910 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 24 ... - VIVA

910 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 24, NO. 7, JULY 2005

Intravital Leukocyte Detection Using the GradientInverse Coefficient of Variation

Gang Dong, Nilanjan Ray, Member, IEEE, and Scott T. Acton*, Senior Member, IEEE

Abstract—The problem of identifying and counting rollingleukocytes within intravital microscopy is of both theoretical andpractical interest. Currently, methods exist for tracking rollingleukocytes in vivo, but these methods rely on manual detectionof the cells. In this paper we propose a technique for accuratelydetecting rolling leukocytes based on Bayesian classification. Theclassification depends on a feature score, the gradient inverse coef-ficient of variation (GICOV), which serves to discriminate rollingleukocytes from a cluttered environment. The leukocyte detectionprocess consists of three sequential steps: the first step utilizes anellipse matching algorithm to coarsely identify the leukocytes byfinding the ellipses with a locally maximal GICOV. In the secondstep, starting from each of the ellipses found in the first step, aB-spline snake is evolved to refine the leukocytes boundaries bymaximizing the associated GICOV score. The third and final stepretains only the extracted contours that have a GICOV scoreabove the analytically determined threshold. Experimental resultsusing 327 rolling leukocytes were compared to those of human ex-perts and currently used methods. The proposed GICOV methodachieves 78.6% leukocyte detection accuracy with 13.1% falsealarm rate.

Index Terms—Active contours, boundary extraction, classifica-tion, leukocyte detection, microscopy.

I. INTRODUCTION

I DENTIFICATION and counting of rolling leukocytes areclinically important tasks conducted in research laboratories

based on animal experiments to study the mechanisms of in-flammation. Adhesion of rolling leukocytes to the vascular en-dothelium is a critical event in inflammation and plays a keyrole in human immune system [1]. The lower velocity of rollingleukocyte is caused by the continuous formation and breakageof bonds between selectin adhesion molecules on the leukocytesurface and their ligands on the endothelium [2]. Several crucialparameters of rolling leukocytes, such as rolling leukocyte flux(number of rolling cells passing per unit time), rolling leuko-cytes per unit length, and the rolling leukocyte volume fractionare usually used to gain information in the inflammatory processand in designing/testing anti-inflammatory drugs [3].

Manuscript received January 21, 2005; revised March 1, 2005. This workwas supported in part by the National Institutes of Health (NIH) under GrantHL68510. The Associate Editor responsible for coordinating the review of thispaper and recommending its publication was E. Meijering. Asterisk indicatescorresponding author.

G. Dong and N. Ray are with the Department of Electrical and ComputerEngineering, University of Virginia, Charlottesville, VA 22904 USA (e-mail:[email protected]; [email protected]).

*S. T. Acton is with the Department of Electrical and Computer Engineering/Biomedical Engineering, 351 McCormick Road, University of Virginia, Char-lottesville, VA 22904 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TMI.2005.846856

As with identification of most cells in microscopic cy-tology, the rolling leukocyte detection process is traditionallyperformed manually by experienced laboratory technicians.However, since each video sequence contains a large numberof rolling leukocytes (100s–1000s), this approach is extremelytedious and time consuming. Additionally, because any humanobserver suffers from inevitable lack of attention when pre-sented with large amount of data, error will inevitably arise,thus limiting the usefulness of the data. As a result, we aremotivated to automate the identification process.

There exist several methods for automatic object detection[4]–[11]. The Hough transform (HT) method [4] is a well-established approach, where the detected edge pixels vote for ashape according to a parametric representation in the parameterspace. The main problem to be faced using the HT is the properidentification of peaks in the parameter space: false peaks maybe caused by spurious line and arc segments. Also, it has beenshown that even the modified HT methods are computationallyexpensive [5]. The edge-radius-symmetry (ERS) transformmethod, developed by Chabat et al. is used for the identificationof bronchi on CT images of the lungs analysis [6]. Basedon local edge strength, radial distribution, and symmetry, theimage pixels are ranked to provide a sorted list of the mostlikely positions of all dominant ellipses. Detection methods,based on mathematical morphology, have been introduced byBatman et al. [7], and by Mukhopadhyay et al. [8]. In the relatedwork of Egmont-Petersen et al. leukocytes in contact with thevessel wall are detected using a neural network that is trainedwith synthetic images generated by a stochastic model [9]. Adifferent approach based on image level sets has been suggestedby Mukherjee et al. where possible cells are captured througha search of the image level lines—boundaries of connectedcomponents within level sets [10]. The temporal informationobtained by optical flow is used in low contrast object detectionby Markandey et al. [11]. The basic assumption of the opticalflow technique is that the motion shift per frame should be lessthan the width of the transition of image edge.

In this paper, we present an automated object detectionalgorithm that aims to identify rolling leukocytes within themicrovasculature using video microscopy in vivo. Typically, adetection algorithm generates hypotheses for a given model bytesting the correspondence between a designed model and imagefeatures, resulting in some kind of a goodness score. Given thegoodness score, a correct correspondence for a leukocyte alwaysresults in a score greater than a known threshold score , and anincorrect correspondence always scores less than . Then thedetection algorithm could test correspondences to see whetherthere are correspondences resulting in good scores. However,

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DONG et al.: INTRAVITAL LEUKOCYTE DETECTION 911

in intravital microscopy imagery, due to the presence of mea-surement noise, occlusion, and clutter (leukocytes moving atthe rate of flow, platelets, erythrocytes, muscle striation), acorrect correspondence might lead to a score less than , whilean incorrect correspondence might yield a score of or greater.The difficulties in detection are further aggravated by the factthat there is no definitive procedure exactly prescribing whatscores should be generated, or what features should be used ineach specific case. This suggests that a formulation of a highlydiscriminative score must be employed, and it should reflect theconfidence in choosing one hypothesis over others. In this paper,we present a novel formulation of a score function, the gradientinverse coefficient of variation (GICOV), which is the ratio of themean and the standard deviation of directional image derivativesover an entire closed contour fitted to a leukocyte boundary.The directional derivatives are taken in the outward normaldirections on contour points.

The inspiration for considering an edge based scoring methodrather than a region based measure comes from our previouswork [12], where we observe that the edges emerging from theintravital microscopy imagery are more consistent visual cuesfor leukocytes than regional measures such as average intensity.Fast moving platelets, blood flow, static muscle striations, as wellas frequent imaging intensity and illumination change causedby the change of focus due to the breathing movement of theliving subject add severe clutter to the intensity profile insideand outside the cells. In fact, a leukocyte appearing bright oftenturns dark in appearance and vice versa during a video sequence,as it moves nearer or moves further from the focal plane in thecourse of its flow through the microvessel. We also observe thatthe intensity profile inside a leukocyte appearing dark closelyresembles that of the background. In these cases, an annularstructure, at times broken, separates inside and outside [12].These observations have led us to propose an edge based scoringstatistic (the GICOV), rather than one based on regional imageintensity statistics. In contrast to our previous work where weconsidered local edge gradients [12], [13] for cell tracking, in thispaper we consider an edge based score that is computed over anentire contour for the purpose of cell detection. The motivationbehind the GICOV is our observational hypothesis that edgestrength of a leukocyte remains somewhat constant around itsboundary. Thus on one hand, we require the fitted contour tohave high directional gradients in the outward normal direction,and on the other hand, we require low variation among thesedirectional gradients. Consequently, we propose the GICOV,which is the ratio of the mean and the standard deviation of thedirectional derivatives (in the outward normal direction).

Thresholding on the scoring statistic for object detection in-herently involves a tradeoff between detection of true positivesand false alarms. Choosing a detection threshold by an ad hoctrial and error procedure will often lead to poor performance andwill lack robustness. In this paper, we analytically determine theGICOV detection threshold used by exploring the underlyingscore density functions.

A three-stage detection process is presented in this paper. Thefirst stage is designed to identify leukocytes coarsely by ellipsematching, under the assumption that rolling leukocytes can beroughly approximated by ellipses [9]. Note that this assumption

does not impose a serious limitation upon the problem becauseit is only an initial estimate. The region of searching for thecells could be determined through the delineation of the vesselboundary [14]. In the second stage we employ active contour orsnake [15] to refine the boundary estimate of the rolling leuko-cyte. Enhanced performance is achieved by way of a B-splineparametric representation of the snake and using the GICOVas the external force to move the snake. In the final stage, athreshold for the GICOV score, obtained via a Bayes classifi-cation approach, is applied for the two-class (here, leukocyteand nonleukocyte) partition problem.

To design the Bayesian classifier we assume a step edgemodel for a leukocyte and homogeneous surroundings in anideal noise-free image. We then consider an additive whitenoise process for image formation, and derive probabilitydensity functions (pdfs) of the GICOV both for leukocyte andnonleukocyte classes associated with a B-spline contour inclosed form. The GICOV threshold selection is then performedin the Bayesian framework and represented as a function ofparameters of the leukocyte edge model and underlying whitenoise process from image formation. We estimate these param-eters from a training set of intravital video frames and utilizethem to compute the GICOV threshold for classification.

Experimental results using 30 intravital microscopy images(containing a total of 327 leukocytes) from temporal sequencesshow the feasibility of the method. In particular, our methodachieves 78.6% leukocyte detection accuracy with 13.1% falsealarm rate, and compares favorably against the currently usedobject detection algorithms, such as the Hough transformmethod, the ERS transform method and level set method.

The remainder of this paper is organized as follows. A briefoverview of the snake model and the B-spline curve represen-tation is given in Section II-A. In Section II-B, we describe theBayes classifier. In Section III, we first define the GICOV scoreused to detect leukocytes. We then discuss the ellipse matchingand B-spline snake procedures used. The pdfs associated withthe GICOV score are developed in Section IV, and threshold se-lection is discussed. Section V demonstrates the performance ofthe detector. Computational expense is addressed in Section VI,and finally in Section VII, we summarize our results and discussfuture research directions.

II. BACKGROUND

In this section we provide necessary background for two es-sential components of the proposed leukocyte detection process:the B-spline snake and basics of the Bayesian classifier.

A. B-Spline Representation

Active contour models (also called ‘snakes’ [15], [16]), whichoptimize/minimize an energy functional, have been recognizedby the computer vision community as a successful strategy forextracting contours and detecting objects [17]. A snake algo-rithm is typically characterized by three parts: 1) a model ofinternal energy that serves to impose smoothness of the con-tour; 2) a model of external energy that ties the contour to theunderlying image by attracting the contour toward applicationdependent image features; 3) an iterative energy optimization


procedure that attempts to simultaneously minimize the internaland external energies. The minimal solution on a given imagewould result in a contour that has desirable geometric proper-ties and is positioned at the areas of interest of the image. Afterdiscretization of the arc length into segments of length , thesnake is described by a number of points, called snaxels.

Instead of using a sequence of evenly spaced discrete pointsto represent the snake, a continuous parametric B-spline repre-sentation of the snake [18] holds advantages over the traditionalparametric model: 1) The B-spline snake is a regular curve,hence the regularity is implicitly built into the model, and the in-ternal energy terms of traditional model are no longer required.2) The B-spline is a suitable match for the leukocyte contour dueto the regular nature of leukocyte boundaries. 3) The B-splinefirst and higher order derivative vectors can be easily and accu-rately determined, which is important in computing the GICOVscore. 4) The combination of the spline representation and ex-ternal energy enables us to suppress jagged corners and obtaina regular contour without losing edge features. 5) Individualmovement of control points will only affect the contour locally.

The 2-D B-spline curve is defined overa parameter by the following equation [19]

(1)

The curve is drawn for various values of , varying from to. Typically, , and . A point on the curve,

or sample point, at the parametric value is denoted by .There are control points, denoted by . These control pointsare points in object space, using which the shape of the B-splinecurve can be controlled. In our geometric model, the positionsof these control points are changed to achieve the required shapeof the curve. The actual curve may not pass through the controlpoints, though B-spline curves always lie within the convex hullof the polygon delineated by the control points. The basis func-tions, or the blending functions, are denoted by . Thesebasis functions will decide the extent to which a particular con-trol point controls the curve at a particular parameter value ,and have a polynomial form. The parameter is the order of thecurve. (For the cubic curve, we use .)

The order of the basis function is determined by the followingcriteria. If the basis function is inadequate in the sense of ap-proximating the boundaries being sought, then the boundariesof objects that are not spanned by the basis function cannot beextracted. On the other hand, if the basis function is constructedby some functions with certain undesirable behavior, such assevere oscillations, which are typical behavior of a high orderpolynomial, then the snake will be sensitive to noise. Within thescope of this detection technique, we utilize a cubic B-spline,i.e., curves that are continuous and have continuous slopesand curvature. The first basis function is given by [19]

(2)

and the other basis functions are simply translated copies gov-erned by [19]

(3)

These bases are bi-infinite, extending to ; however,finite bases are needed for practical applications. The proper-ties of basis functions and B-splines are discussed in detail in[19]. For further reading on the B-splines, [20] and [21] are sug-gested.

B. The Bayesian Classifier

For the problem of leukocyte detection, we need to decide,given a discriminative score from object of interest, if that scoreis high enough to warrant membership in the leukocyte class

, or low enough for the nonleukocyte class . We applythe Bayes classifier for leukocyte detection and cast our problemin terms of this framework, since the Bayes classifier yields theminimum error when the underlying distributions are known.The error, called the Bayes error, is the optimal measure forfeature effectiveness, when classification is of concern, since itis a measure of class separability [22].

Let be the GICOV score obtained from a given con-tour and underlying image. Let the a posteriori probabilities ofthe leukocyte class and the nonleukocyte class given

be and , respectively. The object delineatedby the contour is classified to the leukocyte class or nonleuko-cyte class according the Bayes decision rule for minimum error[22]

ifotherwise.

(4)

The density function of class , , and density func-tion of class , , can be computed from the conditionaldensity functions using Bayes’ theorem [22]

(5)

where and are the a priori probabilities of leuko-cyte class and nonleukocyte class , respectively, andis the mixture density function. The Bayes decision thresholdis set at the point where . The Bayes decisionrule for leukocyte detection is then defined as follows

ifotherwise.

(6)

III. DETECTION METHOD

In order to achieve robust performance in a cluttered envi-ronment, a detection method based on the GICOV is presented.Ellipse matching is performed to find the ellipses with locallymaximal GICOV value. Then, a B-spline snake with a GICOVconstraint is implemented for the refinement of boundary andpossible enhancement of the GICOV score, which is the scoreused for detection.

A. GICOV

In this section, we turn our attention to the formulation of ascore function with discriminating potential for leukocyte detec-tion. The function assigns a score, the GICOV, to each estimatedcontour such that the contour with the highest score correspondsto the true boundary of leukocyte, which coincides with our ob-servation of the leukocyte morphology.


Fig. 1. (a) An example image obtained from an in vivo experiment in a mousecremaster. It shows rolling leukocytes and stationary structures. (b) A typicalbright cell. (c) A typical dark cell.

Fig. 1(b) and (c) depicts two typical leukocyte images—abright cell and a dark cell, drawn from a frame of a temporalsequence [shown in Fig. 1(a)]. The different mean intensitiesare mainly decided by their positions relative to the midplaneof the vessel, which is the focal plane of microscopy duringthe experiment. Note that rolling leukocytes are visible througha heavily cluttered environment—faster flowing blood streamand erythrocytes, static vessel boundary and muscle striations,as well as occasional defocusing due to the breathing movementof the living subject lead to clutter. In such a cluttered environ-ment, it may not be reasonable to assume a uniform intensityprofile within a leukocyte, despite the fact that a leukocyte con-tains a homogenous region of cytoplasm enveloped by a distinctcell membrane. Thus, instead of assuming a homogeneous in-tensity profile, we assume uniform directional edge strengthsalong a leukocyte boundary and this assumption is the principalmotivation for proposing the GICOV, a score defined on a 2-Dcontour, to characterize a rolling leukocyte from intravital mi-croscopy images. We hypothesize that the GICOV will be themaximum within a local neighborhood of a contour, when thecontour is delineating the leukocyte boundary.

Let represent a 2-D closed contour parameter-ized through . If denotes an image then themean of the image gradient over the entire contour computed inthe outward normal direction is given by

(7)

where is the unit outward normal to the contourat and is the length of the contour given by

.The incorporation of directional information yields superior

results when the contour intersects adjacent object boundaries.When the snake model is implemented without respect to direc-tionality, the snake may be attracted to a strong edge that has adifferent gradient direction and could be resultant from anotherobject. The use of the gradient operator as a segmentation/edgedetection tool is substantially enhanced by the utilization of di-rectional information [23].

The variance of the image gradients over the entire contourcomputed in the outward normal direction is given by

(8)

We may expect that if the contour delineates an object,i.e., when it is located on the edge of an object in an image,the statistic given in (7) will yield a high magnitude. The ra-tionale behind this hypothesis is the notion of an “edge” in animage that can be defined as a 2-D curve (not necessarily closed)across which maximum (with respect to some suitable neighbor-hood of the points on the curve) image intensity difference isobserved. Furthermore, if the region inside an object is brighterthan the background outside the object then the sign of wouldbe negative, whereas the sign would be positive when the out-side is brighter than the inside. Now, if the object edge strength,viz., the image gradient magnitude across the object boundary,is sustained along the object boundary, then we can also ex-pect that the statistic given by (8) will exhibit a low value.It is important to note that even if the object edge strength werenot quite uniform along its boundary, in a highly cluttered en-vironment the statistic (8) would be quite effective in rejectingclutter. Taking into account the above considerations, we definethe GICOV as

(9)

where functions as a normalization factor. (Note: is re-placed by , the number of contour samples, in the discretecase.) In order to seek an object boundary in a cluttered envi-ronment, as with detection of a rolling leukocyte from intravitalimagery, we want to locate a closed contour such that

(10)

B. Ellipse Matching

As mentioned in Section I, leukocytes appear to be slightlydeformed ellipses (from teardrop shapes to perfect circles) in theintravital imagery. Thus, while computing (10) for a leukocytewe may restrict the search spaces to deformed ellipses. Sincewe know the approximate size of the leukocytes, and we knowthe blood flow direction (could be computed using optical flow),the orientation of the leukocyte (and hence the orientation of theellipse) should not vary dramatically from the flow direction.Thus we can utilize the problem-specific prior knowledge whilecomputing (10).

Typically, an ellipse is characterized by five parameters:center position coordinates , lengths of the major andminor axes and , and the orientation of the major axis .Let be a point on this clockwise ellipse, whose coordinatecan be expressed as


via a parameter . Note that the unit outward normalat can be written as

(11)

Let us now take discrete points on the ellipse determinedthrough parameter values , then we can rewrite (7)and (8) respectively as their unbiased sample estimate withinthis finite sample

(12)

and

(13)

where is given by

(14)

Furthermore, we can express (10) as shown in (15) at the bottomof the page. Thus, the ellipse with the highest GICOV scoreis denoted by , and the maximal GICOVscore corresponding to this case is denoted by . Welimit the search space as indicated by the values , , etc.,appearing in (15). In fact we perform this maximization forevery pixel location in the image domain to compute

.Next we find locally maximum GICOV scores dictated by a

circular neighborhood of certain size

if ,otherwise

(16)

where is the dilation operator. The resulting binary imageindicates possible leukocyte centers where its value is

nonzero. Note that the size of the structuring element imposesa minimum separation distance between two leukocytes.

C. B-Spline Snake With GICOV Constraint

After the coarse stage of ellipse matching, the ellipse outlinesthe coarse boundary of the object. In the refinement stage, aB-spline snake with a GICOV constraint, which is initializedat the ellipse, is implemented to better delineate details of thecell boundary. As mentioned, the spline representation is par-ticularly well matched with the leukocytes due to the regularnature of leukocyte boundaries.

1) Energy Formulation: Section II-A has provided the linkbetween the spline and the conventional snake. As aforemen-tioned, we do not require internal energy term due to the intrinsicregularity property of B-spline, which reduces the number of re-quired parameters. It should also be noted that the original snaketechnique [15] implements the internal continuity and smooth-ness constraints, which effectively low-pass filter the contour.These constraints result in contraction: a closed contour shrinksto a point without the support of external forces. The B-splineparameterization of the contour removes the unwanted contrac-tion force, producing a solution without contraction bias, asdesired.

One of the strengths of the snake model is the ability to ac-commodate constraints in order to meet a new objective. Let usconsider now the situation where snaxels sampled along thesnake contour are given by

...... (17)

Then, our total energy function is defined as the square ofGICOV associated with contour , and underlying imagewith negative sign, given by (18) shown at the bottom of thepage. With the B-spline parametric representation, the unitoutward normal at can be written as

......

...... (19)

(15)

(18)


Fig. 2. (a) Synthetic image with a set of coarse initial contours (in white) superimposed. (b) Result of B-spline snake with GICOV constraint on (a). (c) Resultof B-spline snake evolution on (a) without GICOV constraint.

Fig. 3. Evolution results of B-spline snake with GICOV constraint on leukocytes in vivo, shown in (a), (c), and (e). Evolution results of B-spline snake withoutGICOV constraint on same images, shown in (b), (d), and (f), respectively. Snakes are shown in white.

where and are corresponding coordinates of controlpoints, and is the number of con-trol points. In general, a spline snake has a sparse set of controlpoints but a large number of sampling points. The negative signis used in (18) as we employ a minimization process for theoptimization.

2) Optimization: The minimization process is implementedby the method of gradient descent. Using variational calculus[24], we can produce a set of partial differential equations thatupdate the contour position in an iterative manner. The B-splineformulation allows an easy computation of the gradient functionof the energy term. The parameters subject to optimization arethe B-spline control points, yielding

(20)

3) Examples: To demonstrate performance in the presenceof clutter, the method is compared with the GICOV without thevariance term in the denominator, that is, only with average gra-dient strength constraint, using synthetic and real images. Ex-perimental results are shown in Fig. 2 for methods applied tothe synthetic image and in Fig. 3 for actual leukocyte images.These images show that as a consequence of variation of con-trast and initialization, the B-spline snake with only the gradientstrength term is sensitive to the influence of high contrast re-gions (in contrast to the snake with the full GICOV constraint).This illustration supports our claim that the GICOV is robust ina cluttered environment.

In Figs. 2 and 3 we used 12 control points along the contourand sampled the contour at 4 points along each segment deter-mined by the control points (i.e., 48 snaxels in total). We notethat the exact number of sampling points is not critical; a less

dense sampling may be used to increase optimization speed. Adeterministic iterative algorithm may be employed to estimatethe parameterization order via a minimum description lengthcriterion [25].

IV. DENSITY DERIVATION AND OPTIMUM

THRESHOLD SELECTION

In this section we derive the densities andanalytically, so that the densities can be utilized in the classifi-cation process—partitioning the extracted contours into leuko-cyte and nonleukocyte classes. Let us begin with the followingdefinitions and notations. A snaxel is called an edge snaxel ifit coincides with the location of an object edge; otherwise, it isa off-edge snaxel. Let denote the set of all possible contourconfigurations regarding the various combinations of edge andoff-edge snaxels. Given a contour, we denote the number of edgesnaxels by and total number of snaxels by . In the distributionderivation, for the Gaussian distribution model, we will use theterminology to denote that the random variable

is normally distributed with mean and variance .denotes the expectation of the random variable .

Now that we have selected a particular algorithm for leuko-cyte extraction and obtained associated GICOV scores, we aimto determine an appropriate threshold of GICOV value for op-timum detection. By “optimum,” we mean that at this thresholdvalue, the Bayes error due to measurement noise and clutter isminimal.

Regarding a contour residing on the underlying image, wefirst discuss the statistical image and noise model. The statis-tics of the image gradient in the outward normal direction ofthe contour are further examined. We then, based on config-urations of contour, derive the pdf of GICOV associated withleukocyte class and nonleukocyte class as a function of param-eters of image and noise model. At last, the optimum detectionthreshold used by the Bayes classifier is decided.


Fig. 4. Quantile-Quantile plots of noise in (a) the bright and (b) the dark regions.

A. Statistical Image and Noise Model

In order to describe the statistics of the normal directional gra-dient, knowledge of underlying image, noise and gradient oper-ators is required. We start with modeling the intensity change atan object boundary point as a step function of unknownamplitude , which is vertically oriented. It features as an abrupttransition from one homogenous region to another but of dif-ferent irradiance from the first. Projected to the image plane,the focal blur of this edge can be modeled as the convolution ofa Gaussian blurring kernel

(21)

with scale constant and the ideal step edge to generate aramp edge. The Gaussian here represents a coarse estimate ofthe point spread function, as it has been used similarly in depth-from-defocus work [26], [27].

To explore the statistical behavior of measurement noisepresent in our imagery, we accessed the Gaussianity of

our noise data empirically with a Quantile-Quantile plot. If thedistribution is Gaussian, then the plot will be approximately astraight line. The noise present in the images is obtained fromsmoothly varying regions by subtracting the intensity averagesfrom the original images. Regions with bright appearance(average gray level is 178) and dark appearance (average graylevel is 85) are used. The Quantile-Quantile plots are shown inFig. 4 using 1000 samples each. We can deduce from the figurethat the noise in our imagery shares very similar statisticalbehaviors with the Gaussian distribution. The slight bending upon the left and bending down on the right of plots mean that theactual noise has little shorter tails than the Gaussian.

The standard deviations of noise computed from the brightand dark region are very close (4.23 and 4.21, respectively),which rejects the hypothesis of multiplicative signal-dependentnature of the noise. Therefore, the measurement noise is mod-eled here as an additive, zero mean white Gaussian noise withstandard deviation. To estimate image gradient, we utilize theGaussian first derivative as follows:

(22)

(23)

where denotes the scale of the first derivative of Gaussian(DOG) operator. Therefore, the gradient intensity is given by

(24)

while the maximum value, occurring on the axis, is given by

(25)

It can be seen that the responses of the (directional) gradientoperator due to noise alone, which we denote as and ,respectively, follow a Gaussian distribution with zero mean andstandard deviation

(26)

The norms of the first derivatives of the Gaussian in (26) aregiven by

(27)

Since the statistics of image and noise are well defined now,the distribution of the normal direction gradient can be derivedas follows. For any desired direction , a directional gradient

can be constructed by computing the two compo-nents of filter response that can be proved to be independent[28], so that if

and

then

(28)


Fig. 5. The correlation coefficients of normal image gradient estimate on neighboring snaxels.

Let us note that the normal directional gradient is used in thecomputation of GICOV. Assume is the angle of an outwardnormal vector of a given contour relative to the positive axis,and the contour resides exactly on the locations of uncontami-nated edges, so that

(29)

By combining (28) and (29), the image gradient in is thus

(30)

Therefore, the normal directional image gradientgiven in (30) follows a Gaussian distribution with mean

and variance .

B. The Independence Assumption

In our paper, the derivations are based on an independenceassumption, which states that the normal directional gradientscomputed along the snake contour are independent of eachother. However, this assumption does not always hold in thestrict sense; the reason is that, for the neighboring snaxelsin small vicinity, the circular areas resulting from Gaussiansmoothing in the derivative estimation may overlap if theGaussian operator scale is not small enough.

Two experiments are conducted to illustrate the situation inour application. In the first experiment, we fix the Gaussiansmoothing kernel scale as 1 pixel while changing the distancebetween the neighboring snaxels. The computed correlationcoefficients of normal gradients are shown in Fig. 5(a). We alsomeasure correlation for different Gaussian scales with fixedintersnaxel distance [see Fig. 5(b)]. It can be observed that1) the dependence of gradients on neighboring snaxels depends

strongly upon the separation of the snaxels and the smoothingfilter scale chosen; and 2) the correlation increases withsmoothing scale. In our application the neighboring snaxels areabout 4 pixels apart, typically, so that the correlation coefficientis about 0.1. In the linear dependence sense, the coefficient ofdetermination [29] is about 0.01, which means that only 1% ofthe variance in a measurement can be explained by variation inits neighboring snaxels. Therefore, we can conclude that if thesnaxels are widely spaced relative to the smoothing scale (suchas our case in this application), the independence assumptioncan be justified in the approximation.

C. GICOV Density Function: None/All of Snaxels on Edges

To derive the conditional density function of the GICOV,, where is the number of edge snaxels, let us first

consider a simple configuration where all snaxels along thecontour locate either on homogenous regions, or on the peaks ofedges, i.e., the edge snaxel number or . The main resultin this section is the density function given in Proposition 2.

Mathematically, the problem can be stated as follows: Letbe a set of independent finite random sam-

ples from a Gaussian distribution , while representsthe normal directional gradient. The value of could be either

as shown in (25) or 0, and is given in(26). Let us rewrite the GICOV in (9) as

(31)

where is sample mean, is the number of snake samples(snaxels), and is estimate of sample standard deviation, de-

noted by . The expres-sion is often called the standard error of the mean in sta-tistical inference. In order to find the distribution of in (31),we need the following Theorem [30].

Theorem 1: Let and be independentrandom variables. Then has a (Student’s) t


Fig. 6. Analytic plots and Monte Carlo simulation plot for pdf. (a) p(V jj = 0) with � = 0, � = 3, n = 24. (b) p(V jj = n) with � = 10, � = 3, n = 24.

distribution with noncentrality parameter and degrees offreedom (dof).

The formula for the density function of the t distribution canbe found in [30]. This distribution resembles a Gaussian withwide tails. To demonstrate that conditions of Theorem 1 are sat-isfied in our problem, we derive Proposition 1.

Proposition 1: Let be independent sam-ples from a Gaussian distribution with mean , vari-ance . Let and

. Then (a) , (b), and (c)

and are independent.Proof: See Appendix A.

From Theorem 1 and Proposition 1, we can get the followingProposition straightly.

Proposition 2: Let be a set of random samplesfrom a Gaussian distribution with mean , variance . Let

and ,then has a noncentral t distribution withdof and noncentrality parameter .

Proof: Taking, , ,and , we have

(32)

Combining Theorem 1, Proposition 1 and (32), we have the dis-tribution of as described in Proposition 2. This completes theproof.

Using Proposition 2, we can determine the conditional den-sity function, , for the GICOV conditioned on the loca-tion of the contour points (whether these points are located onor off actual object edges).

Monte Carlo (MC) simulations were performed to test thevalidity of the derivation. The experiment consisted of 10000 iterations with sample length . In each iteration,a random value was generated from a normally distributedrandom number generator. Fig. 6 plots density functions of the

analytical and the MC simulation with different values of forcomparison. It can be seen that the analytical and MC plotsmatch closely, even where has a rather large value.

D. GICOV Density Function: Portion of Snaxels on Edges

While the GICOV in the last section is modeled as a non-central t distribution, now we discuss a more general situationwhere some snaxels reside on edge, others not, i.e., the numberof edge snaxels . Let us denote the percentageof edge snaxels as and percentage of off-edge snaxels as. Similarly, the problem can be stated as follows: Let

be a set of random samples from a Gaussiandistribution , while for

, and for . The variancesare held the same as those given in the previous Section IV-C.

and are still independent. (The proof is similar to theproof of Proposition 1 and is omitted.) However, since someelements of the sample have nonzero mean, follows a non-central chi-square distribution with dof, which does notsatisfy the conditions in Theorem 1. The distribution in this caseis much more difficult to model. However, one can still derive aGaussian distribution that lies very close to the real underlyingdistribution by exploiting an elementary result from statisticaltheory:

Theorem 2: [31]: Any noncentral chi-square random vari-able with dof can be written as the sum of a noncentral chi-square random variable with one dof and an independent centralchi-square random variable with dof.

Furthermore, it is well known that the central chi-squarestatistic with dof can be expressed as the sum ofsquares of independent random variables, , all drawnfrom a Gaussian population with zero mean and standarddeviation . Following directly from Theorem 2, we should beable to express , a noncentral random variable withdegrees of freedom as

(33)


Fig. 7. Analytic plots and Monte Carlo simulation plot for pdf p(V jj = rn) with edge snaxels percentage r = 50%. (a) Standard deviation � = 1,Kullback Leibler distance = 0:007. (b) Standard deviation � = 3, Kullback Leibler distance = 0:035. (A distance of zero connotes complete agreementin distribution.)

while is a constant, are drawn fromand are independent. The values of the two parameters

and can be determined by solving for the mean of (shownin Appendix B)

(34)

If is expanded in a Taylor series, from (33), then we have

(35)

Terms of order have components proportional to .If the coefficient of variation (the ratio of the variance of to) is small enough, terms of order 3 and higher in (35) may be

dropped. In other words, to the extent that this approximation isaccurate, high signal-to-noise ratio (SNR) is preferred. We tryto approximate the distribution of with a Gaussian as follows.

The mean of , the denominator of , is given by

(36)

and the variance by

(37)

The inaccuracy of the Taylor expansion in (37) for large valuesof is insignificant, as the high order terms are very small.

The numerator of , , is a summation of Gaussian randomvariables; hence, it is also Gaussian distributed. Therefore, thisdistribution is completely described by its mean and variance,given by

(38)

(39)

Finally, is expressed as the ratio of two independentGaussian random variables, approximately. The approxima-tion of its distribution can be obtained using Proposition 3 asfollows.

Proposition 3: Let two independent random variables, . Let the ratio of them

The distribution of can be approximated by a Gaussian distri-bution if the condition

holds true.Proof: See Appendix C.

The above formulations lead to the approximation of the con-ditional density function for . We also ranMonte Carlo simulations to verify the analysis of this section.Figs. 7 and 8 ( , ) plot analytical and Monte Carlosimulation density functions with different combinations ofand . Our analysis yields a good approximation of the exactdistribution, especially when is small. When is large, asshown in Fig. 7(b) and Fig. 8(b), it can be seen that Gaussianapproximation drifts by a small distance from the MC density.


Fig. 8. Analytic plots and Monte Carlo simulation plot for pdf p(V jj = rn) with edge snaxels percentage r = 75%. (a) Standard deviation � = 1,Kullback Leibler distance = 0:012. (b) With standard deviation � = 3, Kullback Leibler distance = 0:093.

The relative entropy (distance) between two densities can bemeasured by Kullback Leibler distance (KLD) [32]. Note thatthe between two Gaussian densities and

. From these plots, we can conclude that the Gaussianapproximation leads to rather accurate statistical models.

E. Optimum Threshold Selection

After modeling the density functions of all possible combina-tions of edge/off-edge snaxels, which we here refer to as patterns,a Bayes classifier is implemented for threshold selection. Thecontour configuration set can be partitioned by a predefinedoverall edge snaxels percentage into two subsets: the leukocyterelated contour set and the nonleukocyte related contour set. Therule for partitioning is that a leukocyte related contour musthave or more snaxels among snaxels residing on actualobject edges; otherwise, we assign it as a member of nonleuko-cyte related contour set. Thus, since class conditional densityfunctions and are, respectively, modeled as

(40)where the prior class probabilities are ,

, then, we have posterior probabilities as

(41)

where denotes the conditional density function ofgiven each pattern , which we obtain in Section IV-C and IV-D.The mixing coefficient represents the prior probability of thepattern , and it can be reasonably assumed to follow binomialdistribution (plotted in Fig. 9) such that .

Fig. 9. Distribution of contour pattern withn = 24. x axis denotes the numberof edge snaxels.

Here, we summarize the following steps used to obtain theassociated parameters of density functions from training data.

Step1) Determine image associated parameters , ,and defined in Section IV-A from training

images. (In our experiments, the parameters usedwere: gray levels, gray levels

, pixels, pixelsfor bright leukocytes; For dark leukocytes,gray levels, gray levels ,

pixels, pixels.) The standard devi-ation of the focal blur is estimated by using thederivatives of the Gaussian PSF [33]. The methodof automatic selection of the smoothing kernelscale can be found in [34].

Step 2) Calculate the mean and standard deviationusing (25) and (26): (In our experiments, forbright-appearing leukocytes, gray


Fig. 10. (a) Bayes decision rule for dark-appearing leukocytes. (b) Bright-appearing leukocytes. The Bayes threshold is indicated as intersection of two densitycurves, with = 7:2 for dark leukocyte and = �7:9 for bright leukocyte.

levels per pixel and gray levels per pixel.For dark-appearing leukocytes, gray levelsper pixel and gray levels per pixel.)

Step 3) Calculate the conditional density function ofGICOV value given each possible contour pattern

using (36)–(39) and Proposition 3.Step 4) Determine the parameter of the prior probability

(binomial) of each con-tour pattern from the training set using ML esti-mation. (On the training sets, the patterns of con-tours extracted by B-spline snake are determinedby the normal gradient intensities on their snaxels,using gradient thresholding. For the training, 200such contours are used.) The parameter is foundto be 0.65 by ML estimation in our application.

Step 5) Determine the edge snaxels percentage through asupervised training set of extracted contours. (Onthe training sets, each of the contours extracted byB-spline snake is manually labeled as leukocyte/nonleukocyte. The number of edge snaxels for eachcontour is then counted. Thus, depending on thenumber of edge snaxels, any percentage dividesthe entire set of contours into two classes for whichthe classification error can also be found with respectto the manually classified leukocyte/nonleukocytelabels. The percentage is set as 75% by minimizingthis classification error for the entire training set ofcontours.)

Step 6) Calculate and (illus-trated in Fig. 10) using (41). Find the optimumthreshold for detection as the intersection pointof and by (5). (Inour experiments, for dark leukocytes and

for bright leukocytes.)

Once we derive the GICOV threshold value for the trainingset we apply the same threshold for the test images and obtainthe results described in the next section.

Fig. 11. An example of leukocyte detection. (a) An example image in the videosequence showing leukocytes and stationary structures. (b) Result of ellipsematching (in white) on (a). (c) Final result of detection by GICOV thresholdingafter using B-spline snake with GICOV constraint from (b).

V. EXPERIMENTAL RESULTS

The problem of intravital leukocyte detection is attacked assummarized with the following steps:

Step 1) Implement ellipse matching algorithm that yields:a) locally maximal GICOV score; and b) corre-sponding ellipses producing the maximal GICOVscore.

Step 2) Implement B-spline snake with GICOV constraintstarting from the best match ellipses (found in step1) to refine the leukocyte boundaries and computeGICOV scores of evolved contours.

Step 3) Implement thresholding for GICOV scores to re-tain the contours that have GICOV scores above thethreshold .


TABLE ICOMPARATIVE LEUKOCYTE DETECTION PERFORMANCE OF THE HOUGH TRANSFORM METHOD, THE ERS METHOD, THE LEVEL SET METHOD, THE GRADIENT

METHOD AND THE PROPOSED GICOV METHOD ON THE TEST DATA SET, WHICH CONTAINS 30 IMAGES AND 327 LEUKOCYTES

As an example of the detection process, resultant images areshown in Fig. 11. The results of ellipse matching are shownin Fig. 11(b), from which boundary refinements are obtainedby B-spline snakes. Rolling leukocytes [Fig. 11(c)] are accu-rately extracted by way of GICOV thresholding. It can be seenfrom Fig. 11(b) that almost all leukocytes are extracted by el-lipse matching algorithm, and the following snakes and Bayesclassifier eliminate most false alarms. Note that our method issuccessful in detecting leukocytes correctly even though someof the cells reside very close to each other and are overlappingin some cases. This fact is evident from the example results inFig. 11(c). Fig. 11 also serves as an example of diversity in ap-pearance of the cells.

For all experiments in our implementation, the B-spline snakealgorithm adopts the spline bases of order of four (i.e., cubicpolynomials), uses six control points along the contour and sam-ples the contour at four points along each segment determinedby the control points (i.e., 24 snaxels totally). Since the objectsof interest are circular or elliptic, it may be argued that moreelements are required to properly delineate a more complicatedboundary.

To evaluate the performance of the proposed detectionmethod, Table I tabulates the comparative leukocyte detectionperformance of the Hough transform method, ERS transformmethod, level set method and proposed method in terms ofdetection rates and false alarms. Further, to illustrate the effectof the variance term in the GICOV measure, detection resultsusing the GICOV without the variance term (referred to here asthe gradient method) are also given in Table I. The test data setcontains 30 images and 327 leukocytes (214 bright leukocytesand 113 dark leukocytes) observed in venules of the mousecremaster muscle. Manually detected leukocytes by an expertact as ground truth for all our experiments. For this detectionexperiment, the detection rate is defined as the ratio betweenthe number of leukocytes correctly detected and the numberleukocytes determined by the expert. The false alarm rate isdefined as the ratio between the number of nonleukocyte objectswrongly identified as leukocytes and the number leukocytesdetermined by the expert.

Experimental results show that the proposed GICOV method,achieving 78.6% leukocyte detection accuracy with 13.1% false

alarm rate, compares favorably against the currently used objectdetection algorithms, such as Hough transform method [4], ERStransform method [6], and level set method [10]. If the varianceterm is removed from the GICOV measure (viz., the gradientmethod), the performance degrades to 71.6% for detection and26.0% for false detection.

VI. COMPUTATIONAL EXPENSE

1) Ellipse matching: One concern in the leukocyte detectionprocess is the computational expense with respect to feasi-bility of implementation in intravital laboratory research.The computation of the two statistics in (12) and (13) canbe performed through convolution operations, where theconvolution kernels are constructed using the ellipse out-ward unit normal. Thus, the complexity for an image of

pixels is the fast Fourier transform. Weused ellipses for matching and each one corre-sponding to a different combination of lengths of majorand minor axes and orientations. The total complexity ofthe algorithm is, therefore, given by .It can be seen that, unfortunately, ellipse matching usingthree flexible parameters is both demanding in storagespace and time consuming. However, in practical situa-tions, the range of can be narrowed according somedomain-specific knowledge.

2) B-spline snake: For a total of snaxels on the contour andcontrol points, it should be noted that the complexity of

one update of the B-spline snake is . The overallcomplexity for the snake model is, therefore, given by

, where denotes an average number of updates(typically ).

Applied to the image (640 165 pixels) shown in Fig. 11the computational time in a Matlab implementation was 3.2 minfor the ellipse matching step, and 6.5 min for the B-spline snakestep. For the three existing detection techniques used for com-parison, the processing time required was 10.3 min for Houghtransform method [4], 7.9 min for ERS transform method [6],and 3.1 min for level set method [10].


VII. CONCLUSION

A new method for detecting leukocytes rolling along the vas-cular endothelium in vivo has been developed and presented. Inour paper, the problems due to artifacts appearing in the intrav-ital imagery are mitigated through seeking a feature score—theGICOV. Ellipse matching and B-spline snakes are used for con-tour extraction and potential rolling leukocyte hypothesis gener-ation. We derive a statistical model to determine a likely GICOVscore of leukocyte, and use the Bayes classifier to distinguishcorrect detections from false alarms.

It is possible to extend this work by: 1) integrating a priorishape information of leukocyte; 2) integrating temporal infor-mation from the video sequence; and 3) applying this approachas a front end for the leukocyte tracking problem [12], [13].

APPENDIX A

Following the linearity of mean and variation of independentrandom variables, we have

(A1)

And obviously it is Gaussian distributed. This proves part (a) ofProposition 1.

Let , while . Without loss of gener-ality, picking for example, we have the covariance functionas

(A2)

So on the case of Gaussian variables, and are inde-pendent. It can be generalized to the fact that all andare independent. Thus, , a function of , is independentwith , proving part (c) of Proposition 1.

It is known that as , the sum of squareof such random variables is chi-square distributed with de-grees of freedom, denoted by . However, here we use samplemean rather than true mean when we compute the variance,which results in loss of one degree of freedom.is, therefore, distributed. This proves part (b) of Proposi-tion 1.

APPENDIX B

Let , for , andindependent, while . Without loss ofgenerality, taking , for example, we have

(B1)

Let , for , and inde-pendent. Taking , for example, similarly, we have

(B2)

Thus

(B3)

Computing the mean of (33), we have

(B4)

Comparing (B3) with (B4) and considering theirstatistical meaning, their values are given by

and .

APPENDIX C

We have , . The ratiomay be expanded in a Taylor series

(C1)

where , . Terms of orderhave components inversely proportional to . If the coefficientof variation (the ratio of the standard deviation to the mean) of

is small and is significantly less than unity, terms oforder 2 and higher may be dropped, yielding


and.

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