1640 ieee transactions on image processing,...

1640 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 12, DECEMBER 2004

Edge Detection in Ultrasound Imagery Using theInstantaneous Coefficient of Variation

Yongjian Yu, Senior Member, IEEE, and Scott T. Acton, Senior Member, IEEE

Abstract—The instantaneous coefficient of variation (ICOV)edge detector, based on normalized gradient and Laplacianoperators, has been proposed for edge detection in ultrasoundimages. In this paper, the edge detection and localization per-formance of the ICOV-squared (ICOVS) detector are examined.First, a simplified version of the ICOVS detector, the normalizedgradient magnitude squared, is scrutinized in order to revealthe statistical performance of edge detection and localization inspeckled ultrasound imagery. Both the probability of detectionand the probability of false alarm are evaluated for the detector.Edge localization is characterized by the position of the peak andthe 3-dB width of the detector response. Then, the speckle-edgeresponse of the ICOVS as applied to a realistic edge model isstudied. Through theoretical analysis, we reveal the compensatoryeffects of the normalized Laplacian operator in the ICOV edgedetector for edge-localization error. An ICOV-based edge-de-tection algorithm is implemented in which the ICOV detector isembedded in a diffusion coefficient in an anisotropic diffusionprocess. Experiments with real ultrasound images have shownthat the proposed algorithm is effective in extracting edges in thepresence of speckle. Quantitatively, the ICOVS provides a lowerlocalization error, and qualitatively, a dramatic improvementin edge-detection performance over an existing edge-detectionmethod for speckled imagery.

Index Terms—Edge detection, instantaneous coefficient of vari-ation, speckle, ultrasonic image.

I. INTRODUCTION

MEDICAL ultrasound (US) has been widely used forimaging human organs (such as the heart, kidney,

prostate, etc.) and tissues (such as the breast, the abdomen, themuscular system, and tissue in the fetus during pregnancy).US imaging is real-time, nonradioactive, noninvasive, andinexpensive. However, US imagery is characterized by lowsignal-to-noise ratio (SNR), low contrast between tissues, andspeckle contamination. In general, medical US imagery is hardto interpret objectively. Thus, automatic analysis and interpre-tation of US imagery for disease diagnostics and treatmentplanning (e.g., in prostate cancer brachytherapy) is desirableand of clinical value. An essential step toward automaticinterpretation of imagery is detecting boundaries of differenttissues. Though, in general, the boundary of an object can be acombination of step edges, ridges, ramp edges, etc., we focus

Manuscript received December 30, 2003; revised January 28, 2004. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Luca Lucchese.

Y. Yu is with the Department of Radiation Oncology, University of VirginiaHealth System, Charlottesville,VA 22908 USA (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TIP.2004.836166

upon detecting the boundaries of human organs that can bemodeled as step edges.

Marr and Hildreth [10], and, later, Haralick [8] have exam-ined the use of zero crossings produced by the Laplacian-of-Gaussian (LoG) operator for the detection of edges. Canny [5]proposed the odd-symmetric derivative-of-Gaussian filter as anear-optimal edge detector, while even-symmetric (sombrero-like) filters have been proposed for ridge and roof detection [14].However, Bovik [3], [4] proved that both the gradient and theLoG operator do not have the property of constant false alarmrate in homogeneous speckle regions of speckled imagery. Ithas been argued that the application of such detectors gener-ally fails to produce desired edges from US imagery [3], [6],[12], [21]. Some constant false alarm rate (CFAR) edge de-tectors for speckle clutter have been proposed, including theratio of averages (ROA) detector [3], the ratio detector [21],the ratio of weighted averages [6], and the likelihood ratio (LR)[7], [12]. Other ratio detectors include the refined gamma max-imum a posteriori detectors [9] and more recent improvements[2], which use a combination of even-symmetric and odd-sym-metric operators to extract step edges and thin linear structuresin speckle. With CFAR edge detectors, the image needs to bescanned by a sliding window composed of several differentlyoriented splitting subwindows. The accuracy of edge locationfor these ratio detectors depends strongly on the orientation ofthe subwindows with respect to edges. For the LR detector, anedge bias expression has been derived in [7]. The bias in edgelocation is deleterious in obtaining quantitative estimates of thevolume of the organs from diagnostic US imagery.

In an attempt to develop a more efficient edge detector withhigh-edge positioning accuracy for US imagery, we turn our at-tention to differential/difference operators that are straightfor-ward to compute in small windows. We believe that the keyproblem in developing differential-type edge detectors is cor-rectly accommodating the multiplicative nature of speckle. In[24], the use of a new partial differential equation (PDE)-basedspeckle-reducing filter for enhancement of US imagery is pro-posed. This filter relies on the instantaneous coefficient of vari-ation (ICOV) to measure the edge strength in speckled images.Denoting the image intensity at position as , the instan-taneous coefficient of variation is given by

(1)

where , , , and are the gradient, Laplacian,gradient magnitude, and absolute value, respectively.

1057-7149/04$20.00 © 2004 IEEE

YU AND ACTON: EDGE DETECTION IN ULTRASOUND IMAGERY 1641

Specifically,

where ,; and

. The derivation of (1) can be found in[24]. It is seen that the ICOV (1) combines image intensity withfirst and second derivative operators, which are well known inthe existing literature.

The ICOV is meant to allow for balanced and well localizededge strength measurements in bright regions as well as in darkregions. Experimentally, the performance of the ICOV has beendemonstrated for edge-preserving speckle-reducing anisotropicdiffusion (SRAD) [24] on US and radar imagery. However, theedge-detection mechanism and performance of the ICOV hasnot been analyzed quantitatively in terms of some figures ofmerit (FoM). The objective of this paper is to provide a solutionto this problem. Because of the complexity of the ICOV, wewill study the ICOV-squared (ICOVS) and make the assumptionthat the edge-detection performance of the ICOV can be derivedfrom that of the ICOVS, taking into account that the squaringoperation is a one-to-one monotonic mapping for any positivefunction.

Quantifying the performance of the ICOVS in a closed, exactform is difficult. In this paper, we first consider a special case ofthe ICOVS, the normalized gradient magnitude-squared oper-ator (NG), for which both the statistical detection performanceand the edge-localization performance are examined (we donot consider popular wavelet-based approaches, such as [18]and [19]). Appropriate statistical models for speckle and edgeprocesses are established for the performance analysis, and theprobability density functions (pdfs) in homogeneous speckle re-gions and edge regions are derived for the NG operator. Theedge-localization performance for the NG operator is charac-terized by the position of the peak and the 3-dB width of themean detector response. Then, the same calculations and anal-ysis are carried out for the ICOVS. In particular, the Laplacianterms in the ICOV are studied carefully to identify their roles inedge detection.

To validate the quantification of the NG and the ICOVS, thetheoretical statistical performances are compared with thosegiven by Monte Carlo simulation. Also, the NG and ICOVSoperators are directly tested on a synthetic one-dimensional(1-D) signal with low speckle. For validating the capability ofthe ICOV edge detector for US images, a practical ICOV-basededge-detection algorithm is implemented, by embedding theICOV detector in a speckle-reducing anisotropic diffusionprocess. Validation using medical B-mode and phantom imagesis provided for the ICOV-based edge-detection algorithm andcompared to the performance of an existing algorithm.

II. NG OPERATOR IN HOMOGENEOUS SPECKLE

In Section II, we analyze the statistical performance of the NG(a special case of the ICOVS) operator as applied to speckle pat-terns in homogeneous regions, following the method of Bovik[3]. First, we summarize the image and speckle models and thebasic assumptions used as the framework for quantifying the

detector performance in speckle. Then, we examine the NG op-erator in homogeneous speckle regions.

A. Models for Image and for Speckle and Basic Assumptions

represents a random process that models the observedintensity at location in an image. Using the multiplicativemodel, we write where is adeterministic function governing the underlying reflectance ofthe object being imaged and is a wide-sense stationaryrandom process describing the normalized speckle process. Ithas been shown [15] that the normalized speckle processis gamma distributed with pdf( ) where is the equivalent number of looks. Ac-cording to the central limit theorem, for large , the gammapdf can be approximated by a Gaussian: . Becausethe rate of convergence of to is ,this Gaussian approximation is not appropriate for small valuesof (usually, when ). For , the pdf of thecube-root of , , can be approximated by a Gaussian:

[23]. The rate of convergence of thepdf of the standardized to the standard normal distribution is

, making the Gaussian a satisfactory approximationfor for [1].

When studying the speckle properties, it is necessary toknow not only the first-order characteristics of the speckledescribed by the single point pdfs, but also the correlationproperties of the speckle. The correlation is described bythe speckle power spectral density (PSD) function, whichis the Fourier transform of the autocovariance function of

. However, there is no general model of the spectraof image speckle for an arbitrary imaging modality. Variousmodels have been derived for the spectra of US, syntheticaperture radar, or laser speckle patterns. A common featureof these speckle processes is that they can be modeled asband-limited processes containing only lower spatial frequen-cies. Without loss of generality, we consider the spectra ofa B-scan US. We approximate the speckle PSD functions of

bywhere (units: cycles/meter) is the noise band-width for the -look data, and are spatial frequencies(units: cycles/meter) in the direction and direction, respec-tively. The noise bandwidth for single-look data , assumedto be equal in both axial and lateral directions, is related tothe transducer dimension in the transverse direction, thewavelength of the illuminating beam, and the distancefrom the transducer to the focal zone by inthe lateral direction [or to the radio frequency (RF) pulseenvelope shape by where is the pulse widthin the axial direction]. Accordingly, the PSD forcan be derived as where

is a factor dependent ononly . The derivation of these relations can be found inAppendix I.

B. Statistics of the NG-Filtered Homogeneous Speckle Process

In a homogenous speckle region, the reflectivity function isuniform: where is a positive constant. Denoting


the NG operator as , we find that the NG edge-detectionresponse yields

(2)

where is a scaling constant, which can be set to unity if weonly deal with images in the continuous domain. However, when(2) is applied to a sampled image defined on a grid with a sam-pling frequency in both directions, since finite differencesare used to approximate the derivatives in (2), the discrete formof (2) will contain a factor . Therefore, for convenience,we preset the constant to , so that the discrete form of(2) takes a simple form to facilitate use in digital domain. Notethat (2) is a simplification to the squared ICOV presented in (1).The analysis of (2) sheds light on and provides instructive basisfor the analysis of the ICOV squared, since the edge-detectionperformance will be approximately identical in both cases. InSection II-B, based on the properties of , we first derive a pdffor a normalized and then present the formulae for eval-uating and bounding the probability of falsely detecting an edgein homogeneous speckle regions using the detector . Fi-nally, typical probabilities of false edge detection withare evaluated and plotted.

Instead of seeking the pdf for , we derive the pdf

for the statistic , so as to takeadvantage of a known pdf. By approximating the speckleprocess by a Gaussian process with mean ( )and variance and a Gaussian power spectrum den-sity having the cutoff frequency , it has been shown that

and are mu-tually independent zero-mean Gaussian random processes withvariance [13]. InAppendix II, it is shown that is independent of

(or ), leading to the conclusion that and

are independent.

According to [20], we know that is

distributed with two degrees of freedom and

is noncentral distributed with one degree of freedom andnoncentrality parameter . Collectingthese facts, we know that the statistic is the ratio of twomutually independent random variables: the numerator being

distributed and the other having a noncentral distribution.Therefore, follows a special case of the doubly noncentral

(Fisher) distribution, i.e., , using the notationof [20]. By means of the series representation of the pdf of thedoubly noncentral -distribution, we can write the pdf foras follows:

(3)

Fig. 1. Plots of the false alarm probability of NG in homogeneous speckleareas versus threshold for different numbers of looks. (a) Curves for L = 1 and2. (b) Curves forL = 4, 8, and 16. The solid lines are the theoretical curves andthe dashed curves are obtained using a Monte Carlo method.

Using (3), we find that the probability that is greater than avalue is given by

(4)

We denote the hypothesis of no edge being present by .Given a threshold , from (4), we find that , the false edge-detection probability of given the hypothesis that noedge is present is given by

(5)where . From (4) and (5), it is seen that the quantity

does not depend upon the local mean , indicating that theNG-edge detector is a CFAR detector. If the threshold is low-ered adaptively with the decrease of , as is increasedby a speckle filtering or diffusion, then will decrease ex-ponentially with . Fig. 1 depicts plots (in solid line) of theprobability of false alarm in homogeneous speckle areas as afunction of the threshold for different , using our analyticalexpressions (4) and (5). The parameter needed to computethe theoretical is given by since

and where is an arbitrary factor thatshould be greater than two, according to the Shannon samplingtheorem on aliasing. We empirically set the factor to 3. Ona synthetic (spatially discretized) image of correlated specklewith the same statistical characteristics as our speckle model, aMonte Carlo simulation (see Section V for details) is performedusing the discrete NG operator: . The resultingplots from the Monte Carlo simulation are also shown in Fig. 1.It may be observed that the theoretical prediction and MonteCarlo results are in good agreement.

III. DETECTING SPECKLED EDGES USING THE NG OPERATOR

In Section II, we discussed the performance of the NG oper-ator as applied to homogeneous speckle regions. In Section III,the performance of the detection of edges in speckle using theNG operator is examined. First, a realistic speckled-edge imagemodel is introduced. Then, the statistics of the NG operator onedges are derived. Finally, the geometric characteristics of theNG operator are addressed.


A. Realistic Speckled-Edge Process

Consider the nonstationary random field

(6)

where is a deterministic edge function used to model theunderlying image intensity changes across structure boundaryand is given by

(7)

where is the contrast of an edge defined as the ratio ofand and is the average of and .

; isthe error function, is the edge scale parameter (width ofthe edge transition zone) and is the stationary specklerandom process as defined in Section II. In contrast to theidealized step edge process adopted by Bovik [3] and Touzi[21], the edge process we utilize contains ramp edges that havetransition zones. The idealized step edge is a special case ofthe edge process used here (when ). Because the NG(or ICOV) edge detector is rotation-invariant and the specklepower spectral density function is assumed to be identical in

direction and direction, the edge performance obtainedwith vertically oriented edge function (7) applies to arbitrarilyorientated edges as well.

Two reasons for using (7) to model the underlying edge func-tion are as follows. Assume that there is a 1-D discontinuity at

in the physical parameter (e.g., the ground reflec-tivity, the tissue density, etc.) of an object being imaged. Let

, where is a 1-D unit stepfunction and constants and are, respectively, the param-eter values on both sides of the discontinuity. Consider a non-coherent imaging device with an illuminating beam of nonzerowidth. Then, the image of the parameter can be related tothe parameter by a convolution ,where is the point spread function of the imaging devicenormalized such that the area of is unity. Letting bea Gaussian function with a standard deviation , we find that

, which can berewritten as (7). Therefore, the edge scale in (7) can be pro-portional to the scanning beam width or the transmitting pulsewidth, depending on the orientation of the edge. Equation (7)also allows the modeling of Gaussian linear filtering applied toa step edge. In this case, the edge transition parameter wouldbe the standard deviation of the Gaussian kernel of the filter.

B. Statistics of the NG-Filtered Edge Process

Now, the NG filtered speckle-corrupted edge process, ac-cording to (2), is calculated by

(8)

where . To evaluate the statistics of theNG-filtered edge process, we first need the cumulative distri-bution of . Since random variables , (bothbeing zero mean Gaussian ) and (being

Fig. 2. Probability of NG edge detection on edges as a function of threshold(starred line: L = 1; dashed line: L = 2; dotted line: L = 4; dashed-dottedline: L = 8; and solid line: L = 16). (a) and (b): Theoretical plots. (c) and (d):Monte Carlo simulation plots. (a) and (c): � = 0:1. (b) and (d): � = 0:2.

) are mutually independent, we areable to find the joint pdf of and(see Appendix III)

(9)

where , , ,

, and . As a cross check, we haveproven that expression (5) can be derived from (9). Therefore,the cumulative distribution of can be computed by

(10)

where The region of the doubleintegral in (10) is a circular disk in the plane centered at( , 0) with a radius of .

Given a threshold , the conditional probability of correctlydetecting an edge when one is actually present (hypothesis )is given by

(11)

where . Fig. 2 plotsedge-detection performance for different values of and asobtained using (11) and using a Monte Carlo simulation (see


Section V.A). For the theoretical plots, the double integral (10)is numerically evaluated using the simplest Riemann sum ap-proximation. It is shown that the theoretical prediction agreeswith and Monte Carlo-simulated results. We can observe thatfor a given value of , an edge that has a larger value (higheredge contrast) is more likely to be correctly detected. Also, fora given edge, the more speckle that is removed, the wider therange of becomes over which the probability of edge detec-tion approaches unity. Of course, the range of has a limit thatis proportional to the contrast of edge.

It is seen that for , a deviation exists between the MonteCarlo and analytical results (later, in Fig. 4, the same effect canalso be observed). This discrepancy is due to the fact that theGaussian model for the cube-root transformed speckle becomesless accurate as decreases.

C. Edge-Localization Characteristics of the NG Operator

We have examined the statistical performance of the NG op-erator for edge detection in the presence of speckle. In this sub-section, we characterize the edge localization of the normalizedgradient detector in terms of the position of the peak of and thewidth of the edge response. We focus on the mean of the edgeresponse of the NG edge detector in speckle.

First, we find the mean of the NG response to edges using (8).According to Appendix II, we know that the random variable

is (even) symmetrically distributed around the origin, hence the mean of is equal to zero. Therefore, the mean

of the NG filtered edge process [i.e., the mean of (8)] is

(12)where , and is a

constant obtained by averaging over a large homoge-neous speckled region.

Using the properties of the error function, we have the fol-lowing series expansions:

(13)

where . With (13), and retaining terms up to thesecond order in , (12) can be approximated by

for (14)

From (14), we find that the peak of the average edge NG re-sponse appears approximately at the position

(15)

To illustrate the accuracy of (15), we present a numerical ex-ample. For , we have . Using (15),we find that ; while using the rigorous formula(12), we can compute numerically that .

When the contrast of an edge is lower than(i.e., 12 dB), is real and

Fig. 3. NG edge localization as function of the edge contrast parameter b.(a) Peak response bias x (b). (b) 3-dB main lobe width �x (b).The solid lines are obtained using approximate formulae and the starred linesare computed using the rigorous formulae. The bias and width measures are inunits of the edge scale �.

negative, meaning that the peak of the edge strength is bi-ased toward the darker side of the edge. So, the quantityquantifies the edge-location bias of the NG operator. Since

for , andis proportional to the normalized edge contrast , we find

that higher edge contrast corresponds to higher edge-locationbias. Besides, it is evident that the location bias is proportionalto the edge scale . It is also worthwhile to mention that theconclusions regarding the position of the edge strength peakand the higher edge contrast are in line with [5] and [7].

The gradient magnitude creates a bell shaped response at anedge, even-symmetric with respect to the edge center; and, onthe other hand, the normalizing function is odd-symmetricat the edge center. As the result of the division of the even-symmetric function by the odd-symmetric one, the edge strengthpeak gets skewed toward the darker side of the image.

If , then (15) generates complex values, meaning that(15) does not hold for edges with high contrast. This is due tothe fact that for high-contrast edges, the edge-location bias bythe normalized gradient exceeds , the transition zone width ofthe edge, and, thus, the approximation (13) is invalid.

The edge localization of the NG detector can be further quan-tified, approximately, by the 3-dB (or half-peak-value) width ofthe NG response

(16)

where is the maximum ofin the interval [ ,1], given by

(17)

Fig. 3 shows plots of the peak bias and 3-dB width of the NG-fil-tered edge signals as functions of the equivalent edge contrast. It can be observed from Fig. 3 that the peak bias formula (15)

represents a good approximation to the true peak bias, and thatthe 3-dB width expression (16) underestimates the true width

by an approximately fixed offset 0.1. Wealso observe that is insensitive to the contrast param-eter over a wide range of . We will use this fact in Section IV


to show how the Laplacian terms in the ICOV detector affectedge localization.

IV. ICOVS RESPONSE TO EDGES

WITH AND WITHOUT SPECKLE

In Sections II and III, we discussed the edge detection andlocalization performance of the NG operator. We found that theNG operator exhibits certain edge-location errors. In Section IV,we examine the ICOVS response to edges and speckle (sinceour target images are ultrasound images) and compare the edgelocalization of the ICOVS and the NG operator.

A. ICOVS Response to Speckled Edges

In the continuous domain, the ICOVS-filtered speckle-cor-rupted edge process is defined by

(18)

Using (6),and

we have (19), shown at the bottom of the page, where, ,

, ,, , and

.From Section II, we recall that is a Gaussian process

, that and are mutually indepen-dent Gaussian with zero mean and variance ,and that and or are independent. In the same manner,we know that (or ) and (or ) are mutually in-dependent Gaussian processes. The variance of or isgiven by .

Since and are mutually independent Gaussian pro-cesses, we get that the Laplacian is with

. Given these relationships, we can derivethe joint pdf, , for , , and (seeAppendix II). Hence, with the ICOVS, the probability of edgedetection at the edge position given the hypothesis that atrue edge is present is given by

(20)

Fig. 4. Probability of ICOVS edge detection on edges as a function ofthreshold (starred line: L = 1; dashed line: L = 2; dotted line: L = 4;dashed-dotted line: L = 8; and solid line: L = 16). (a) and (b): Theoreticalplots. (c) and (d): Monte Carlo simulation plot. (a) and (c): � = 0:1. (b) and(d): � = 0:2.

where is the volume bounded by

with . Similarly, the probability of false detec-tion under hypothesis for the ICOVS operator in homoge-neous speckle regions is givenby the triple integral of over the volume

Fig. 4 plots edge-detection performance for different valuesof and , as obtained using two methods: the triple integralof and a Monte Carlo simulation. Fig. 5illustrates the plots of the probability of false alarm in homoge-neous speckle areas as a function of the threshold for different

from the two methods. For the theoretical plots, all triple in-tegrals are numerically evaluated using the simplest Riemann

(19)


Fig. 5. False alarm probability of ICOVS in homogeneous speckle areas versusthreshold for different numbers of looks. (a) Curves forL = 1 and 2. (b) Curvesfor L = 4, 8, and 16. The solid lines are the theoretical results and the dashedcurves are obtained using a Monte Carlo method.

sum approximation. It can be seen that, once again, the theoret-ical and Monte Carlo plots match closely. From these plots andthe plots in Fig. 1, it is concluded that ICOVS and NG operatorshare similar statistical edge-detection performance in homoge-neous speckle regions and on edges in terms of probability offalse alarm and probability of edge detection.

B. ICOVS Response to Edges Without Speckle

We have considered the statistical responses of ICOVS atedge positions and in the regions far away from edges. Now,we turn our attention to how ICOVS responses in regions nearedges. To make the problem tractable, we restrict ourselves onlyto the edges where the speckle level is negligible. First, we for-mulate a more general form of ICOVS in the continuous domain.Then, we discuss two limiting cases of the general ICOVS, bywhich we show that the Laplacian term in the denominator ofthe ICOVS serves to increase edge-location accuracy, while therole of the Laplacian term in the numerator of the ICOVS issharpening the edge response. Finally, we consider the generalcase where both Laplacian terms are neither zero nor infinite.It is shown that the ICOVS provides superior edge-localizationperformance relative to the NG operator.

To conduct the analysis, we generalize the continuous ICOVS(18)

(21)where , and are nonnegative weighting parameters (alldimensionless). Since the represents an isotropic dif-fusion process, in (21) is simplya smoothed version of the image . Recalling the rela-tionship between a diffusion process and Gaussian filtering [8],we may approximate for smallvalues of with where

. Therefore, (21)is reformulated as

(22)

Adopting this formula for the ICOVS and lettingin (6), we can write the ICOVS response to a speckle-free edgelying along the axis in the following form

(23)

where , , and, which is the Gaussian convolution of the edge function

(24)

Within a narrow band along the edges, (24) and, thus, (23)can be further simplified. Using the series expansions

where and, and and

, and retaining terms up to the second orderin , we can approximate (23) by

for (25)

Given (25), we can analyze the performance of the ICOVS de-tector near the edge transition in cases where it is difficult toderive an analytical solution using the more rigorous formula(23).

1) Two Limiting Cases of the ICOVS Edge Response: First,we examine the edge response of ICOVS when is set to zero.Restricting ourselves to a narrow band around the edge where

, from (25), we have the approximation. We find that the edge-

location bias (the position of the peak) ofequals . Since, we have that , implying that the ICOV operator

is capable of achieving higher edge-location accuracy comparedwith the NG edge detector, i.e., the Laplacian term in the denom-inator in (21), associated with weight alone, compensates forthe edge bias of NG. With the approximation thatfor , the edge-location accuracy ofis improved approximately by a factor of comparedwith that of the NG. As an example, if , the improve-ment factor will be 1.22 for an edge with , which corre-sponds to an edge having a 2-pixel-wide transition zone.

The 3-dB width of the signal, our secondedge-localization measure, is given by

. As illustrated by

Fig. 3(b), the function for the NG edge detectoris insensitive to . Since has the same func-tional form with respect to , we have the approximation

. Therefore, we infer thatthe different values do not alter significantly the edge re-sponse width of the edge detector. Thus, we conclude thatthe Laplacian term in the denominator of the ICOVS operator


Fig. 6. Q(1; �; x; y) response, normalized to unity at x = 0, to the e(x)edge signal. The solid and dashed curves correspond to � = 2=3 and � = 0,respectively.

serves to reduce the edge position bias without increasing thelocalization error relative to the ICOVS response width.

In the second limiting case, we scrutinize

(see Fig. 6 for its plots) for which

the expression for the ICOVS, (23), and are used.Intuitively, the gradient magnitude-squared (GM) ( )generates one lobe centered on the edge, while the Laplaciansquared ( ) generates double lobes at both sides ofan edge. The absolute value of the difference of the GM and theappropriately weighted LS results in a sharpened lobe with twosmall side lobes. It is obvious that the peak edge response occurson the edge, i.e., , and the null-to-null main lobe width ofthe is given by , showing that in-creasing results in shrinkage of the main lobe ofregardless of the edge contrast . Though a larger value willgenerate a sharper edge response, the parameter cannot bechosen arbitrarily large because the height of the side lobe in-creases with parameter . We find that the peak of the side lobeis determined by , which is a mono-tonically increasing function of . The peak height of the sidelobes is greater than that of the main lobe if . When

, the peak height of the side lobe is one half of thatof the main lobe. Strong side lobes introduce edge-alike arti-facts, resulting in an increased probability of false alarms nearan edge. In practice, should be chosen such that the side lobeheight is at least less than or equal to , the average speckleresponse in homogeneous speckle regions, in order to avoid anincrease in false alarms. This illustrates the tradeoff betweenedge response width and false alarms.

2) ICOVS Edge Responses: Through two limiting cases, wehave shown that the Laplacian term associated with weightalone compensates for edge bias, and that the Laplacian term as-sociated with alone tends to narrow edge responses. Now, weaddress the edge-localization performance of the ICOVS oper-ator where the two Laplacian terms associated with the weights

and work jointly.Based upon the conclusions drawn from the above dis-

cussion, we expect that the ICOVS would achieve betteredge-localization performance than either

or . Let the main lobe of the ICOVS responselie in an interval centered on the edge where

. Within the interval , wefind via the first derivative of (25) that the position of the peakICOVS signal is a root of the following:

(26)

Since the ICOVS peak lies close to the edge (where ), weadopt the Newton’s root-finding method to solve (26). Choosingthe initial guess root as , we find that the edge-locationbias of the ICOV, up to the first-order approximation, is givenby

(27)

From (27), it is seen that the edge-location accuracy of theICOVS is improved by a factor of , com-pared with the normalized gradient edge response where thepeak response is at for

. For instance, given that , and ,the improvement factor is 1.53. Therefore, an increase in eitherLaplacian weight ( or ) reduces the edge-location error.

Following the same steps for deriving the 3-dB width of theNG operator, we find that the 3-dB width of the ICOVS edgeresponse can be approximated by

(28)where is the maximum of the function

near . Noting that

and using , from (28), we findthat the 3-dB width of the ICOV response is bounded fromabove by

(29)

Expression (29) shows explicitly that a larger value of and asmaller value of are preferable for decreasing .Fig. 7 plots the speckle-free edge-localization performance ofthe ICOVS (with and ) as function ofthe edge contrast parameter . It is seen that the edge-locationbias curve obtained using the approximate formula agrees withthat is computed using the rigorous formula, and that the 3-dBwidth curves obtained using the approximate expression (28)and the upper bound expression (29) have fixed offsets to thecurve derived from the rigorous model.

In summary, our theoretical analysis shows that the Laplacianterm associated with the weight tends to sharpen the edgeresponse of the ICOVS but may increase the probability of falsealarm near an edge if is too large (see the second limit casein Section IV-A), while the Laplacian term associated with theweight tends to reduce the bias of the ICOVS, but also maywiden the response when is too large. The ICOVS, namelyICOV, operator seeks to optimize the edge detection in speckle


Fig. 7. ICOVS speckle-free edge-localization performance as function ofedge contrast parameter b. (a) Edge position x (b) (solid line: approximated;starred line: exact). (b) 3-dB main lobe width �x (b) (solid line: upperbound; starred line: rigorous; and dotted line: approximated). The bias andwidth measures are in units of the edge scale �.

imagery in terms of low false edge-detection probability andhigh edge-localization accuracy.

C. Edge-Localization Comparison Between the ICOVSOperator and the NG Operator

As a comparison, Fig. 8 plots the analytical edge-localizationproperties of the NG and ICOVS (with and )edge detectors on speckle-free edges. In terms of the edge-loca-tion bias, the ICOVS achieves an improvement factor of approx-imately 1.5, compared with the NG. Regarding the 3-dB width,the ICOVS operator achieves an improvement of approximately10% over a wide range of values, relative to the NG operator.

V. VALIDATION

In Section V, we validate the essential statistical character-istics of the NG and ICOVS operators, the performance of theNG and ICOVS operators as applied directly to speckled US im-agery, and the performance of a practical ICOV-based edge-de-tection algorithm. A Monte Carlo method is employed to verifythe statistical characteristics of the NG and ICOVS operators.The performance of the NG and ICOVS edge detectors is ver-ified using a synthetic 1-D signal with low speckle. For vali-dating the performance the practical ICOV-based edge-detec-tion algorithm, some real ultrasound image results are exempli-fied. Furthermore, a comparative validation of the ICOV-basededge-detection algorithm is made by comparing it with an ex-isting edge-detection method.

A. Validation by Monte Carlo Method

In Sections II–IV, we have quantified the edge-detection per-formance of the NG operator and the ICOVS operator under hy-pothesis or using mathematical analysis and numericalintegration methods. It is necessary to verify those quantifica-tions using at least one alternative method. To this end, we em-ploy a Monte Carlo method, which requires generation of sim-ulated speckle pattern. Next, we will describe how speckle pat-tern is synthesized, followed by our validation procedures andresults.

1) Speckle-Pattern Synthesis: Synthetic 1-look B-scanspeckle intensity data is formed by the simulator describedin [24, Subsection A, Section IV]. We chose the simulator

Fig. 8. Edge-localization performance comparison between the NG andICOVS operators. (a) Comparison of edge position bias. (b) Comparison of3-dB edge response width. The solid lines are for NG while the dashed forICOVS. (c) and (d) Edge-localization improvement of ICOVS over NG (lengthmeasures are in units of the edge scale �).

Fig. 9. Comparison of theoretical and measured statistics of synthetic 1-lookspeckle. (a) Theoretical and measured (in both lateral and axial directions)autovariance functions versus lags. (b) Normalized histogram and exponentialpdf.

parameters as m s (the speed of sound in tissue),MHz (the center frequency), 2 pixels (the pulse

width of transmitting ultrasonic wave), and 2 pixels (thebeam width of transmitting ultrasonic wave). The ultrasoundcross-section distribution where isa Gaussian white noise field with zero mean and unity variance.

2) Validation of the Probability of False Edge Detection inHomogeneous Speckle Regions: To calculate the probabilitiesof false alarm of the NG or the ICOVS operator in -lookspeckle, a single-look speckle image of 256 256 pixels is firstsynthesized. The -look dataset is then generated by filteringthe single-look data with a Gaussian filter with a standarddeviation [where is determined by (AI.5)]. For the NG(or ICOVS) operator, the Monte Carlo-predicted probability offalse edge detection at a given threshold is the fraction of


Fig. 10. One-dimensional speckled-edge detection. (a) A speckled-edge signal in 1-D. (b) Discrete gradient magnitude (GM) signal and normalized gradient(NG) signal. (c) Edge signal from the 1-D discrete ICOVS.

pixels whose (or ) values are greaterthan the threshold among the total number of pixels in theimage, i.e., 256 . The probabilities of false edge detection forthe NG operator and the ICOVS operator under hypothesisresulted from the Monte Carlo method have been plotted inFigs. 1 and 5, respectively.

To ensure correct generation of speckle, we monitor theaverage empirical autocovariance functions (ACF) in lateraland axial directions as well as the histogram of synthesized1-look speckle pattern. To this end, empirical autocovariancefunctions [3] are computed over ten 90 90 windows in the1-look speckle image. Fig. 9(a) illustrates the averaged, radialand lateral empirical ACFs, together with theoretical ACF (thesame in radial and lateral directions), versus lag (in units ofpixels), and Fig. 9(b) shows the intensity histogram (normalizedto unity at ) and the exponential pdf. The figures exhibitclose matches between the computed statistics and the statisticspredicted by theory.

3) Validation of the Probability of Edge Detection at EdgePositions: A strip-shaped image (2048 32 pixels) with edgesalong the long axis of symmetry is synthesized to assess theedge-detection performance at edge positions under hypothesis

. Specifically, the image dataset is created by multiplyingthe edge function [see (7)] with a synthesized multilookspeckle pattern. It can be seen that possible edge positions areat the 2048 2 pixels on both sides of the long symmetric axisof the image. With the NG (ICOVS) detector, given a giventhreshold , the edge-detection probability under hypothesis

is estimated as the fraction of possible edge pixels with(or ) values being greater than the

threshold . The probabilities of edge detection at edge posi-tions have been plotted in Figs. 2 and 4.

B. Experimental Validation: Edge Detection From 1-D Signal

Section V-B and the Section V-C are devoted to experimentalvalidation of the ICOV operator. In Section V-B, we considerdetecting the edge from a 1-D synthetic digital signal withrelatively low speckle:where . According to (7), the edgescale parameter for this signal. Fig. 10(a) plots the1-D speckled signal and Fig. 10(b) the corresponding edgestrength signals generated by the discrete gradient magnitudesquared (GM) ( , where is a differenceapproximation to the derivative with respect to ) operator and

the discrete NG ( operator, respectively. InFig. 10(b), it is seen that the edge strength signal computed byNG is well balanced on both high- and low-mean signal sidesof the edge indicating the constant false alarm rate (CFAR)property, while those generated by the GM are not CFAR inhomogeneous speckle areas of different means. However, themain lobe of the NG response is skewed with the position ofpeak edge being biased toward the darker side of the edge.Fig. 10(c) is obtained by applying the 1-D discrete ICOVS

to the1-D signal. It is seen that this edge detector allows for bal-anced and well localized edge strength measurements in brightregions, as well as in dark regions.

It is interesting to compare the edge localization obtained bydiscrete NG and ICOVS with theoretically-predicted edge lo-calization of NG and ICOVS. Zooming in Fig. 10(b) and (c),we find that the edge-location bias is 1 pixel for discrete NGand zero for discrete ICOVS. For the 1-D signal in Fig. 10(a),we know that (or, equivalently, ), giving rise tothat the edge-location bias of the NG is equal to

pixels. Using and , we findpixels, the edge-location bias of ICOVS. For this example, it isseen that the theoretical edge localization of the NG and ICOVSoperators agree satisfactorily with experimental localization ofthe discrete NG and ICOVS operators, considering the finite dif-ference approximation to derivative.

C. Experimental Validation: Edge Detection From US Imagery

We continue experimental validation of the efficacy of theICOV operator using real US images. In Section V-B, theICOV operator was applied directly to the speckled signalin which the speckle is relatively low in magnitude. Directlyapplying the ICOV operator to US images with high degrees ofspeckle usually produces numerous spurious edges and missesweak edges, a scenario similar to when directly performinggradient- or Laplacian-based edge detection in noisy opticalimagery. Thus, an indirect method is applied. Our methodutilizes anisotropic diffusion with the ICOV as an implicit edgedetector (in the diffusion coefficient), analogous to anisotropicdiffusion with a gradient-based diffusion coefficient for im-agery with additive noise. In Section V-C, we first present sucha practical ICOV-based edge-detection algorithm for speckledUS imagery, and then test the algorithm on four medical USimages. Finally, we compare ICOV with a ratio-edge detector.


1) ICOV-Based Edge-Detection Algorithm: The algorithmbegins with solving the following partial differential equation(PDE)

(30)

where represents the divergence, denotes the gradient,is the intensity of an input image at location

in a Cartesian coordinate system, denotes the border of theimage domain , is the outward normal vector to , andis the diffusive coefficient.

(31)

with the edge detector being defined by

(32)

where , and are positive parameters properly chosen suchthat the discrete version takes the form of (1). The scale function

serves as a threshold governing the magnitude of the ICOVrequired for an edge. We utilize tools from robust statistics toautomatically estimate as in [17]

(33)where

median mediandenotes the median absolute deviation and the constant( ) is derived from the fact that the MAD of azero-mean normal distribution with unit variance is 1/1.4826.The estimator (33) is derived by taking into account thesimilarity of the square root of the NG and the ICOV andassuming that the logarithm of image intensity is a piecewiseconstant function that has been corrupted by zero-meanGaussian noise. Then, the edge strength image is extracted by

(34)

where denotes a binary image obtained by thresh-olding at level and the is the pointwise multiplicationoperation. Finally, an edge map is formed bywhere is a predetermined threshold.

The partial differential (30) is solved numerically using aJacobi iterative method. Choosing a sufficiently small time-step

and a grid-size in both and directions, we discretizethe time and space coordinates as , ,

, , , ,where is the area of the image domain. Let

. A numerical approximation to (30) is given bythe following:

(35)

where

(36)

(37)

(38)

(39)

(40)

(41)

(42)

To avoid distortion at the image boundaries, symmetricboundary condition is required by which we mean that theimage intensity function has equal values at both sides of theboundary. The numerical solution becomes stationary when

where is a preset small positive number. The setcalculated from the stationary

diffused image forms an edge strength image in which edgesare points of significant values.

2) Experimental Results: In Section V-C 2), we demon-strate the performance of the proposed ICOV-based edge-de-tection algorithm. Fig. 11 illustrates four experiments usingB-scan images of a human throat, a human prostate, a phantomprostate with implanted radioactive seeds (as in brachytherapyof prostate cancer), and the left ventricle of a murine heart.The first row shows, from left to right, the B-mode image ofa human throat, its diffused version, and the extracted ICOV,respectively. The second to the fourth rows illustrate the imageand corresponding diffused image and edge strength image fora human prostate, a prostate phantom with implanted radioac-tive seeds, and the left ventricle of a murine heart, respectively.Since the proposed algorithm needs envelope-detected intensityimagery as its input, the dynamically-compressed B-modeimages are decompressed approximately by taking the expo-nential of the B-mode image divided by 25 before being inputinto the algorithm. In our numerical experiments, we generallychoose the parameters as follows: , , and

. It is seen qualitatively that significant edgestrength is distributed along the actual boundaries in all fouredge strength images consistently.

D. Comparison

To better justify the usefulness the ICOV-based edge-detec-tion algorithm, it is necessary to compare the algorithm withexisting algorithms for detecting edges in speckle imagery. InSection V-D, we compare the edges obtained by the RoA edgedetector of Bovik [3] with those obtained by the ICOV-basededge-detection algorithm. The ratio edge detector of Touzi et al.[21] and the likelihood edge detector of Oliver et al. [12] hadsimilar performance as the RoA (as tested empirically in our re-search); therefore, we report the results of only one of such ratiodetector.

The RoA operator [3] measures the edge strength by

(43)


Fig. 11. Experimental results of edge detection from four B-mode US images using the ICOV-based detection algorithm. First column: Images of a human throat,a human prostate, a prostate phantom with implanted radioactive seeds, and the left ventricle of a murine heart, respectively, from top to bottom. Second column:Corresponding diffused images. Third column: ICOV edge-strength images.

with

(44)

(45)

where , , , and are the mean in-tensities in the subwindows immediately to the right, left, top,and bottom of image coordinate , respectively. Fig. 12 il-lustrates the left, right, top, and bottom subwindows of a 7 7analyzing window. For an arbitrary window (where isan odd number), the subwindows are either or

. An edge is declared to be present at coordinateif where is a predetermined threshold.

Fig. 12. Four subwindows of a 7� 7 analyzing window centered at the crossedpixel.

The procedure of the comparison is as follows. (1) Run twoalgorithms on an input image to obtain two edge strength im-ages. (2) Enhance each edge strength image by full-scale con-trast stretching (with 256 gray levels). (3) Identify edges as top


Fig. 13. Comparison of the ICOV-based edge-detection algorithm and the RoA detector on a phantom US image. (a) Phantom US image (lefthand side of theimage is closer to US transducer; scan is in the vertical direction). (b) ICOV edge strength image. (c)–(e) ICOV detected edge maps (top 5%, 1%, and 0.6% brightestin ICOV edge strength image). (f) Ground true edges. (g) RoA edge strength using a 39� 39 window. (h)–(j) RoA detected edge maps (top 5%, 1%, and 0.6%brightest in RoA edge strength image).

Fig. 14. Sensitivity of algorithm performance with respect to parameters. (a) For the proposed method. (b) For RoA.

% of the brightest pixels in each edge strength image. (4) Quan-tify the detection performance in terms of Pratt’s FoM [16].

The comparison is made using a 2-D slice of real data ob-tained from imaging an ellipsoidal phantom. The data were ac-quired using a Sonos 5500 US system (Agilent Technologies,San Jose, CA). One full frame of raw (RF) data contains 259lines axially, with each line having 3680 pixels in the axial direc-tion, and each pixel signal coded with 16 bits. An envelope-de-tected amplitude image is first formed. To reduce the volumeof the data, we down sample (using a 7 tap FIR low-pass fil-tering and 4:1 decimation) the image by a factor of 4 in therange direction, yielding an image of 259 920 pixels. Then,we extract a subimage of 220 396 pixels as the test image forwhich a time-gain-compensation (TGC) is also carried out tocalibrate residual propagation attenuation losses. Fig. 13 depictsthe edge strength images obtained by two edge-detection algo-rithms using the phantom image. The first row shows, respec-tively and from left to right, the noisy, attenuation-calibratedphantom image, the ICOV edge strength image, and thre re-

sultant edge maps. The second row shows from left to right,the manually-drawn boundaries representing the ground truthedges, the RoA edge strength image, and three resultant edgemaps, respectively. In the ICOV-based algorithm, the time stepshould be chosen so that the numerical implementation is stableand converges rapidly. We choose to optimize thealgorithm performance in terms of Pratt’s FoM. To implementthe RoA edge detector, the size of the window needs to be spec-ified. A larger window will reduce spurious edges at the costof degraded edge localization on real edges. On the contrary, asmaller window size will increase edge localization but producemore spurious edges. Depending on the size of the interestingobject in an image, the filter window is chosen to achieve a bal-ance between the allowable number of spurious edges and de-sired edge-localization performance. The elliptical shape to bedetected has a long axis of 200 pixels and minor axis of 120pixels, so we choose a 39 39 analyzing window, an optimal inthe empirical sense in the range of 9 9 through 57 57 win-dows. Since the ground-truth edge pixels in Fig. 13(f) makes


up 0.53% of the test image, we set the threshold at the levelsuch that top 0.6% brightest pixels are detected as edges in theedge strength image. As a reference, we also show edge mapsthat are detected as top 1% or 5% most significant pixels in edgestrength images.

In addition to visually comparing the ICOV-based detectionalgorithm and the RoA operator, we use Pratt’s FoM [16] toquantify their edge-detection performance. Metric FoM penal-izes both the number of incorrect edge detections and the errorsof edge location in an edge map based on a map of ground-trueedges. For the phantom image, the ground truth edges shouldbe on the boundary of the elliptical shape which is drawn manu-ally and shown in Fig. 13(f). From the top 0.6% most significantedge maps and the ground-true edge map, we find a FoM valueof 0.64 for the proposed algorithm and a FoM value 0.27 withthe RoA operator for the example image.

In our comparison, we have considered the sensitivity of thetwo algorithms to changes in their parameters and have chosenthe optimal parameters for each algorithm. Fig. 14 shows theplots of FoM versus time step for ICOV-based algorithm andFoM versus window size for the RoA operator. In the left plot,it is seen that when the performance of the ICOV-based algorithm is best for the test image; in the right plot, wesee that is optimal for the RoA detector. The steepdropoff of FoM (the top 0.6% curve) after indicatesthat the ICOV-based algorithm has become errant due to numer-ical instability.

From the comparison, we see that the ICOV-based algorithmoutperforms the RoA detector in terms of significantly higherFoM. We also observe that the ICOV-based algorithm is sensi-tive to the time step chosen for discrete implementation.

VI. CONCLUSION

The edge-detection and edge-localization performance of theNG and ICOVS operators have been examined. The study showsthat the NG and ICOVS operators are constant false alarm edgedetectors, and that the edges detected by the ICOVS detectormanifest reduced localization errors compared with those de-tected by the NG operator. Quantitative improvements of theICOVS in terms of the location of the peak and the width ofthe ICOVS response have been derived relative to those of theNG operator. The Laplacian operator in the ICOVS operator hasbeen identified as the compensating factor for edge-localizationerror. A Monte Carlo simulation is performed to validate theanalysis of the NG and ICOVS and shows that the analysis is inagreement with the Monte Carlo simulated results. A practicalICOV-based edge-detection algorithm for US imagery has beenimplemented by embedding the ICOV operator in anisotropicdiffusion process. Encouraging experiments have been obtainedwith the ICOV-based edge detection on US images, as comparedto a ROA-type edge-detection method for speckled imagery.

APPENDIX I

Using the method described in [22], and approximating theaxial point spread function (PSF) of the RF pulse envelope shape

and the transverse PSF of the returned echo amplitude to beGaussian with standard deviations and , respec-tively, we find the autocovariance function for B-scan single-look speckle-intensity image is

(AI.1)Note that the noise power spectrum density function (PSD) ofthe single-look speckle is the Fourier transform of

, so we have

(AI.2)-look speckle-intensity can be obtained by con-

volving with a 2-D Gaussian kernel having standarddeviation (where is to be determined); the autocovariancefunction of the -look intensity fluctuation is then given by

(AI.3)

where and .

Taking the Fourier transform of yields the PSDfor the -look speckle intensity

(AI.4)Noting that the variance of the multilook speckle

, we get the relationhsip be-tween and

(AI.5)

Therefore, if , (AI.3) can be reduced to

(AI.6)

Next, we derive the noise power spectrum for the Wilson-Hilferty transformed speckle pattern . If

and are the -look intensities of speckle at two pointsand , then the second-order pdf is

(AI.7)

where is the modified Bessel function of th order.is the (correlation coefficient) modulus of the autocorrelation


function of the normalized to unity at lag valuesand , i.e.

Expanding the Bessel function in the joint pdf and integrating,we obtain the autocorrelation function

where is a hypergeometric function. Since, we have

Now

Retaining terms up to the first order in , we get

(AI.8)

Taking the Fourier transform of (AI.8) yields

(AI.9)

APPENDIX II

For a real-valued Gaussian random process with auto-correlation function , the cross correlation of the productof and its first derivative is given by

Denoting the expectation of by , we find that

since is zero mean, indicating that and are

uncorrelated at any time . Since at a given time, both andare Gaussian random variables, we know that they must

be mutually independent.

APPENDIX III

Suppose that three Gaussian random variables, , and are

mutually independent. The mean of , , is nonzero.The joint pdf for , and can be expressed by

.We now define two ratio random variables and

. Substituting for and for , we find thatthe joint pdf for , and is given by

. The marginal pdf for and can beobtained by doing the following integral [13]

in which term is the Jacobian of the variable transformation.Inserting the pdfs of the random variables , and intoabove integral, after simplification, yields

where and .Similarly, for four mutually independent Gaussian random

variables , , ,and , if defining that , ,and , we can find that the joint pdf function of ,

and is given by

where ,and .

REFERENCES

[1] F. Abramovich and P. Bayvel, “Some statistical remarks on the deriva-tion of BER in amplified optical communication systems,” IEEE Trans.Commun., vol. 45, pp. 1032–1034, Sept. 1997.

[2] A. Baraldi and F. Parmiggiani, “A refined gamma MAP SAR specklefilter with improved geometrical adaptivity,” IEEE Trans. Geosci. Re-mote Sensing, vol. 33, pp. 1245–1257, Aug. 1994.


[3] A. C. Bovik, “On detecting edges in speckle imagery,” IEEE Trans.Acoust., Speech, Signal Processing, vol. 36, pp. 1618–1627, Oct. 1988.

[4] R. A. Brooks and A. C. Bovik, “Robust techniques for edge detectionin multiplicative Weibull image noise,” Pattern Recognit., vol. 23, pp.1047–1057, 1990.

[5] J. Canny, “A computational approach to edge detection,” IEEE Trans.Pattern Anal. Machine Intell., vol. 8, pp. 679–698, June 1986.

[6] R. FjØrtoft, A. Lopés, P. Marthon, and E. Cubero-Castan, “An optimalmultiedge detector for SAR image segmentation,” IEEE Trans. Geosci.Remote Sensing, vol. 36, pp. 793–802, May 1998.

[7] O. Germain and P. Réfrégier, “Edge localization in SAR images: Perfor-mance of the likelihood ratio filter and accuracy improvement with anactive contour approach,” IEEE Trans. Image Processing, vol. 10, pp.72–78, Jan. 2001.

[8] R. M. Haralick, “Zero crossing of second directional derivative edgeoperator,” in Proc. SPIE Symp. Robot Vision, vol. 336, Washington, DC,May 1982, pp. 91–99.

[9] A. Lopes, E. Nezry, R. Touzi, and H. Laur, “Structure detection and sta-tistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens.,vol. 14, pp. 1735–1758, 1993.

[10] D. Marr and E. Hildreth, “Theory of edge detection,” Proc. Roy. Soc.,vol. B207, pp. 187–217, 1980.

[11] J. L. Morel and S. Solimini, Variational Methods in Image Segmenta-tion. Boston, MA: Birkhauser, 1995.

[12] C. J. Oliver, I. McConnell, D. Blacknell, and R. G. White, “Optimal edgedetection in SAR,” in Proc. Int. Conf. Remote Sens., vol. 2584, Paris,France, 1995, pp. 152–163.

[13] A. Papoulis, Probability, Random Variables, and Stochastic Processes,2nd ed. New York: McGraw-Hall, 1984, p. 137.

[14] M. Petrou, “Optimal convolution filters and an algorithm for the detec-tion of wide linear features,” Proc. Inst. Elect. Eng., vol. 140, no. 5, pp.331–339, Oct. 1993.

[15] L. J. Porcello, N. G. Massey, R. B. Innes, and J. M. Marks, “Specklereduction in synthetic-aperture radars,” J. Opt. Soc. Amer., vol. 66, pp.1305–1311, 1976.

[16] W. K. Pratt, Digital Image Processing. New York: Wiley, 1977.[17] P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outlier Detec-

tion. New York: Wiley, 1987.[18] M. Simard, S. Saatchi, and G. DeGrandi, “The use of decision tree

and multiscale texture for classification of JERS-1 SAR data overtropical forest,” IEEE Trans. Geosci. Remote Sensing, vol. 38, no. 5,pp. 2310–2321, Sept. 2000.

[19] M. Simard, G. DeGrandi, and K. Thomson, “Adaptation of the wavelettransform for the construction of multiscale texture maps of SAR im-ages,” Can. J. Remote Sens., vol. 24, pp. 264–265, Sept. 1998.

[20] A. Stuart, J. K. Ord, and S. Arnold, Kendall’s Advanced Theory of Sta-tistics , 6th ed. New York: Oxford Univ. Press, 1984, vol. 2A, pp.563–564.

[21] R. Touzi, A. Lopès, and P. Bousquet, “A statistical and geometrical edgedetector for SAR images,” IEEE Trans. Geosci. Remote Sensing, vol. 26,pp. 764–773, Nov. 1988.

[22] R. F. Wagner, S. W. Smith, J. M. Sandrik, and H. Lopez, “Statistics ofspeckle in ultrasound B-scans,” IEEE Trans. Son. Ultrason., vol. 30, pp.156–163, May 1983.

[23] E. B. Wilson and M. M. Hilferty, “The distribution of chi-square,” Proc.Nat. Acad. Sci., vol. 17, pp. 684–688, 1931.

[24] Y. Yu and S. T. Acton, “Speckle reducing anisotropic diffusion,” IEEETrans. Image Processing, vol. 11, pp. 1260–1270, Nov. 2002.

Yongjian Yu (SM’04) received the B.S. degree inphysics and the M.S. and Ph.D. degrees in electricalengineering from the University of Electronic Sci-ence and Technology of China (UESTC), Chengdu,Sichuan, China, in 1982, 1985, and 1989, respec-tively, and the Ph.D. degree in electrical engineeringfrom University of Virginia, Charlottesville, in 2003.

He was a member of the faculty of UESTC from1989 to 1999, where he held the position of AssociateProfessor of electronic engineering until 1993. FromAugust 1993 to December 1995, he was a Visiting

Scholar with the Remote Sensing Group, Alenia Spazio SpA, Rome, Italy. From1999 to 2000, he was a Research Assistant with the Department of Electricaland Computer Engineering, Oklahoma State University, Stillwater. Since 2003,he has been a Postdoctoral Fellow with the Department of Radiation Oncology,University of Virginia Health System, Charlottesville. His current research inter-ests include medical imaging, and signal, image, and video processing systems.

Dr.Yu received the Science and Technology Progress Award from the Min-istry of Electronics Industry of China in 1994 and the National Scientific andTechnological Progress Award of China in 1999.

Scott T. Acton (SM’99) received the B.S. degree inelectrical engineering from Virginia Tech, Blacks-burg, in 1988, and the M.S. and Ph.D. degreesin electrical and computer engineering from theUniversity of Texas, Austin, in 1990 and 1993,respectively, where he was a Microelectronics andComputer Development Fellow.

He was in industry with AT&T, Oakton, VA, theMITRE Corporation, McLean, VA, and Motorola,Inc., Phoenix, AZ, and in academia with OklahomaState University, Stillwater. He is the 2004 Technical

Program Chair for the 38th Asilomar Conference on Signals, Systems, andComputers. His research interests include anisotropic diffusion, active contours,biomedical segmentation problems, biomedical tracking problems, and war.

Dr. Acton received an ARO Young Investigator Award for his research invideo tracking. He also received the Halliburton Outstanding Young FacultyAward in 1998, he was named the Eta Kappa Nu Outstanding Young Elec-trical Engineer in 1997—a national award that has been given annually since1936—and, at the University of Virginia, Charlottesville, he was named the Out-standing New Teacher in 2002, elected a Faculty Fellow in 2003, and holds theWalter N. Munster Chair in electrical and computer engineering and biomedicalengineering. He is the recipient of a Whitaker Foundation Biomedical Engi-neering Research Grant for work in cell detection and tracking. He served as anAssociate Editor for the IEEE TRANSACTIONS ON IMAGE PROCESSING and cur-rently serves as an Associate Editor for the IEEE SIGNAL PROCESSING LETTERS.

1640 ieee transactions on image processing,...

Documents