blood vessel segmentation in complex-valued magnetic resonance images with snake active contour...

8/4/2019 Blood Vessel Segmentation in Complex-Valued Magnetic Resonance Images With Snake Active Contour Model

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International Journal of E-Health and Medical Communications, 1(1), 41-52, January-March 2010 41

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Keywords: Blood Vessel, Phase-Contrast Magnetic Resonance Imaging, Segmentation, Simulated Annealing,Snake Active Contour Model

introduCtion

Magnetic Resonance Imaging (MRI) has playedincreasing significant roles in today medical

practice. Among the most prominent clinicalstrength of MRI are its non-hazardous natureand its ability to provide high contrast betweenvarious types of tissue. Nowadays the applica-

tion of MRI has expanded beyond conventionalstructural mapping into chemically-specified

B V s gm C mp x-V M g c

r c im g w h sac v C M

Astri Handayani, Institut Teknologi Bandung, Indonesia

Andriyan B. Suksmono, Institut Teknologi Bandung, Indonesia

Tati L. R. Mengko, Institut Teknologi Bandung, Indonesia

Akira Hirose, University of Tokyo, Japan

aBstraCt

Accurate blood vessel segmentation plays a crucial role in non-invasive blood ow velocity measurement based on complex-valued magnetic resonance images. We propose a speci c snake active contour model-

based blood vessel segmentation framework for complex-valued magnetic resonance images. The proposed framework combines both magnitude and phase information from a complex-valued image representationto obtain an optimum segmentation result. Magnitude information of the complex-valued image provides a

structural localization of the target object, while phase information identi es the existence of owing matterswithin the object. Snake active contour model, which models the segmentation procedure as a force-balancing

physical system, is being adopted as a framework for this work due to its interactive, dynamic, and customizablecharacteristics. Two snake-based segmentation models are developed to produce a more accurate segmentationresult, namely the Model-constrained Gradient Vector Flow-snake (MC GVF-snake) and Stochastic-snake.MC GVF-snake elaborates a prior knowledge on common physical structure of the target object to restrict and guide the segmentation mechanism, while Stochastic-snake implements the simulated annealing stochastic

procedure to produce improved segmentation accuracy. The developed segmentation framework has beenevaluated on actual complex-valued MRI images, both in noise-free and noisy simulated conditions. Evaluationresults indicate that both of the developed algorithms give an improved segmentation performance as well asincreased robustness, in comparison to the conventional snake algorithm.

DOI: 10.4018/jehmc.2010010104


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42 International Journal of E-Health and Medical Communications, 1(1), 41-52, January-March 2010


and flow-related imaging. Advanced applica-tions of MRI require supports of sophisticatedsignal processing procedure to infer the desiredinformation from typical Magnetic Resonanceimages. This chapter presents the developmentof blood vessel segmentation algorithms dedi-cated to support the quantitative flow imaging

based on Phase Contrast Magnetic ResonanceAngiography (PC-MRA). PC-MRA requiresthe consideration of both the magnitude and

phase information derived from the MRInatural complex-valued signals, rather thanrelies solely on the magnitude information ascommonly practiced in the MRI application for

structural mapping.MRI is physically based on the Nuclear Magnetic Resonance (NMR) phenomenondiscovered at 1946 by Felix Bloch and EdwardPurcell. This phenomenon notes the resonanceof magnetic systems when triggered with radiowaves with frequencies corresponding to itsnatural magnetic frequencies; that is the gyro-scopic precession frequency of the magneticmoment of the nuclei under the influence of an external static magnetic field. By observingthe externally measured NMR signals, MRI

produces images of the internal physical andchemical characteristics of an object. This ismade possible by the spatial encoding principlesdeveloped by Paul Lauterbur on 1972, whichin turn served as a foundation for the currentMRI systems.

The potential of MRI for flow-relatedimaging is due to its coherent imaging nature.As a coherent imaging system, MRI produces

complex-valued signal where magnitude and phase images can be derived. The general formof complex-valued image can be written as:

I Ix y x y j x y , , exp ,( )= ( ) ( )( )f (1)

where j = - 1 is the imaginary number. Themagnitude part of equation (1) I x y ,( ) corre-sponds to the magnitude image, while the phase

part f x y ,( ) corresponds to the phase image.

The magnitude and phase images of MRIhave its own physical and clinical interpretation.MRI magnitude image represents the structuralinformation of the object, while the phase imageis related to the internal physical and chemicalcharacteristics. Among the simultaneous useof the MRI magnitude and phase images inclinical applications is the Magnetic ResonanceAngiography (MRA) procedure, first proposed

by Charles Dumoulin in 1987. This procedureallows imaging and measurement of bulk

blood flow in large vessels without the needfor contrast agents.

Quantitative flow measurement with MRI

is based on the observation of phase shifts ac-quired by magnetic moments (spins) movingalong a magnetic field gradient in comparisonto stationary spins. For linear field gradients,the amount of this phase shift is proportionalto the velocity of the moving spin. Phase shiftsin MRI systems however not only occur frommovement of magnetic moments. The originallystationary tissue may also experience phaseshift come from chemical shift or externalmagnetic field inhomogeneity. To measure thenet phase shift between moving and stationaryspins, NMR signals observation is repeated in a

bipolar gradient mode. Measurement in bipolar gradient mode will eliminate the phase shiftaccounted by another cause than movement of magnetic moments.

As a consequence of the MRI phase im-age formation, phase shift should lies betweenthe ranges of . Therefore, the calculation of velocity from phase shift information requires

a procedure to tune the maximum phase shiftmeasured by the system to the peak velocity ex - pected in the vessel (encoding velocity). Encod-ing velocity determination is very crucial in thePC-MRA procedure. Wrapping of the velocityinformation will occur if the encoding velocity istoo low. In the other hand, measurement will be

prone to error and fail to detect slow movementif encoding velocity is too high. Peak velocityvaries under different physiological conditions,and its exact determination prior to the measure -ment itself is not possible. However, its valuecan be approximated with various preliminary


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techniques, either based on MRI or another measurement modality such as real time Dop-

pler Ultrasound (Lotz et al., 2002).The next step required in the PC-MRA pro -

cedure is the calculation of blood flow volumewithin the vessel in every time fraction. Thiscalculation will require an accurate estimationof vessel boundaries, so that the cross-sectionalarea of the vessel of interest can be correctly de-termined and the flowing volume fraction can beanalyzed. Segmentation technique then becomeof high importance in this matter. Blood vessel

boundaries in magnitude images correspondto the contour which is formed by pixels with

large intensity gradient, as intensity gradientin magnitude images indicate tissue changes.On the other hand, blood vessel boundariesshould also be the contour that encloses as muchflow as possible, as appeared in the intensityof phase images. Therefore, the determinationof blood vessel boundaries is required to take

both the magnitude and phase information intoaccount.

The segmentation algorithm presented inthis chapter is developed based on the GVF-Snake active contour model. GVF-Snake ActiveContour Model is a segmentation algorithm thathas found its vast application in medical imageanalysis due to its ability to handle delicatecomplicated structures, in addition to its high-adaptability and ease of implementation nature.Since its first introduction by Chenyang Xuand Jerry Prince in 1998, GVF-snake has beenone of the most cited works in medical imagesegmentation. Its utilization for segmenting

sophisticated anatomical structures in two andthree dimension imagery has been very popular.However, little attention has been devoted toits extension for implementation in originallycomplex-valued images.

A snake is a curve x s x s y s s , , ,0 1 in a two dimensional image field. The snakeis deformable and capable of moving throughan image, seeking for minimum of energyfunctional defined by

E E s ds

E s E s ds

snake

ext

= ( )( )= ( )( )+ ( )( )( )

x

x x

0

1

0

1

int (2)

where E int is the internal energy of the snakedue to bending, and E ext is an external energy

produced by the image. Since the internal en-ergy correspond to snakes tension and rigiditythat are related to the first derivative x( s) andsecond derivative x( s) of the snake, we maywrite (2) as

E s s E s ds ext = ( ) + ( )

+ ( )( ) 12

2 2

0

1a b x x x' ''

(3)

where and are the weight of snakes tensionand rigidity, respectively. The external energydepends on the image field I ( x, y), which isconsidered as a continuous function, and typi-cally is one of the following types:

E x y I x y ext 1

2, ,( )= - ( ) (4.a)

E x y G x y I x y ext 2

2

, , ,( )= - ( )* ( )( )s (4.b)

E x y I x y ext 3 , ,( )= ( ) (4.c)

E x y G x y I x y ext 2 , , ,( )= ( )* ( )s (4.d)

where * denotes convolution, is the gradi-

ent operator, and G (x,y) is a two-dimensionalGaussian function whose standard deviationis . Convolution by the Gaussian functionis equivalent to image blurring. The last twoformulations, i.e. (4.c) and (4.d), are suitablefor processing an image of line drawing or a

black and white image.The snake energy functional in (3) can

be minimized by solving the Euler-Lagrangeequation of:

a b s x s x E

x ss ssss ext ( ) - ( ) - = 0 (5.a)


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a b s y s y E

y ss ssss

ext ( ) - ( ) -

= 0 (5.b)

where x ss and x ssss are the second and fourthderivative of the snake along the x-axis and y ss and y ssss are the second and fourth derivative of the snake along the y-axis.

In some cases, it is necessary to initiateobjects boundary search within a large area.Usually, one may increase of the Gaussianfunction so that the boundary can be sensedfrom a large distance. However, this procedure

is not always effective (Xu & Prince, 1997,1998). A new formulation of energy minimi-zation called gradient vector flow (GVF) fieldis defined as:

e m + + +( )+ - u u v v f f dxdy x y x y 2 2 2 2 2 2v (6)

which, by using calculus of variations, can besolved by the following Euler equations:

m - -( ) +( )=2 2 2 0u u f f f x x y (7.a)

m - -( ) +( )=2 2 2 0u u f f f y x y (7.b)

In equation (7) 2 is the Laplacian operator.Those equations can be solved further by treatingu and v as functions of time and solving:

( )

( ) ( ) ( )( ) ( ) ( )( )2 22, ,

, , , , , , ,

t

x x y

u x y t

u x y t u x y t f x y f x y f x y

=

+

(8.a)

( )

( ) ( ) ( )( ) ( ) ( )( )2 22, ,

, , , , , , ,

t

y x y

v x y t

v x y t v x y t f x y f x y f x y

=

+

(8.b)

These equations are called the general-ized diffusion equation where the steady-statesolution (as t ) converges to the Euler equation.

BaCkGround

Complex-valued image processing can beformulated in the original domain of complex-

valued signal, i.e. real and imaginary pair of values, or in the domain of real-valued mag-nitude and modulo-2 phase intensity signals,i.e. magnitude and phase intensity domain.The Complex-valued Markov Random Field(CMRF) phase unwrapping algorithm devel-oped in Suksmono and Hirose (2002) andthe snake formulation for object detection in

phase images given in Suksmono et al. (2006)treat the complex-valued information as a

compound magnitude and modulo-2 phaseintensity images.However, in most complex-valued imag -

ing modalities, the observable information is believed to reside in the magnitude and phaseintensity representation instead of in the rawreal and imaginary signal pair. Therefore,complex-valued image ssegmentation is usually

performed on the magnitude and phase intensityimages. This is the common pattern used inmost of MRI chemical-shift and flow-relatedimaging applications, where MRI magnitudeimage is related to object structure and MRI

phase image is related to the internal chemicaland physical characteristics of the object. Thesegmentation procedure usually treats thesemagnitude and phase intensity images as twoseparated source of information. The aim of elaborating the two sources of information isto obtain accurate structural localization of the chemical or physical quantity of interest.

Segmentation is then performed based on con-straints that are derived from both magnitudeand phase image. The previous works dedicatedin PC-MRA are conducted in similar approach.Among the most prominent works in this fieldare those of Solemn et al. (2004) and Chunget al. (2002).

Solemn et al. (2004) uses the constraintderived from the magnitude intensity gradientand the velocity-based term inferred from the

phase intensity image to perform segmentation.The method shows higher numerical stabilityand an increasing tendency of segmentation


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performance in areas of high velocity noise, dueto the inclusion of magnitude intensity term. Onthe other hand, in cases where the magnitude rep-resentation of blood vessel is practically equal tothe surrounding pixels, the velocity based termstill makes the segmentation become possible.However, magnitude intensity gradient valuemay not yield an accurate structural constraintdue to the existence of stronger non-vessel object

boundaries. This is the main problem faced onthe simultaneous use of magnitude and phaseintensity information.

Chung et al. (2002) gives the definition of speed image, i.e. the compound image of mag-

nitude and phase information. It is formed bymultiplying the magnitude image with the differ-ence image of two phase intensity images takenin a subsequent pair of bipolar gradient. Seg-mentation is then performed in this compoundimage domain. High intensity stationary objectwill become less prominent in the speed image,while low intensity moving object will appear clearer. To obtain noise-free phase information,

preprocessing by phase denoising filter is sug-gested prior to the formation of speed image.Phase unwrapping step may also be considerednecessary in case the encoding velocity used inthe measurement procedure were too low (Lotzet al., 2002). The speed image provides a better input formulation for image segmentation withconstraints derived from both magnitude and

phase intensity information.When dealing with snake active contour

model, the problems which are most likely to be found are sensitivity to contour initializa-

tion, spurious edges or existence of stronger non-targeted boundaries, and local minima of energy functional. In cases such as PC-MRA

blood vessel segmentation where the targetedobject tends to reside in a relatively constantcoordinates for every image in the data set,the problem can be solved by determining anoptimum contour initialization through a seriesof experiment. The matter of spurious edges or stronger misleading boundaries in the other handis trickier. The existence of such edges is mostlikely to prevent the snake from converging tothe targeted contour, as these edges give lower

value of snake energy. Therefore, the targetedcontour configuration becomes a local, insteadof global minimum of snake energy functionalas expected.

Even when the targeted contour configura-tion does correspond to the global minimum of the energy functional, there is still a chance thatsnake does not converge to this configurationdue to the downhill nature of the gradient-descent snake energy minimization scheme.This scheme searches the minimum value of afunction iteratively according to the directionwhich gives the lowest descent of the functiongradient. It guarantees the next solution to be

more of a minimum than the previous one, butfails to find the global minimum of the functiononce it has reached and stuck in a local minimum.This characteristic also accounts for the snakesensitivity to initialization, since the initial pointwhere the gradient descent numerical methodstarts to evaluate the function to be minimizedmay greatly affect whether the scheme wouldconverge into a desired minimum value.

ProPosed MetHodThe method proposed in this chapter addressestwo main problems found in the implementationof conventional GVF Snake in the segmentationof blood vessel from a complex-valued image.The first problem is the local minimum energyvalue, and the second problem is the downhillcharacteristics of the gradient descent numericalmethod adopted in the GVF Snake iteration.

In the case where the targeted object boundaries were formed by the pixels with thehighest intensity gradient in the whole image,GVF Snake has given satisfactory performance.However, in blood vessel segmentation fromcomplex-valued MRI image, the vessel itself is not the most prominent object existed in theimage. There are objects with higher inten -sity gradient in its boundaries which naturallycorrespond to stronger minima of the energyfunction. Since GVF-snake tends to converge

to a stronger minimum, segmentation in suchcondition is likely to fail. We propose the ad-


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dition of geometric shape constraint into theGVF-snake energy function, in order to makethe targeted contour configuration becomea stronger minimum. This additional shapeconstraint forces the contour to converge notonly to the location of strong edges but also to ageometrical configuration that suits the charac-teristics of the targeted segmentation object. Wenamed the approach as the Model-constrainedGVF Snake (MC-GVF Snake).

Conventional snake algorithm utilizesgradient-descent numerical method to find theminimum value of its energy function. The

problem with this numerical method is that it

stops its searching once a local minimum isreached. In many cases, however, the targetedcontour configuration may not correspond to thefirst minimum found in the search. To overcomethis problem, we attempted to use stochasticmethod in implementing the energy minimiza-tion procedure. Stochastic-snake uses randomsampling with simulated annealing to estimatethe minimum energy value related with targetedobject boundaries. The stochastic nature givesthis scheme a hill-climbing capability whichmakes it possible for the procedure to findstronger and more suitable minima once it hasreached a local minimum.

M -C GVf-s

In many natural images, target object is correlat -ed to a local, instead of global, energy minimum.In order to produce an accurate segmentationresult, a snake energy function modificationis required to make the target object boundary

become a global minimum. One approach to the problem is to include a model constraint intothe energy function formulation. The constraintshould characterize a distinct feature of thetarget object. Furthermore, it is necessary for the constraint not to be susceptible to noise. Inthe case of blood vessel segmentation, due tothe shape-similarity nature of the blood vesseltransversal projections along the body medialaxis, we propose the incorporation of a shape-

preserving model constraint into the snakeenergy minimization scheme. The model shape

constraint formulation is given as an additionalconstraint which urges the snake to preserve acircular shape during its deformation.

For every snake contour which varies totime v(s,t) , we define a closest circle templatevc(s,t) which has its centre in the centroid of thecontour and its radius as the square root of thearea enclosed by the contour after divided by .Figure 1 gives an illustration of the relationship

between snake contour and its correspondingcircle template.

The model constraint external force F c(v(s,t)) will direct the snake control points intotheir closest circle template points, as given by

the following mathematical formulation:

F v s t w v s t dist v s t v s t C c C , , arg min , , ,( )( )= ( )( ) ( ) ( )( )( )

(9)

In this formulation, w c(v(s,t)) denotes theweighting parameter for the additional externalforce F c, while [arg min (dist(v(s,t), v c(s,t))) ]denotes the closest distance between eachsnake control points and the correspondingcircle template.

The GVF-snake energy minimizationscheme, then become as follows:

a b ks x s x s w u w F ss ssss GVF C Cx ( ) - ( ) + ( ) + ( ) =( ) 0 (10.a)

a b ks y s y s w v w F ss ssss GVF C Cy ( ) - ( ) + ( ) + ( ) =( ) 0 (10.b)

where wGVF and wC are the GVF and model-constraint force weighting parameters, respec-tively given by:

w v s t e GVF dist v s t v

C s t

,arg min ( , ), ( , )

( )( )=- ( )( )

2

(11)

w v s t w v s t C GVF , ,( )( )= - ( )( )1 (12)


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and u and vare the GVF force vector component

in x and y direction, while F Cx and F Cy are themodel constraint force vector component alsoin x and y direction.

The weighting scheme of the external forces provides a dynamic ratio between GVF forceand model-constraint force. Model constraintdominates the external force term when snakecontour is far from a circular shape, while GVFforce dominates the external force term whensnake has already resembled a circle.

s ch c s

Stochastic-snake utilizes simulated annealingstochastic procedure to estimate minimumenergy value related to targeted object con -tour. This scheme is based on Metropolis-Hastings acceptance criteria in Markov ChainMonte-Carlo random sampling scheme. Inthis case, snake energy from various contour configurations is considered as an unknown

distribution defined in solution space, fromwhich Markovian random samples with the

distribution of P(x) are to be taken. Utilizing

the Metropolis-Hastings criterion, the distribu-tion of these random samples is expected toapproach the actual snake energy distribution asmore samples are taken, such that the minimumvalue of the actual snake energy distributioncould be estimated from the minimum value of the random samples distribution. As mentionedearlier, the stochastic nature of this proceduregives a hill-climbing ability which provideschances to find stronger minima value as thesearch proceeds. However, it is the stochasticnature also which cause the procedure not to

be able to guarantee the convergence to globalminimum value.

In a simulated annealing procedure, theMarkov chain of the actual distribution of inter-est is modeled as a Gibbs distribution of:

P x Z

e T T

E x T ( )( )

=-1

(13)

where the partition function Z T is given by:

Figure 1. Illustration of circle template for shape preservation


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Z e T

E x T

x

=-

"

( )

W(14)

In both equation, T is the annealing temperature and E is the energy functional to beminimized, or in this case the snake energy func-tional. At high temperature, the probability of any samples to be accepted in the Markov chainis equally likely. As the annealing temperaturedecreases, the distribution of Markovian randomsamples will converge to a uniform distributionover the global minimum of E . The probabilityof making a move from one random sample x(t) to another candidate sample x* is given by a

sampling distribution Q(x) . In this research, thesampling distribution is a Gaussian probabilitydensity function. The sampled snake contour configuration in the other hand is represented

by two parameters, i.e. the coordinate of thereference point from which the radial distanceof snake nodes in various angular orientations ismeasured and the set of radial distance of eachavailable snake nodes with respect to the refer-ence point. In this research, three sampling sce-narios have been developed. The first scenariosamples the coordinate of the reference point,assuming a constant uniform radial distance.The second scenario samples the set of radialdistance, assuming a constant reference point.The last scenario samples both the reference

point and the set of radial distance.The criterion of accepting or rejecting

a random sample in the simulated annealing procedure is based on the Metropolis-Hastingscriterion:

a T t

T t

t

P x

P x

Q x x

Q x x =

min ,

* ( *)* ( )

( ; *)( *; )( )

/ ( )

( )1

1

(15)

For the definition of P(x) given in (13),the Metropolis-Hastings criterion can be re-written as:

a T t

T t

t E x x

Q x x

Q x x = - ( ){ }( )

min , exp ( *,( ; *)( *; )

( )/ ( )

( )11

D

(16)

where E is the difference of energy functionvalue between candidate sample x* and the lastaccepted random sample x(t). For a complete,more detailed explanation regarding the simu -lated annealing procedure, readers are suggestedto refer to Mc.Kay (2003).

Initial temperature, cooling schedule, andfinal temperature determination are of impor-tant role in a simulated annealing scheme. Thedetermination of the three parameters is givenin the next paragraphs:

Initial temperature is expected to provide avast hill-climbing opportunity. In this research,initial temperature is designed to enable themaximum acceptance probability of p for random samples with considerably maximumhill-climbing energy value difference. Metrop-olis-Hastings criterion for the acceptance for the correspondent hill-climbing samples can

be written as follows:

exp / ( )-{ }DE T pinit (17)

T E

pinit ( )

ln> -

( )D (18)

Cooling schedule is designed such that theacceptance probability of hill-climbing samples

decreases with the factor of 1/r as the simulatedannealing temperature decreases. The criterioncan be expressed as follows:

T E

E T r T t HC

t

HC t t

t ( )( )

( ) ( )

( )

ln+ =

+ + ( )( )

1 D

D(19)

where(t)

HC is the average energy value of hill-climbing samples accepted in temperature

T(t).


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Final temperature is defined as a tempera-ture that gives temperature decrement gradientof less than a particular threshold value of thresh . Simulated annealing final temperaturetherefore can be given as follows:

T T

T thresh

fin fin

fin

( ) ( )

( )

-

--

blood vessel segmentation in complex-valued magnetic resonance images with snake active contour...

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