected to a conditional mutation

before going for regular selection. However, larger search

space some times leads to unrealistic matches, resulting poor

output, even after the final optimization of the flow field. The

mean absolute deviation (MAD) of Euclidean distance between

the true and predicted marker and respective standard deviation

(STD) is measured (Table 1).

Fig. 5. Mapping of samples on initial contour on to the next frame. Sharp

discontinuity indicates wrap around.

Table 1

Mean absolute deviation (MAD) and corresponding standard deviations from the actual marker location using wavelet, fuzzy and bending energy model

Sl. no. Wavelet model BE model Fuzzy model

MAD STD MAD STD MAD STD

1 5.6221 1.1307 7.1167 3.2121 6.2224 0.7301

2 6.1456 0.8223 5.3820 2.0826 5.5827 0.7930

3 5.9824 0.7355 6.3414 2.4535 5.1631 0.6121

4 6.1095 1.3845 7.1590 2.7155 4.9233 0.7122

5 7.0120 0.9729 8.0354 3.5457 5.2322 0.6812

6 4.7786 0.9184 4.7356 2.2324 5.0345 0.9765

7 3.7430 1.9508 5.9591 2.0245 4.7165 0.7552

8 4.1164 1.3370 7.0515 1.9477 5.1329 0.5120

9 5.5228 0.7438 7.9358 2.1758 4.2117 0.4822

10 3.0309 0.6710 8.5327 1.8824 3.8214 0.6812

A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446442

The consistency of the results in accurately analyzing the

detail non-rigid motion from contour data is studied by varying

the number of samples in a separate study. The total number of

samples is varied between 88 and 24 in discrete steps of 8 by

suitably choosing the number of samples in the initial and final

level. In this experiment we have implanted four markers on

the surface. The mean correlation (PPMC) between the actual

trajectories of the marker and that obtained by different

methods are computed. The mean correlation coefficient for

different value of number of samples in to which the contours

are divided is presented for comparison. The correlation can be

maximum when the trajectories are parallel and can only be

parallel when they identically follow the same path because

they start from the same location on the initial contour. Fig. 6

shows the correlation vs. the number of samples using both the

methods.

6.2. Sequential cardiac image analysis (in vivo)

The input images for this analysis are a sequence of MR

images acquired on a healthy volunteer (29 years, without

pathology) with a Siemens Magnetom Impact 1.5-T Scanner

(Fig. 7). LV was imaged with five cross-sectional tomographic

slices and each slice is imaged at 16 cardiac phases. Gradient

echo axial image sequences were acquired with a section

thickness of 5mm (Field of view 350!350 mm). The data

acquisition was R wave triggered with a repetition time of

40 ms and an echo time of 7 ms. Image frames over the cardiac

Fig. 6. PPM correlation between the actual trajectory of motion obtained using

different methods.

cycle were stored in BMP format. Out of 16 sequential images

four alternate frames from end diastole (ED) to end systole

(ES) were selected for visualization of motion.

A trained observer extracted the contours for this particular

experiment (Fig. 7). Fig. 8 shows the optimized contours with

internal energy alone by applying micro-genetic algorithm

[12,13]. The contours presented for this analysis are divided

into 64 uniform samples (number of samples with which the

final flow field has to be estimated). A Micro-genetic algorithm

is computationally in expensive and can result smooth curves

because the image force is ignored in this case. This algorithm

is faster and works as a low pass filter for the contour shape.

However, a global optimum is not guaranteed for the solution.

A similar analysis to study the consistency of the results

with respect to the total number of samples is performed and

compared with the wavelet and bending energy model. Due to

the non-availability of full proof technique to validate the

algorithm based point-wise motion of the endocardial

boundary, we consider few corresponding point sets as virtual

markers (Fig. 9). The virtual markers are indexed by their

corresponding positions while tracking them with 60 samples.

The mean absolute deviation (MAD) of the marker trajectories

is estimated by varying number of samples between 32 and 96

in discrete steps of eight. The final flow field optimization is

performed with same values of a, b and PT as in case of the

phantom image. The averages of MAD at four reference

locations are shown in Fig. 10.

6.3. Algorithm performance analysis

Experiments with sequence of automatically delineated

contours using two different modalities (Echocardigraphy and

MRI) are carried out to study the robustness of the algorithm

w.r.t parametric variation. The results were compared by

correlating it with the motion of virtual markers as in the

previous case. The average correlation between the reference

trajectories and that obtained varying the number of samples is

presented in Table 2. Fig. 11 shows the typical case with six

sequential contours extracted by our automated approach. The

findings are quite similar in nature to the results obtained with

manually out lined data in Section 6.2. The consistency of the

motion field when they are evaluated in the reverse direction is

Fig.7.(a)Axialview

ofthe4th

slicefrom

theapex

ofheartshowingtheleftandrightventricle

intheim

age.(b)–(e)Manually

outlined

endocardialwallofleftventricle

forfouralternatefram

es.

Fig. 8. (a) Manually outlined contours of gradient echo MRI sequence from ED

to ES images (Fig. 7 overlaid). (b) Contours optimized minimizing the internal

energy alone with the bending and stretching coefficients w1Z0.4 and w2Z0.7.

These parameters were used for initial point matching.

A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446 443

important even though cardiac muscle does not follow exactly

the same path in two phases. An accurate reverse tracking of

the contour data in both phases provides a rich testing ground

for algorithm development. Eight sets of sequential contours in

a particular phase were chosen for analysis (gradient echo

images with four axial locations in two cardiac cycles from ED

to ES). Fig. 11 shows a particular set with 12 frames in a

sequence (third axial slice from the apex). The flow vectors are

shown in Fig. 12 including our previous wavelet approach. The

accuracy in tracking the exact reverse motion field is evaluated

by analyzing the trajectories of the virtual markers (Fig. 11).

The mean correlation between the trajectories of the markers in

both the phases for the eight sets of data is presented in Fig. 13.

In another study, we have tested 10 sets patient data

(echocardiograpy images). These images were acquired by

Hewlett Packard (HP) Sonos 1500 machine, 2.5 MHz

transducer frequency with an imaging depth of 16 cm s.

Acquisition was conducted with the help of radiologists at

B.M. Birla Heart Research Center, Calcutta in a transthorasic

position. There was minimum relative motion between the

transducer and the body under study to reduce motion artifacts.

The continuous video images were converted into discrete

frames (12 frames in a cycle) in a Silicon Graphics workstation.

Fig. 9. Motion trajectory with 60 samples. Four virtual marker positions are

shown as the reference to study the robustness of the algorithm as the number of

samples are varied.

Fig. 10. Mean absolute deviation (MAD) at four reference locations in Fig. 9.

The mean deviation is calculated based on the Euclidean distance from the

reference or the virtual marker position.

Table 2

Correlation of virtual markers on automatically delineated boundaries in

Fig. 11

Sl. no. Number of

samples

DWT model BE model Fuzzy model

1 96 0.9271 0.9113 0.9245

2 88 0.9291 0.9276 0.9221

3 80 0.9113 0.9212 0.9278

4 72 0.9473 0.9072 0.9410

5 64 0.9232 0.8378 0.9547

6 56 0.9376 0.8056 0.9451

7 48 0.9112 0.7421 0.9234

8 40 0.9356 0.7567 0.9214

9 32 0.9345 0.7104 0.9451

Fig. 12. Flow vectors in a complete cardiac cycle. 1, 2 Fuzzy model 3, 4 wavelet

model.

A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446444

The boundaries of the images were delineated by an automated

approach. A similar procedure was applied to track the point-

wise motion. The mean correlation between a virtual marker

trajectory in each set and that obtained with different number of

samples in the analysis are shown for both methods in Table 3.

Virtual marker (Fig. 14) is considered as the reference with 50

Fig. 11. Motion trajectory of a cardiac cycle analyzed for 60 samples with

values of a, b andPT remaining same as in case of manually outlined images.

The axial images were obtained by an automated approach in 12 frames

gradient-echo MRI sequence. (a) ED to ES (b) ES to ED with equal number of

temporal phases. The virtual markers are indexed on samples 1, 15, 34 and 50.

samples in the study. The samples were varied between 40 and

72 in discrete steps of 4.

6.4. Genetic parameters

Selection of genetic parameters (population size R,

crossover pC and mutation rate pM) for final flow field

evaluation is an important criterion to make it efficient in

handling constrained multivariable problems. As an example a

small population may encourage premature convergence where

a large population requires more evaluations per generation.

Fig. 13. Correlation coefficients between trajectories of virtual markers in both

phases using fuzzy, wavelet and bending energy approach. Number of samples

were fixed at 56.

Fig. 14. Motion field with 50 samples in case of echocardiography image from

ES to ED.

Table 3

Mean correlation coefficient with respect to virtual marker

Sl. no. Number of

samples

DWT model BE model Fuzzy model

1 72 0.9034 0.9023 0.8919

2 68 0.9205 0.9172 0.9211

3 64 0.9450 0.9023 0.9108

4 60 0.9372 0.9245 0.9412

5 56 0.9245 0.9047 0.9337

6 52 0.9405 0.8629 0.9214

7 48 0.9024 0.7872 0.8724

8 44 0.8927 0.7689 0.9020

9 40 0.9012 0.7732 0.8845

A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446 445

A user defined portion m!R (mZ0.7) is generated in the

current population to present the next generation for final flow

field evaluation. The population fitness function variance s and

maximum iterations were fixed at 0 and 60 as the termination

criteria. A small value of sZ0 represents absolute conver-

gence, which occurs rarely hence the iteration terminate when

maximum iterations is reached. The typical values of pC, pM,

and R are fixed at 0.5, 0.08 and 40.

6.5. Response to optimization parameters

The GA optimizes the flow vectors by incorporating a

conditional mutation scheme. The candidate solutions having

Fig. 15. Computational time vs. PT to evaluate the motion field in case of

Echocardiography images.

at least a pair of corresponding samples where PT (0.2 (Eq.

(10))) are subjected to a conditional mutation. With the above

modification we observed the energy function E in Eq. (9)

converges well within 60 iterations and the optimum value is

consistent each time the iterations start with a different seed.

A similar test has been conducted to study the performance in

terms of computational time by varying PT between 0.1 and

0.3. The computational time vs. PT is shown in Fig. 15

when the program runs in a 1.5 GHz PC loaded with Matlab

in a particular case (echocardiography images Fig. 14). The

slight decrease in computational time is due to because the

candidates are subjected to less conditional mutations and

ultimately the final flow field adheres to matches with less

shape similarity.

7. Conclusion

Even though fuzzy shape based motion evaluation is

computationally expensive in comparison to the wavelet and

BE model, the results presented for the actual marker motion

on sets of phantom images are found to be attractive. In

addition to that a consistent correlation between the actual and

predicted trajectory of markers is observed using this

algorithm. There is a decrease in correlation with wavelet

and BE model when the number of samples in the analysis is

less. This is the prime motivation behind the work.

This is our initial attempt to provide quantitative evaluation

of motion field in 2D using fuzzy shape descriptors. The shape

correspondence is optimized with genetic algorithm, taking the

non-rigid and non-uniform motion of endocardial muscle in to

consideration. The initial level shape correspondence is based

on a smooth boundary, optimized with internal energy alone.

The basic idea is to establish a course shape correspondence,

which is robust to the interobserver and parameter specific

anomalies in case of manually outlined and computer

generated contours as well. Correspondence in the finer level

depends on the true perceived boundary unlike our previous

work [24] where each level of correspondence was associated

with level specific curvature information. The point matching

based on constrained closeness criteria for final flow field

optimization makes it more attractive over our previous work,

because it reliably adhere to shape matched templates to

produce a smooth motion field during the final optimization.

The robustness of the algorithm has been tested varying

number of samples and the consistency of the flow vector

trajectories in both the direction. Even though the cardiac

motion does not retain the same path in the systolic and

diastolic phases it provides a good testing ground for the

algorithm performance analysis. The smoothness and closeness

criterions play a decisive role in establishing the final shape

correspondence. However, optimal setting of parameters a and

b, were decided based on an experimental verification of the

results presented in the work [31]. The scope of 3D

visualization with necessary modifications in the algorithm

may be a better aid for diagnostic radiology.

A. Mishra et al. / Image and Vision Computing 24 (2006) 436–446446

Acknowledgements

The authors would like to thank to B.M. Birla Heart

Research Center and Eko Imaging Institute, Calcutta, India for

their assistance in acquiring the data. Authors also acknowl-

edge Dr N. Chakraborty and Dr S. Kundu for their valuable

suggestions in manually outlining the contours in the data sets.

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