Edge Detection in Ultrasound Imagery using

the Instantaneous Coefficient of Variation

Yongjian Yu and Scott T. Acton†

Abstract: The instantaneous coefficient of variation (ICOV) edge detector, based on normalized gradient and Laplacian

operators, has been proposed for edge detection in ultrasound images. In this paper, the edge detection and localization

performance of the ICOV-squared (ICOVS) detector are examined. First, a simplified version of the ICOVS detector, the

normalized gradient magnitude-squared (NG), is scrutinized in order to reveal the statistical performance of edge detection and

localization in speckled ultrasound imagery. Both the probability of detection and the probability of false alarm are evaluated

for the detector. Edge localization is characterized by the position of the peak and the 3 dB width of the detector response.

Then, the speckle edge response of the ICOVS as applied to a realistic edge model is studied. Through theoretical analysis, we

reveal the compensatory effects of the normalized Laplacian operator in the ICOV edge detector for edge localization error. An

ICOV-based edge detection algorithm is implemented in which the ICOV detector is embedded in a diffusion coefficient in a

anisotropic diffusion process. Experiments with real ultrasound images have shown that the proposed algorithm is effective in

extracting edges in the presence of speckle. Quantitatively, the ICOVS provides a lower localization error, and qualitatively, a

dramatic improvement in edge detection performance over an existing edge detection method for speckled imagery.

Index terms ― Edge detection, instantaneous coefficient of variation, speckle, ultrasonic image.

Submitted to IEEE Transactions on Image Processing ____________________________________ Yongjian Yu is with the Dept. of Radiation Oncology, UVA health system, Medical Center, P.O. Box 800375, 1335 Lee Street, University of Virginia, Charlottesville,Virginia 22908. Phone (434) 243-9423, Fax (434) 982-3520, yy6b@virginia.edu. †S.T. Acton is with the Dept. of Electrical and Computer Engineering and the Dept. of Biomedical Engineering, 351 McCormick Road, University of Virginia, Charlottesville, Virginia 22904. Corresponding author: Phone (434)982-2003, Fax (434) 924-8818, Email acton@virginia.edu.

Permission to publish this abstract separately is granted.

1

I. INTRODUCTION

Medical ultrasound (US) has been widely used for imaging human organs (such as the heart,

kidney, prostate, etc.) and tissues (such as the breast, the abdomen, the muscular system, and tissue in the

fetus during pregnancy). US imaging is real-time, non-radioactive, non-invasive and inexpensive.

However, US imagery is characterized by low signal to noise ratio, low contrast between tissues and

speckle contamination. In general, medical US imagery is hard to interpret objectively. Thus, automatic

analysis and interpretation of US imagery for disease diagnostics and treatment planning (e.g., in prostate

cancer brachytherapy) is desirable and of clinical value. An essential step toward automatic interpretation

of imagery is detecting boundaries of different tissues. Though in general, the boundary of an object can

be a combination of step edges, ridges, ramp edges, etc., we focus upon detecting the boundaries of

human organs that can be modeled as step edges.

Marr and Hildreth [10] and, later, Haralick [8] have examined 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 a near-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 the LoG operator do not have the property of constant false alarm rate in

homogeneous speckle regions of speckled imagery. It has been argued that the application of such

detectors generally fails to produce desired edges from US imagery [3][6][12][21]. Some constant false

alarm rate (CFAR) edge detectors for speckle clutter have been proposed, including the ratio of averages

(ROA) detector [3], the ratio detector [21], the ratio of weighted averages [6], or the likelihood ratio (LR)

[7][12]. Other ratio detectors include the refined gamma maximum a posteriori detectors [9] and more

recent improvements [2], which use a combination of even-symmetric and odd-symmetric operators to

extract step edges and thin linear structures in speckle. With CFAR edge detectors, the image needs to be

scanned by a sliding window composed of several differently oriented splitting sub-windows. The

accuracy of edge location for these ratio detectors depends strongly on the orientation of the sub-windows

2

with respect to edges. For the LR detector, an edge bias expression has been derived in [7]. The bias in

edge location is deleterious in obtaining quantitative estimates of the volume of the organs from

diagnostic US imagery.

In an attempt to develop a more efficient edge detector with high edge positioning accuracy for US

imagery, we turn our attention to differential/difference operators that are straightforward to compute in

small windows. We believe that the key problem in developing differential type edge detectors is one of

correctly accommodating the multiplicative nature of speckle. In [24], the use of a new partial differential

equation (PDE) based speckle reducing filter for enhancement of US imagery is proposed. This filter

relies on the instantaneous coefficient of variation (ICOV) to measure the edge strength in speckled

images. Denoting the image intensity at position (i, j) as I , the instantaneous coefficient of variation

is given by

ji,

jiq ,

( )

( )2,

2,

2,

22,

,

)4/1(

)16/1()2/1(

jiji

jiji

ji

II

IIq

∇+

∇−∇= , (1)

where , , ∇ 2∇ and | | are the gradient, Laplacian, gradient magnitude and absolute value, respectively.

Specifically, 2

, jiI∇ = [ ]2,

2,5.0 jiji II +− ∇+∇ where [ ]1,,,1,, , −−− −−=∇ jijijijiji IIIII

jiI ,1 4−

,

; and ∇ . The derivation of (1) can

be found in [24]. It is seen that the ICOV (1) combines image intensity with first and second derivative

operators, which are well-known in the existing literature.

[ ]jijijiji IIII ,1,,, , −−∇ ++ jiI ,1+= jiji II ,1, + −+jijiij III ,1,12 ++= −+

The ICOV is meant to allow for balanced and well localized edge strength measurements in bright

regions as well as in dark regions. Experimentally, the performance of the ICOV has been demonstrated

for edge-preserving speckle reducing anisotropic diffusion (SRAD) [24] on US and radar imagery.

However, the edge detection mechanism and performance of the ICOV has not been analyzed

quantitatively in terms of some figure of merits. The objective of this paper is to provide a solution to this

3

problem. Because of the complexity of the ICOV, we will study the ICOV-squared (ICOVS) and make

the assumption that the edge-detection performance of the ICOV can be derived from that of the ICOVS,

taking into account that the squaring operation is a one-to-one monotonic mapping for any positive

function.

Quantifying the performance of the ICOVS in a closed, exact form is difficult. In this paper, we

firstly consider a special case of the ICOVS, the normalized gradient magnitude-squared operator (NG),

for which both the statistical detection performance and the edge localization performance are examined.

(We do not consider popular wavelet based approaches such as [18] and [19].) Appropriate statistical

models for speckle and edge processes are established for the performance analysis, and the probability

density functions in homogeneous speckle regions and edge regions are derived for the NG operator. The

edge localization performance for the NG operator is characterized by the position of the peak and the 3

dB width of the mean detector response. Then, the same calculations and analysis are carried out for the

ICOVS. In particular, the Laplacian terms in the ICOV are studied carefully to identify their roles in edge

detection.

To validate the quantification of the NG and the ICOVS, the theoretical statistical performances

are compared with those given by Monte Carlo simulation. Also, the NG and ICOVS operators are

directly tested on a synthetic 1-D signal with low speckle. For validating the capability of the ICOV edge

detector for US images, a practical ICOV-based edge detection algorithm is implemented, by embedding

the ICOV detector in a speckle reducing anisotropic diffusion process. Validation using medical B-mode

and phantom images is provided for the ICOV-based edge detection algorithm and compared to the

performance of an existing algorithm.

II. THE NG OPERATOR IN HOMOGENEOUS SPECKLE

In this section, we analyze the statistical performance of the NG (a special case of the ICOVS)

operator as applied to speckle patterns in homogeneous regions, following the method of Bovik [3]. First

4

we summarize the image and speckle models and the basic assumptions used as the framework for

quantifying the detector performance in speckle. Then, we examine the NG operator in homogeneous

speckle regions.

A. Models for Image and for Speckle and Basic Assumptions

I(x, y) represents a random process that models the observed intensity at location (x, y) in an

image. Using the multiplicative model, we write I ),(),(),( yxSyxRyx = where R(x, y) is a deterministic

function governing the underlying reflectance of the object being imaged and S(x, y) is a wide-sense

stationary random process describing the normalized speckle process. It has been shown [15] that the

normalized speckle process S(x, y) is gamma distributed with PDF ( ) LsLL esLL −−Γ= 1)(S sρ ( ∞≤≤ s0

)(s

)

where L is the equivalent number of looks. According to the central limit theorem, for large L the gamma

PDF can be approximated by a Gaussian: N (1, 1/L). Because the rate of convergence of Sρ to N (1,

1/L) is O , this Gaussian approximation is not appropriate for small values of L (usually, when L <

10). For L<10, the PDF of the cube-root of S, , can be approximated by a Gaussian: N (1-1/(9L),

1/(9L)) [23]. The rate of convergence of the PDF of the standardized Ŝ to the standard normal distribution

is O , making Gaussian a satisfactory approximation for Ŝ for L<10 [1].

( 2/1−L

)2/3−

)

3/1S=S

(L

When studying the speckle properties, it is necessary to know not only the first-order

characteristics of the speckle described by the single point PDFs, but also the correlation properties of the

speckle. The correlation is described by the speckle power spectral density (PSD) function, which is the

Fourier transform of the autocovariance function of S(x, y). However, there is no general model of the

spectra of image speckle for an arbitrary imaging modality. Various models have been derived for the

spectra of US, synthetic aperture radar, or laser speckle patterns. A common feature of these speckle

processes is that they can be modeled as band-limited processes containing only lower spatial frequencies.

Without loss of generality, we consider the spectra of a B-scan US. We approximate the speckle PSD

5

functions of S(x, y) by [ ]22222 /)(2exp/2),( Ω+−= yxcyxS fffff ππW where Lfc=Ω (units:

cycles/meter) is the noise bandwidth for the L-look data, and are spatial frequencies (units:

cycles/meter) in the x-direction and y-direction, respectively. The noise bandwidth for single look data f

xf yf

c,

assumed to be equal in both axial and lateral directions, is related to the transducer dimension D in the

transverse direction, the wavelength λ of the illuminating beam, and the distance y0 from the transducer to

the focal zone by )0y(Dfc λ=

( xS fW ,)2 ⋅

in the lateral direction (or to the RF pulse envelope shape by =2/ξcf p

where ξp is the pulse width in the axial direction). Accordingly, the PSD for Ŝ(x, y) can be derived as

where )yfSyxS ffW (),( ˆˆ ≅ µ [ ]))(3()3/1( 3/1ˆ LLLS Γ+Γ=µ is a factor dependent on only L.

The derivation of these relations can be found in Appendix I.

B. Statistics of the NG-Filtered Homogeneous Speckle Process

In a homogenous speckle region, the reflectivity function is uniform: µ≡),( yxR where µ is a

positive constant. Denoting the NG operator as F(x, y), we find that the NG edge detection response yields

( )[ ] [ ]2

2

2

2

22

2

22

),(ˆ

),(ˆ9

),(

),(

),(

),(,

yxS

yxSK

yxS

yxSK

yxI

yxIKyxF

∇=

∇=

∇= , (2)

where K is a scaling constant, which can be set to unity if we only deal with images in the continuous

domain. However, when (2) is applied to a sampled image defined on a grid with a sampling frequency

in both directions, since finite differences are used to approximate the derivatives in (2), the discrete

form of (2) will contain a factor ( . Therefore, for convenience, we preset the constant K to

sΩ

2)sΩ sΩ/1 ,

so that the discrete form of (2) takes a simple form to facilitate use in digital domain. Note that (2) is a

simplification to the squared ICOV presented in (1). The analysis of (2) sheds light on and provides

instructive basis for the analysis of the ICOV-squared, since the edge detection performance will be

approximately identical in both cases. In this subsection, based on the properties of Ŝ, we first derive a

probability density function for a normalized F(x, y) and then present the formulae for evaluating and

6

bounding the probability of falsely detecting an edge in homogeneous speckle regions using the detector

F(x, y). Finally, typical probabilities of false edge detection with F(x, y) are evaluated and plotted.

Instead of seeking the PDF for F(x, y), we derive the PDF for the statistic

[ ]2ˆ

2 )ˆ(18ˆ SSW S Ω∇≡ µ

),(ˆ yxS

, so as to take advantage of a known PDF. By approximating the speckle

process by a Gaussian process with mean (1-1/(9L)) and variance 1/(9L) and a Gaussian power

spectrum density having the cutoff frequency Ω, it has been shown that xyxSyxx ∂∂= ),(ˆ),(S and

yyyxS y ∂),,(ˆ

∫∫R yx ff ),4 2π

xS∂= (ˆ)

2 Sx Wf (ˆ2

are mutually independent zero-mean Gaussian random processes with variance

=yxdfdf LS2

ˆ )( Ωµ [13]. In appendix II, it is shown that is independent of

(or ), leading to the conclusion that

),(ˆ yxS

) (S y,(ˆ yxS x ), yx2

),(ˆ yxS∇ and [ ]2),(ˆ yxS are independent.

According to [20], we know that ( )LS S2

ˆ2

)(ˆ Ω∇ µ is χ2 distributed with two degrees of freedom

and is non-central χ[ 2),(ˆ)9( yxSL

[ )9/(119 LL −=

]

]

2 distributed with one degree of freedom and non-centrality parameter

. Collecting these facts, we know that the statistic W is the ratio of two mutually

independent random variables: the numerator being χ

2L′

2 distributed and the other having a non-central χ2

distribution. Therefore, W follows a special case of the doubly non-central F (Fisher) distribution, i.e.,

F″(2, 1, 0, L′ ), using the notation of [20]. By means of the series representation of the probability density

function of the doubly non-central F-distribution, we can write the PDF for W as follows

kk

kL

Ww

kk

Lew+

∞

=

′−

+

+′= ∑ 2/3

0

2/

)21()21(

!)2()(ρ . (3)

Using (3), we find that the probability that W is greater than a value is given by 0w

+

′−

+=>

12exp

211Pr

0

0

0

0w

wLw

wW . (4)

7

We denote the hypothesis of no edge being present by H0. Given a threshold T, from (4) we find

that PFA, the false edge detection probability of F(x, y) given the hypothesis H0 that no edge is present is

given by

[ ] 2ˆ0 )(162Pr|),(Pr Ω>=>= SFA TWHTyxFP µ (5)

where sΩΩ=Ω . From (4) and (5), it is seen that the quantity PFA does not depend upon the local mean

µ, indicating that the NG-edge detector is a CFAR detector. If the threshold T is selected adaptively with a

decrease of 2ˆ )ΩSµ( , when L is increased by a speckle filtering or diffusion, then PFA will decrease

exponentially with L. Fig. 1 depicts plots (in solid line) of the probability of false alarm in homogeneous

speckle areas as a function of the threshold T for different L, using our analytical expressions (4) and (5).

The parameter Ω needed to compute the theoretical PFA is given by )(1 Ls ζ=ΩΩ≡Ω since

Lfc=Ω and cs fζ=Ω where ζ is an arbitrary factor that should be greater than two, according to the

Shannon sampling theorem on aliasing. We empirically set the factor ζ to 3. On a synthetic (spatially

discretized) image of correlated speckle with the same statistical characteristics as our speckle model, a

Monte Carlo simulation (see Section V. VALIDATION for details) is performed using the discrete NG

operator: )2ijI2(2

ijI∇ . The resulting plots from the Monte Carlo simulation are also shown in Fig.1. It

may be observed that the theoretical prediction and Monte Carlo results are in good agreement.

(a) (b)

8

Fig.1. Plots of the false alarm probability of NG in homogeneous speckle areas versus threshold for

different numbers of looks (a): L = 1, 2 (top to bottom); (b): L = 4, 8, 16 ((top to bottom)). The solid lines

are the theoretical curves and the dashed curves are obtained using a Monte Carlo method.

III. DETECTING SPECKLED EDGES USING THE NG OPERATOR

In the previous section, we discussed the performance of the NG operator as applied to

homogeneous speckle regions. In this section, the performance of the detection of edges in speckle using

the NG operator is examined. First a realistic speckled edge image model is introduced. Then the statistics

of the NG operator on edges are derived. Finally, the geometric characteristics of the NG operator are

addressed.

A. A Realistic Speckled Edge Process

Consider the non-stationary random field

),()(),( yxSxeyxI ⋅= , (6)

where is a deterministic edge function used to model the underlying image intensity changes across

structure boundary and is given by

)(xe

+=σ

ρ xerfcxe 1)( , (7)

where ρ is the contrast of an edge defined as the ratio of )(∞e and )(−∞e and c is the average of )(∞e and

. )(−∞e )1()1( +−= ρρρ ; ( txerfx

∫ −=0

2exp2)(π

(

)dt

), yx

is the error function, σ is the edge scale

parameter (width of the edge transition zone), and S is the stationary speckle random process as

defined in last section. In contrast to the idealized step edge process adopted by Bovik [3] and Touzi [21],

the edge process we utilize contains ramp edges that have transition zones. The idealized step edge is a

special case of the edge process used here (when σ = 0). Because the NG (or ICOV) edge detector is

rotation-invariant and the speckle power spectral density function is assumed to be identical in x-direction

9

and y-direction, the edge performance obtained with vertically oriented edge function (7) applies to

horizontally orientated edges as well.

Two reasons for using (7) to model the underlying edge function are as follows. Assume that there

is a one-dimensional discontinuity at x = 0 in the physical parameter )(xη (e.g., the ground reflectivity,

the tissue density, etc.) of an object being imaged. Let )(()( 21 xx )1 uηηηη −+= , where u(x) is a 1-D unit

step function and constants 1η and 2η are, respectively, the parameter values on both sides of the

discontinuity. Consider a non-coherent imaging device with an illuminating beam of non-zero width.

Then the image e(x) of the parameter can be related to the parameter by a convolution:

, where B(x) is the point spread function of the imaging device normalized

such that the area of B(x) is unity. Letting B(x) be a Gaussian function with a standard deviation σ, we find

that

∫∞

∞−−= duuxBuxe )()()( η

σx

−

++

=ηηηη

erfx22

)( 1212e , which can be rewritten as (7). Therefore, the edge scale σ in (7) can

be proportional to the scanning beam width or the transmitting pulse width, depending on the orientation

of the edge. Equation (7) also allows the modeling of Gaussian linear filtering applied to a step edge. In

this case, the edge transition parameter σ would be 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, according to (2), is calculated by

( )

′+

∇+

′=

),(ˆ)(

),(ˆ)(6

),(ˆ

),(ˆ9

)(

)(),(

2

2

2

22

yxSxe

yxSxe

yxS

yxS

xe

xeKyxF x , (8)

where dxxdexe )()( =′ . To evaluate the statistics of the NG-filtered edge process, we first need the

cumulative distribution of F(x, y). Since random variables , (both being zero mean Gaussian N xSK ˆySK ˆ

10

(0, LS2

ˆ )( Ωµ )) and S ( being N (1-1/(9L), 1/(9L)) are mutually independent, we are able to find the

joint PDF of

ˆ

SSK ˆ≡U and xˆ SSK y

ˆˆ≡V (see Appendix III):

(u

21σ

=)

(e′

)0

++

+

+−=

2/522/32

222

221

222

21

22

,)1(

1)1()(

)1(2exp

2),

wwwwvVU

γγ

µσ

γσ

µ

πσ

µρ , (9)

where , 222 vuw += LS2

ˆ )( Ω= µ , )9/(112 L−=µ , )9(122 L=σ , and [ 12

ˆ )(9 −Ω= Sµγ ] . As a cross

check, we have proven that expression (5) can be derived from (9). Therefore, the cumulative distribution

of F(x, y) can be computed by

∫∫ <++=≤++

9/)]([ ,22

),( 22),(9)]([Pr(

fvxu VUyxF dudvvufVxUfPβ

ρβ . (10)

where ].)())[3/()( xexKx =β The region of the double integral in (10) is a circular disk in the (u, v)

plane centered at (- β(x), 0) with a radius of 3f .

Given a threshold T, the conditional probability of correctly detecting an edge when one is actually

present (hypothesis H1) is given by

( )9/1),0(Pr ),(1 TPHTyFP yxFD −=>= , (11)

where )3(2])0(()[3/( σπρβ =′= eeK . Fig. 2 plots edge detection performance for different values

of L and β as obtained using (11) and using a Monte Carlo simulation (See part A, section V). For the

theoretical plots, the double integral (10) is numerically evaluated using the simplest Riemann sum

approximation. It is shown that the theoretical prediction agrees with and Monte Carlo simulated results.

We can observe that for a given value of L, an edge that has a larger β value (higher edge contrast) is

more likely to be correctly detected. Also, for a given edge, the more speckle that is removed, the wider

the range of T becomes over which the probability of it detection approaches unity. Of course the range of

T has a limit that is proportional to the contrast of edge.

11

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

Fig.2. Probability of NG edge detection on edges as a function of threshold (the starred line: L =1; the

dashed: L=2; the dotted: L=4; the dash-dot: L=8 and the solid: L=16). (a) and (b): theoretical plots. (c)

and (d): Monte Carlo simulation plots. (a) and (c): β = 0.1; (b) and (d): β = 0.2.

It is seen that for L=1, a deviation exists between the Monte Carlo and analytical results. (Later in

Fig. 4, the same effect can also be observed). This discrepancy is due to the fact that the Gaussian model

for the cube-root transformed speckle becomes less accurate as L decreases.

C. Edge Localization Characteristics of the NG Operator

We have examined the statistical performance of the NG operator for edge detection in the

presence of speckle. In this subsection, we characterize the edge localization of the normalized gradient

detector in terms of the position of the peak of and the width of the edge response. We focus on the mean

of the edge response 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 SS x is (even) symmetrically distributed around the origin x=0, hence the mean

of SS x is equal to zero. Therefore, the mean of the NG filtered edge process (i.e., the mean of (8)) is

( )[ ] 0

2

0 22

2

2

22

)(11

)()()( F

xFF

xerfeb

xexeKx µ

ρσµµ +

+

=+

′= − , (12)

12

where K/σσ = , σxx = and 2220

SSKF ∇=µ is a constant obtained by averaging F(x, y) over a

large homogeneous speckled region.

Using the properties of the error function, we have the following series expansions:

)(31)( 53 xOxcbxcbcxe +−+= , )()1()( 42 xOxbcxe +−=′

σ, (13)

where πρ2

=b . With (13), and retaining terms up to the second order in x, (12) can be approximated by

022

222

)1()1()( FF

xbxbx µ

σµ +

+

−≈ . for σ<x (14)

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

σb

bxm

211 −+−≅ . (15)

To illustrate the accuracy of (15), we present a numerical example. For ρ =0.7, we have

πρ2=b =0.79. Using (15), we find that σ49.0−≅mx ; while using the rigorous formula (12), we can

compute numerically that σ48.0−=mx .

When the contrast of an edge is lower than ( ) ( )2/12/10 ππρ −+= (i.e. , is real

and negative, meaning that the peak of the edge strength is biased toward the darker side of the edge. So,

the quantity x quantifies the edge location bias of the NG operator.

Since

)12 dB≈ mx

m

2)11( 2 σσ bbbxm −≈−+−= for 10 <≤ b , and b is proportional to the normalized edge

contrast ρ , we find that higher edge contrast corresponds to higher edge location bias. Besides, it is

evident that the location bias is proportional to the edge scale σ. It is also worthwhile to mention that the

conclusions regarding the position of the edge strength peak and the higher edge contrast are in line with

[5] and [7].

13

The gradient magnitude creates a bell shaped response at an edge, even-symmetric with respect to

the edge center; and on the other hand, the normalizing function e(x) is odd-symmetric at the edge center.

As the result of the division of the even-symmetric function by the odd-symmetric one, the edge strength

peak gets skewed towards the darker side of the image.

If b>1, then (15) generates complex values, meaning that (15) does not hold for edges with high

contrast. This is due to the fact that for high contrast edges, the edge location bias by the normalized

gradient exceeds σ, the transition zone width of the edge, and thus the approximation (13) is invalid.

The edge localization of the NG detector can be further quantified, approximately, by the 3 dB (or

half-peak-value) width of the NG response:

( )12)(42

)()(

2

3 −−≅∆ bfbfb

bx pp

dB σ , (16)

where is the maximum of )(bf p 2

22

)1()1()(

bxxxf

+

−= in the interval [-1,1], given by

( )( )24

222

1114)(bb

bbbf p−

−−−= . (17)

Fig. 3 shows plots of the peak bias and 3 dB width of the NG-filtered edge signals as functions of the

equivalent edge contrast b. It can be observed from Fig. 3 that the peak bias formula (15) represents a

good approximation to the true peak bias, and that the 3 dB width expression (16) underestimates the true

width ≈ 1.2σ by an approximately fixed offset 0.1. We also observe that ∆ is insensitive

to the contrast parameter b over a wide range of b. We will use this fact in Section IV to show how the

Laplacian terms in the ICOV detector affect edge localization.

)(3 bx dB∆ )(3 bx dB

14

(a) (b)

Fig. 3. The NG edge localization as function of the edge contrast parameter b. (a) Peak response bias

. (b) 3 dB main lobe width )(bxm )(3 bx dB∆ . The solid lines are obtained using approximate formulae and

the starred lines are computed using the rigorous formulae. The bias and width measures are in units of

the edge scale σ.

IV. ICOVS RESPONSE TO EDGES WITH AND WITHOUT SPECKLE

In the proceeding two sections, we discussed the edge detection and localization performance of

the NG operator. We found that the NG operator exhibits certain edge location errors. In this section, we

examine the ICOVS response to edges and speckle (since our target images are ultrasound images) and

compare the edge localization of the ICOVS and the NG operator.

A. ICOVS Response to Speckled Edges

In the continuous domain the ICOVS-filtered speckle-corrupted edge process is defined by

( )( )

( )[ ]222

222222

22

)4/1(1

)8/1(),(,

IIK

IIKIyxIKyxq

∇+

∇−∇= . (18)

Using (6),22

2

2

ˆ

ˆ9

)(3

)(

ˆ

ˆ9

+

′+=

∇

S

S

xe

xe

S

S

I

I yx , and)(

)(

ˆ

ˆ

)(

)(6

ˆ

ˆ6

ˆ

ˆ3

2

222

xe

xe

S

S

xe

xe

S

S

S

S

I

I x ′′+

′+

∇+

∇=

∇ ,

we have

15

( )( )[ ] [ ]

[ ] 222

2113

222

2113

22

212

)()(2)(2)4/3(1

)()(2)(2)8/1()(9,

xZZZxZ

xZZZxZZxZyxq

φβ

φββ

+++++

++++−++= , (19)

where ),(ˆ),(ˆ1 yxSyxSKZ x≡ , ),(ˆ),(ˆ

2 yxSyxSKZ y= , ),(ˆ),(ˆ223 yxSyxSKZ ∇= ,

)()()3/()( xexeKx ′=β , )()()3/()( 2 xexeKx ′′=φ , ( )22)( σ

σπρ xecxe −=′ and ( )2

3

4)( σ

σπρ xxecxe −−=′′ .

From Section II, we recall that S is a Gaussian process N (1-1/(9L), 1/(9L)), that S and are

mutually independent Gaussian with zero mean and variance

ˆx

ˆyS

LS2

ˆ21 )( Ω= µσ

xxS yyS

, and that S and or

are independent. In the same manner, we know that (or ) and (or ) are mutually independent

Gaussian processes. The variance of S or is given by

ˆxS yS

xS yS

xxˆ

yyS 3L4Ω2S

S

ˆ ( fS44 ),)4 dfdffWf xyxx µπ

2

3y =

∇

( .∫∫ 2R

Since and are mutually independent Gaussian processes, we get that the Laplacian is N(0,

) with

xxS yyS

23σ 342

ˆ23 LSΩσ 6= µ . Given these relationships, we can derive the joint PDF, )3z,(

1zZ , 21 z, 32 ZZ,ρ ,

for Z1, Z2, and Z3 (see Appendix II). Hence, with the ICOVS, the probability of edge detection at the edge

position given the H1 hypothesis that a true edge is present is given by

∫∫∫−=>=1

321 321321,,12 ),,(1),(Pr

D ZZZD dzdzdzzzzHTyxqP ρ , (20)

where D1 is the volume bounded by ( )[ ] ( )[ ] <+++−++22

22113

22

21 228/1 zzzzzz ββ

with ([ ] 222

2113 224/319/ zzzzT ++++ β ) )3(2 σπρβ = . Similarly, the probability of false detection

under hypothesis H0 for the ICOVS operator in homogeneous speckle regions

02 ),(Pr HTyxqPFA >= is given by the triple integral of ),,( 321,, 321

zzzZZZρ over the volume D0 :

( ) [ ] ≥++−+22

2213

22

21 )(2)8/1( zzzzz [ ] 22

2 )z213 (2)4/3(1)9/ zzT +++( .

Fig. 4 plots edge detection performance for different values of L and β as obtained using two

methods: the triple integral of ),,( 321,, 321zzzZZZρ and a Monte Carlo simulation. Fig. 5 illustrates the plots

16

of the probability of false alarm in homogeneous speckle areas as a function of the threshold T for

different L from the two methods. For the theoretical plots, all triple integrals are numerically evaluated

using the simplest Riemann sum approximation. It can be seen that once again, the theoretical and Monte

Carlo plots match closely. From these plots and the plots in Fig. 1, it is concluded that ICOVS and NG

operator share similar statistical edge detection performance in homogeneous speckle regions and on

edges in terms of probability of false alarm and probability of edge detection.

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

Fig.4. Probability of ICOVS edge detection on edges as a function of threshold (the star line: L =1; the

dashed: L=2; the dotted: L=4; the dash-dotted: L=8 and the solid: L=16). (a) and (b): theoretical plots. (c)

and (d): Monte Carlo simulation plot. (a) and (c): β = 0.1; (b) and (d): β = 0.2.

(a) (b)

Fig.5. The false alarm probability of ICOVS in homogeneous speckle areas versus threshold for different

numbers of looks (a): L = 1, 2 (top to bottom); (b): L = 4, 8, 16 ((top to bottom)). The solid lines are the

theoretical results and the dashed curves are obtained using a Monte Carlo method.

17

B. ICOVS Response to Edges without Speckle

We have considered the statistical responses of ICOVS at edge positions and in the regions far

away from edges. Now we turn our attention to how ICOVS responses in regions near edges. To make the

problem tractable, we restrict ourselves only to the edges where the speckle level is negligible. First, we

formulate a more general form of ICOVS in the continuous domain. Then we discuss two limiting cases

of the general ICOVS, by which we show that the Laplacian term in the denominator of the ICOVS serves

to increase edge location accuracy, while the role of the Laplacian term in the numerator of the ICOVS is

sharpening the edge response. Finally, we consider the general case where both Laplacian terms are

neither zero nor infinite. It is shown that the ICOVS provides superior edge localization performance

relative to the NG operator.

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

( )

[ ]2222

224222

2

),()2/(),(

),(),(),;,(

yxIKyxI

yxIKyxIKyxq

∇+

∇−∇=

α

λλα , (21)

where α, λ and K are non-negative weighting parameters (all dimensionless). Since the ∇

represents an isotropic diffusion process, I in (21) is simply a smoothed

version of the image . Recalling the relationship between a diffusion process and Gaussian filtering

[8], we may approximate I for small values of αK with

),(2 yxI

),()2/(),( 222 yxIKyx ∇+ α

),()2/( 222 yxIK ∇α

),( yxI

,(

),( yx +

)),(),(~ yxg KαyxIyxI Kα ∗≡ where [ ])2( 2α(exp)2(1), 222πα yxy +−=(α xg . Therefore, (21) is

reformulated as

( )

[ ]2

224222

),(~

),(),(),;,(

yxI

yxIKyxIKyxQ

Kα

λλα

∇−∇= . (22)

Adopting this formula for the ICOVS and letting 1),( ≡yxS in (6), we can write the ICOVS response to a

speckle-free edge in the following form

18

( )( ) ( )

22

2222

2

24222

)])/(1/(1[

)2exp()2(1

)](~[

)()(,;,

σαρ

λ

σ

λλα

α ++

−−

=

′′−′=

xerf

xxbxe

xeKxeKyx

K

Q , (23)

where σ/xx = , K/σσ = , σλλ = and )()()(~ xgxex KK ααe ∗≡ . Completing the Gaussian convolution

of the edge function yields

++=

22)()(~

σαρα

K

xerfccxe K . (24)

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

series expansions )(3/)(~ 53 xOxBcxcBcxK +−+=αe where 21 α+= bB and σαα /= , and

)()/(/)( 42 xOxcbbcxex +−= σσ and )()/(2)( 32 xOxcbxexx +−= σ , and retaining terms up to the

second order in x, we can approximate (23) by

( )2

2222

2

2

1

)2()1(),;,(

xB

xxbyxQ+

−−≅

λ

σλα for σ<x . (25)

Given (25), we can analyze the performance of the ICOVS detector near the edge transition in cases

where it is difficult to derive 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 λ is set to zero. Restricting ourselves to a

narrow band around the edge where σ<x , from (25), we have the approximation

( )2

22

2

2

1)1(),;0,(

xBxbyxQ

+

−≅=σ

λα

),;0,( yxQ =

. We find that the edge location bias (the position of the peak) of

λα equals BBx p )11( 2−+−= σ . Since 21 α+= bB b< , we have that 0≤≤ pm xx ,

implying that the ICOV operator is capable of achieving higher edge location accuracy compared with the

NG edge detector, i.e., the Laplacian term at denominator in (21), associated with weight α alone,

compensates for the edge bias of NG. With the approximation that 2σBpx −≈ for 10 <≤ b , the edge

19

location accuracy of Q ),;0,( yx=λα is improved approximately by a factor of 21 α+ compared with

that of the NG. As an example, if =1/2, the improvement factor will be 1.22 for an edge with 2α 1=σ ,

which corresponds to an edge having a 2- pixel wide transition zone.

,( λα

( 2)B

∆≈ x

2

The 3 dB width of the Q ),;0 yx= signal, our second edge localization measure, is given by

1( −f p

(3x dB

. As illustrated by Fig. 3(b), the function ∆ for the NG

edge detector is insensitive to x. Since

)(3 xx dB

)B∆ has the same functional form with respect to B, we

have the approximation σ2.1)() 3(3 ≅∆x dB bB dB . Therefore, we infer that the different α values do not

alter significantly the edge response width of the edge detector. Thus, we conclude that the Laplacian term

in the denominator of the ICOVS operator serves to reduce the edge position bias without increasing the

localization error relative to the ICOVS response width.

)42

)()(

2

3 −=∆BfB

Bx pdB σ

In the second limiting case, we scrutinize 22)(2

2

2

)2(1),;,(2

xebyx x λQσ

λ −=∞ −

cx

(See Fig. 6 for an

example) for which the expression for the ICOVS, (23), and e =∞ )(~ are used. Intuitively, the gradient

magnitude-squared (GM) ( I∇≡ ) generates one lobe centered on the edge, while the Laplacian

squared (LS ) generates double lobes at both sides of an edge. The absolute value of the

difference of the GM and the appropriately weighted LS results in a sharpened lobe with two small side

lobes. It is obvious that the peak edge response occurs on the edge, i.e.,

22 )( I∇≡

0=px , and the null-to-null main

lobe width of the Q )y,; x,( λ∞ is given by λσ2=∆ nnx , showing that increasing λ results in shrinkage

of the main lobe of Q ),;, yx( λ∞ regardless of the edge contrast b. Though a larger λ value will generate a

sharper edge response, the parameter λ cannot be chosen arbitrarily large because the height of the side

lobe increases with parameter λ. We find that the peak of the side lobe is determined by

)2(12 2λλ −e2 )2()( σ eb , which is a monotonically increasing function of λ . The peak height of the side

20

lobes is greater than that of the main lobe ifλ >1.35. When 04.1=λ , the peak height of the side lobe is

one half of that of the main lobe. Strong side lobes introduce edge-alike artifacts, resulting in an increased

probability of false alarms near an edge. In practice, λ should be chosen such that the side lobe height is

at least less than or equal to 0Fµ , the average speckle response in homogeneous speckle regions, in order

to avoid an increase in false alarms. This illustrates the tradeoff between edge response width and false

alarms.

,; xλ

Fig.6. The ),( yQ ∞ and the gradient magnitude response to the e(x) edge signal.

2) ICOVS edge responses

Through two limiting cases, we have shown that the Laplacian term associated with weight α

alone compensates for edge bias, and that the Laplacian term associated with λ alone tends to narrow edge

responses. Now, we address the edge localization performance of the ICOVS operator where the two

Laplacian terms associated with the weights α and λ work jointly.

Based upon the conclusions drawn from the above discussion, we expect that the ICOVS would

achieve better edge localization performance than either Q ),;0,( yx=λα or ),;,( yxQ λ∞ . Let the main

lobe of the ICOVS response lie in an interval [-x1, x2] centered on the edge where 0)2( 22 >xλ)1 22 −− x( .

Within the interval [-x1, x2], we find via the first derivative of (25) that the position of the peak ICOVS

signal is a root of the equation: px

21

( ) 04)1(2 222 =+−++ xxBxxB λ . (26)

Since the ICOVS peak lies close to the edge (where 0=x

0

), we adopt the Newton’s root-finding method to

solve (26). Choosing the initial guess root as , we find that the edge location bias of the ICOV, up

to the 1

)0( =px

st order approximation, is given by

242 λσ

+−≈

Bx p . (27)

From (27), it is seen that the edge location accuracy of the ICOVS is improved by a factor of

22 1)21( αλ ++ , compared with the normalized gradient edge response where the peak response is at

211 2 σσ b

bbxm −≈

−+−= for b<1. For instance, given that , and 2/12 =α 8/12 =λ 1=σ , the

improvement factor is 1.53. Therefore, an increase in either Laplacian weight (α or λ) reduces the edge

location error.

Following the same steps for deriving the 3 dB width of the NG operator, we find that the 3 dB

width of the ICOVS edge response can be approximated by

( ) ( )( )2/),(42

2/),(2/),(1422,

22

22

3λλ

λλλσλ

BgB

BgBBgBx

p

ppdB

++

+−+≅∆ (28)

where ),( λBg p is the maximum of the function 2

2222

)1(

)2()1()(

xB

xxx

+

−−=

λg near x= 0. Noting

that 1),( ≈λBg p and using 21 α+= bB , from (28) we find that the 3 dB width of the ICOV response

is bounded from above by

( ))]1(2[42

2,222

3αλ

σλα+++

≤∆b

x dB . (29)

Expression (29) shows explicitly that a larger value of λ and a smaller value of α are preferable for

decreasing ( )λ,3 Bx dB∆ . Fig. 7 plots the speckle-free edge localization performance of the ICOVS (with

22

2α =1/2 and 2λ =1/8) as function of the edge contrast parameter b. It is seen that the edge location bias

curve obtained using the approximate formula agrees with that is computed using the rigorous formula,

and that the 3 dB width curves obtained using the approximate expression (28) and the upper bound

expression (29) have fixed offsets to the curve derived from the rigorous model.

(a) (b)

Fig. 7. ICOVS speckle-free edge localization performance as function of edge contrast parameter b. (a):

Edge position (solid line: approximated; starred line: exact). (b) 3 dB main lobe width )(bx p )(3 bx dB∆

(Solid line: upper bound ; starred line: rigorous; and dotted line: approximated). The bias and width

measures are in units of the edge scale σ.

In summary, our theoretical analysis shows that the Laplacian term associated with the weight λ

tends to sharpen the edge response of the ICOVS but may increase the probability of false alarm near an

edge if λ is too large (see the second limit case in subsection A); while the Laplacian term associated with

the weight α tends to reduce the bias of the ICOVS but also may widen the response when α is too large.

The ICOVS, namely ICOV, operator seeks to optimize the edge detection in speckle imagery in terms of

low false edge detection probability and high edge localization accuracy.

C. Edge Localization Comparison between the ICOVS Operator and the NG Operator

As a comparison, Fig. 8 plots the analytical edge localization properties of the NG and ICOVS

(with 2α =1/2 and 2λ =1/8) edge detectors on speckle-free edges. In terms of the edge location bias, the

ICOVS achieves an improvement factor of approximately 1.5, compared with the NG. Regarding the 3 dB

23

width, the ICOVS operator achieves an improvement of approximately 10% over a wide range of b

values, relative to the NG operator.

(a) (b) (c) (d) Fig. 8. Edge localization performance comparison between the NG and ICOVS operators. (a): Edge

position bias; (b): 3 dB Edge width. The solid lines are for NG while the dashed for ICOVS. (c) and (d):

the edge localization improvement of ICOVS over NG (length measures are in units of the edge scale σ).

V. VALIDATION

In this section, we validate the essential statistical characteristics of the NG and ICOVS operators,

the performance of the NG and ICOVS operators as applied directly to speckled US imagery, and the

performance of a practical ICOV-based edge detection algorithm. A Monte Carlo method is employed to

verify the statistical characteristics of the NG and ICOVS operators. The performance of the NG and

ICOVS edge detectors is verified using a synthetic 1D signal with low speckle. For validating the

performance the practical ICOV-based edge detection algorithm, both synthesized and real US images are

tested and results are quantified. Furthermore, a comparative validation of the ICOV-based edge detection

algorithm is made by comparing it with three other existing edge detection methods.

A. Validation by Monte Carlo Method

In sections II through IV, we have quantified the edge detection performance of the NG operator

and the ICOVS operator under hypothesis H0 or H1 using mathematical analysis and numerical integration

methods. It is necessary to verify those quantifications using at least one alternative method. To this end,

24

we employ a Monte Carlo method, which requires generation of simulated speckle pattern. Next, we will

describe how speckle pattern is synthesized, followed by our validation procedures and results.

1) Speckle pattern synthesis

Synthetic 1-look B-scan speckle intensity data is formed by the simulator described in Subsection

A, Section IV of [24]. We chose the simulator parameters as c=1500 m/s (the speed of sound in tissue),

=100MHz (the center frequency), 0f xσ = 2 pixels (the pulse-width of transmitting ultrasonic wave), and

yσ = 2 pixels (the beam-width of transmitting ultrasonic wave). The ultrasound cross-section distribution

where is a Gaussian white noise field with zero mean and unity variance. ( ) ( yxGy ,, = )xT ),( yxG

2) Validation of the probability of false edge detection in homogeneous speckle regions

To calculate the probabilities of false alarm of the NG or the ICOVS operator in L-look speckle, a

single-look speckle image of 256 by 256 pixels is first synthesized. The L-look dataset is then generated

by filtering the single-look data with a Gaussian filter with a standard deviation t2 (where t is

determined by (AI.5)). For the NG (or ICOVS) operator, the Monte Carlo predicted probability of false

edge detection at a given threshold T is the fraction of pixels whose )2( 22ijijij IIF ∇≡ (or ) values

are greater than the threshold T among the total number of pixels in the image, i.e., 256

2ijq

2. The

probabilities of false edge detection for the NG operator and the ICOVS operator under hypothesis H0

resulted from the Monte Carlo method have been plotted in Fig. 1 and Fig. 5, respectively.

To ensure correct generation of speckle, we monitor the average empirical autocovariance

functions (ACF) in lateral and axial directions as well as the histogram of synthesized 1-look speckle

pattern. To this end, empirical autocovariance functions [3] are computed over ten 90×90 windows in the

1-look speckle image. Fig. 9(a) illustrates the averaged, radial and lateral empirical ACFs, together with

theoretical ACF (the same in radial and lateral directions), versus lag (in units of pixels), and Fig. 9(b)

shows the intensity histogram. The figures exhibit close matches between the computed statistics and the

statistics predicted by theory.

25

(a) (b)

Fig. 9. (a): Autovariance functions, and (b): histogram of the single-look the synthetic speckle intensity.

3) Validation of the probability of edge detection at edge positions

A strip-shaped image (2048 by 32 pixels) with edges along the long axis of symmetry is

synthesized to assess the edge detection performance at edge positions under hypothesis H1. Specifically,

the image dataset is created by multiplying the edge function e(x) (see (7)) with a synthesized multi-look

speckle pattern. It can be seen that possible edge positions are at the 2048×2 pixels on both sides of the

long symmetric axis of the image. With the NG (ICOVS) detector, given a given threshold T, the edge

detection probability under hypothesis H1 is estimated as the fraction of possible edge pixels with

)2( 22ijijij IIF ∇= (or ) values being greater than the threshold T. The probabilities of edge

detection at edge positions have been plotted in Fig. 2 and Fig. 4.

2ijq

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

This subsection and the next one are devoted to experimental validation of the ICOV operator. In

this subsection, we consider detecting the edge from a 1-D synthetic digital signal with relatively low

speckle: [ ] [ )2sin(1.01)(5.01 iierfI i + ]⋅+= where 20,19,19,20 ⋅⋅⋅−−=i . According to (7), the edge

scale parameter σ =1 for this signal. Fig. 10(a) plots the 1-D speckled signal and Fig. 10(b) the

corresponding edge strength signals generated by the discrete gradient magnitude-squared (GM)

26

(GM 2ix I∇= where ∇ is a difference approximation to the derivative with respect to x) operator and

the discrete NG (NG=

ix I

22ii IIx∇ ) operator, respectively. In Fig. 10(b) it is seen that the edge strength

signal computed by NG is well balanced on both high and low mean signal sides of the edge indicating

the constant false alarm rate (CFAR) property, while those generated by the GM are not CFAR in

homogeneous speckle areas of different means. However, the main lobe of the NG response is skewed

with the position of peak edge being biased toward the darker side of the edge. Fig. 10(c) is obtained by

applying the one-dimensional discrete ICOVS ( )

( )22

222

2

)2/1(

)4/1(

ixi

ixix

i

II

II

∇+

∇−∇=q to the 1-D signal. It is

seen that this edge detector allows for balanced and well localized edge strength measurements in bright

regions as well as in dark regions.

0=ρ

=σ3

It is interesting to compare the edge localization obtained by discrete NG and ICOVS with

theoretically-predicted edge localization of NG and ICOVS. Zooming in Figs. 10(b) and (c), we find that

the edge location bias is -1 pixel for discrete NG, and zero for discrete ICOVS. For the 1-D signal in Fig.

10(a), we know that 5. (or equivalently b=0.56), giving rise to that the edge location bias of the NG

is equal to −−= .0mx 0.3 pixels. Using α =1 and 2λ =1/4, we find x 0.14 pixels, the edge

location bias of ICOVS. For this example it is seen that the theoretical edge localization of the NG and

ICOVS operators agree satisfactorily with experimental localization of the discrete NG and ICOVS

operators, considering the finite difference approximation to derivative.

−=p

27

(a) (b) (c)

Fig.10. (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.

C. Experimental Validation: Edge Detection from US Imagery

We continue experimental validation of the efficacy of the ICOV operator using real US images.

In the last subsection, the ICOV operator was applied directly to the speckled signal in which the speckle

is relatively low in magnitude. Directly applying the ICOV operator to US images with high degrees of

speckle usually produces numerous spurious edges and misses weak edges, a scenario similar to when

directly performing gradient- or Laplacian-based edge detection in noisy optical imagery. Thus, an

indirect method is applied. Our method utilizes anisotropic diffusion with the ICOV as an implicit edge

detector (in the diffusion coefficient), analogous to anisotropic diffusion with a gradient-based diffusion

coefficient in imagery with additive noise. In this subsection, we first present such a practical ICOV-

based edge detection algorithm for speckled US imagery, and then test the algorithm on a synthetic image

and a real image, respectively. Finally, we quantify the quality of the detected edges.

1) ICOV-based edge detection algorithm

The algorithm begins with solving the following partial differential equation (PDE):

( ) ( ) ( )[ ]( ) ( ) ( )

=∂∂=

∇=∂∂

Ω∂ 0);(,0;;);(;

0 ntIIItItqcdivttI

rxxxxxx

(30)

28

where div represents the divergence, ∇ denotes the gradient, ( )x0I is the intensity of an input image at

location x=(x, y) in a Cartesian coordinate system, Ω∂ denotes the border of the image domain Ω, nr is

the outward normal vector to , and c(q) is the diffusive coefficient. Ω∂

( )( )[ ] ( )[ ])(1)()(;1

1;(20

20

20

2 tqtqtqtqtqc

+−+=

xx (31)

with the edge detector being defined by

( )

[ ]);();(

);();();(

2

2/1222

tItI

tItItq

xx

xxx

∇+

∇−∇=

γ

βα (32)

where α, β and γ are positive parameters properly chosen such that the discrete version takes the form of

(1). The scale function q serves as a threshold governing the magnitude of the ICOV required for an

edge. We utilize tools from robust statistics to automatically estimate as in [17]:

)(0 t

)(0 tq

∇=

);(2

);()(0

tI

tIMADctqx

x

( ) [ ] );(ln);(ln2 tImediantImedianc xx ∇−∇= (33)

where “MAD” denotes the median absolute deviation and the constant c (=1.4826) is derived from the

fact that the MAD of a zero-mean normal distribution with unit variance is 1/1.4826. The estimator (33) is

derived by taking into account the similarity of the square root of the NG and the ICOV and assuming that

the logarithm of image intensity is a piece-wise constant function that has been corrupted by zero-mean

Gaussian noise. Then, the edge strength image is extracted by

( ) ( ) ( ) ( )[ ]∞>∞⋅∞= 0;; qqqICOV xxx (34)

where [U(x) >V] denotes a binary image obtained by thresholding U(x) at level V and the ‘•’ is the

pointwise multiplication operation. Finally, an edge map is formed by [ ] where T is a

predetermined threshold.

TICOV >)(x

29

The partial differential equation (30) is solved numerically using a Jacobi iterative method.

Choosing a sufficiently small time step ∆t and a grid size h in both x and y directions, we discretize the

time and space coordinates as: t , n=0,1,···,tn∆= 1,,1,0,, −=== Mijhyihx L , 1,,2,1,0 −⋅⋅⋅= NJ ;

where is the area of the image domain. LetNhMh× ( )tnjhihII nji ∆= ;,, . A numerical approximation to

(30) is given by the following update equation

nji

nji

nji tdII ,,1

,

4

1∆+=+ , (35)

where

( ) ( ) ( ) ([ ]nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji

nji IIcIIcIIcIIc

hd ,1,,,1,1,,,1,,,1,1

2,

1−+−+−+−= −++−++ ) , (36)

( )[ ] ( ) ( )( )[ ]nqnqnqq

cn

ji

nji

20

20

20

2,

,

1)(1

1

+−+= , (37)

[ ]n

jin

ji

nji

njin

ji

II

IIq

,2

,

2/12

,22

,

,

)4/1(

)])[17/1()2/1(

∇+

∇−∇= , (38)

( 2/12,

2,,

2

1 nji

nji

nji III −+ ∇+∇=∇ ) , (39)

[ nji

nji

nji

nji

nji IIII

hI ,1,,,1, ,1

−−±= ±±± ]∇ , (40)

[ ]nji

nji

nji

nji

nji

nji IIIII

hI ,,1,11,1,

2,

2 41−+++= −+−+∇ , (41)

[ ]njiIMADcnq ,0 ln2)( ∇= . (42)

In (41), the central differencing scheme is utilized. To avoid distortion at the image boundaries,

symmetric boundary condition is required by which we mean that the image intensity function has equal

values at both sides of the boundary. The numerical solution I becomes stationary when nji , ε≤)(0 nq

30

where ε is a preset small positive number. The set [ ]ε>⋅= nji

njiji qqICOV ,,, calculated from the

stationary diffused image forms an edge strength image in which edges are points of significant values.

2) Experimental results

In this subsection, we demonstrate the performance of the proposed ICOV-based edge detection

algorithm. Fig. 11 illustrates four experiments using B-scan images of a human throat, a human prostate, a

phantom prostate with implanted radioactive seeds (as in brachytherapy of prostate cancer), and the left

ventricle of a murine heart. The first row shows, from left to right, the B-mode image of a human throat,

its diffused version, and the extracted ICOV, respectively. The second to the four rows illustrate the

image and corresponding diffused image and edge strength image for a human prostate, a prostate

phantom with implanted radioactive seeds, and the left ventricle of a murine heart, respectively. Since the

proposed algorithm needs envelope-detected intensity imagery as its input, the dynamically-compressed

B-mode images are decompressed approximately by taking the exponential of the B-mode image divided

by 25 before being feed into the algorithm. In our numerical experiments, we generally choose the

parameters as follows: h=1, ε = 0.02, and ∆t = 0.05 ~ 0.25. It is seen qualitatively that significant edge

strength is distributed along the actual boundaries in all four edge strength images consistently.

31

(a) (b) (c)

Fig.11. Experiments 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 down. Second column:

corresponding diffused images. Third column: ICOV edge strength images.

32

D. Comparison

To better justify the usefulness the ICOV-based edge detection algorithm, it is necessary to

compare the algorithm with existing algorithms for detecting edges in speckle imagery. In this subsection,

we compare the edges obtained by the ratio of averages (RoA) edge detector of Bovik [3] with those

obtained by the ICOV-based edge detection algorithm. The ratio edge detector of Touzi et al [21] and the

likelihood edge detector of Oliver et al. [12] had similar performance as the RoA (as tested empirically in

our research); therefore, we report the results of only one of such ratio detector.

The ratio of average (RoA) operator [3] measures the edge strength by

),(),(),( 22 yxVyxHyxRoA += (43)

with

),(),(),,(),(max),( yxRyxLyxLyxRyxH = , (44)

),(),(),,(),(max),( yxUyxDyxDyxUyx =V , (45)

where R(x, y), L(x, y), U(x, y) and D(x, y) are the mean intensities in the sub-windows immediately to the

right, left, up and down of image coordinate (x, y), respectively. Fig.12 illustrates the left, right, up and

down sub-windows of a 7×7 analyzing window. For an arbitrary L×L window (where L is an odd

number), the sub-windows are either (L-1)/2×L or L×(L-1)/2. An edge is declared to be present at

coordinate (x, y) if where T is a predetermined threshold. TyxRoA >),(

Fig.12. Four sub-windows of a 7×7 analyzing window centered at the crossed pixel.

The procedure of the comparison is as follows: (1) Run two algorithms on an input image to

obtain two edge strength images. (2) Enhance each edge strength image by full-scale contrast stretching

33

(with 256 gray levels). (3) Identify edges as top p % of the brightest pixels in each edge strength image.

(4) Quantify the detection performance in terms of Pratt’s figure of merit [16].

The comparison is made using a 2-D slice of real data obtained from imaging an ellipsoidal

phantom. The data were acquired using a Sonos 5500 US system (Agilent Technologies, San Jose). One

full frame of raw (RF) data contains 259 lines axially, with each line having 3680 pixels in the axial

direction, and each pixel signal coded with 16 bits. An envelope-detected amplitude image is first formed.

To reduce the volume of the data, we down-sample (using a 7 tap FIR low-pass filtering and 4:1

decimation) the image by a factor of 4 in the range direction, yielding an image of 259 by 920 pixels.

Then we extract a subimage of 220 by 396 pixels as the test image for which a time-gain-compensation

(TGC) is also carried out to calibrate residual propagation attenuation losses. Figure 13 depicts the edge

strength images obtained by two edge detection algorithms using the synthesized image. The first row

shows, respectively and from left to right, the noisy, attenuation-calibrated phantom image, the ICOV

edge strength image, and resulting two edge maps. The second row shows from left to right, the

manually-drawn boundaries representing the ground truth edges, the RoA edge strength image, and two

resulting edge maps, respectively. In the ICOV-based algorithm, the time step should be chosen so that

the numerical implementation is stable and converges rapidly. We choose ∆t =0.25 to optimize the

algorithm performance in terms of Pratt’s FOM. To implement the RoA edge detector, the size of the

window needs to be specified. A larger window will reduce spurious edges at the cost of degraded edge

localization on real edges. On the contrary, a smaller window size will increase edge localization but

produce more spurious edges. Depending on the size of the interesting object in an image, the filter

window is chosen to achieve a balance between the allowable number of spurious edges and desired edge

localization performance. The elliptical shape to be detected has a long axis of 200 pixels and minor axis

of 120 pixels, so we choose a 39×39 analyzing window, an optimal in the empirical sense in the range of

9×9 through 57×57 windows. Since the ground-truth edge pixels in Fig. 13(e) makes up 0.53% of the test

image, we set the threshold T at the level such that top 0.6% brightest pixels are detected as edges in the

34

edge strength image. As a reference, we also show edge maps that are detected as top 1% or 5% most

significant pixels in edge strength images.

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

(f) (g) (h) (i) (j)

Fig.13. Comparison of the ICOV-based edge detection algorithm and the RoA detector on a phantom US

image. (a) Phantom US image (Left hand side of the image is closer to US transducer. Scan is in the

vertical direction); (b) ICOV edge strength image; (c), (d) and (e) ICOV detected edge maps (top 5%, 1%

and 0.6% brightest in ICOV edge strength image). (f) Ground true edges. (g): RoA edge strength using a

33x33 window. (h), (i) and (j): RoA detected edge maps (top 5%, 1% and 0.6% brightest in RoA edge

strength image).

In addition to visually comparing the ICOV-based detection algorithm and the RoA operator, we

use Pratt’s figure of merit (FoM) [16] to quantify their edge detection performance. Metric FoM penalizes

both the number of incorrect edge detections and the errors of edge location in an edge map based on a

map of ground-true edges. For the phantom image, the ground truth edges should be on the boundary of

the elliptical shape which is drawn manually and shown in Fig 13 (e). From the top 0.6% most significant

edge maps and the ground-true edge map, we find a FOM value of 0.64 for the proposed algorithm and a

FOM value 0.27 with the RoA operator for the example image.

35

In our comparison, we have considered the sensitivity of the two algorithms to changes in their

parameters and have chosen the optimal parameters for each algorithm. Fig. 14 shows the plots of FOM

versus time step for ICOV-based algorithm and FOM versus window size for the RoA operator. In the left

plot, it is seen that when ∆t = 0.25 the performance of the ICOV-based algorithm is best for the test

image; in the right plot, we see that L=39 is optimal for the RoA detector. The steep dropoff of FOM (the

top 0.6% curve) after ∆t >0.27 indicates that the ICOV-based algorithm has become errant due to

numerical instability.

Fig. 14. Sensitivity of algorithm performance with respect to parameters.

From the comparison, we see that the ICOV-based algorithm outperforms the RoA detector in

terms of significantly higher FOM. We also observe that the ICOV-based algorithm is sensitive to the

time step chosen for discrete implementation.

VI. CONCLUSIONS

The edge detection and localization performance of the NG and ICOVS operators have been

examined. The study shows that the NG and ICOVS operators are constant false alarm edge detectors, and

36

that the edges detected by the ICOVS detector manifest reduced localization errors compared with those

detected by the NG operator. Quantitative improvements of the ICOVS in terms of the location of the

peak and the width of the ICOVS response have been derived relative to those of the NG operator. The

Laplacian operator in the ICOVS operator has been identified as the compensating factor for edge

localization error. A Monte Carlo simulation is performed to validate the analysis of the NG and ICOVS

and shows that the analysis is in agreement with the Monte Carlo simulated results. A practical ICOV-

based edge detection algorithm for US imagery has been implemented by embedding the ICOV operator

in anisotropic diffusion process. Encouraging experiments have been obtained with the ICOV-based edge

detection on US images, as compared to ratio-of-average-type edge detection method for speckled

imagery.

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38

Appendix I

Using the method described in [22], and approximating the axial point spread function (PSF) of

the radio frequency (RF) pulse envelope shape and the transverse PSF of the returned echo amplitude to

be Gaussian with standard deviations 1 and 1 , respectively, we find the autocovariance function

for B-scan single-look speckle intensity image is

cxf/ cyf/

),(0 yxS

( ) ( )[ ] ( )[ ]2exp2exp, 22S0

yfxfyx cycx ∆−∆−=∆∆C . (AI.1)

Note that the noise power spectrum density function (PSD) of the single look speckle, ( )yxS ff ,0

W , is the

Fourier transform of C , so we have ( yxS ∆∆ ,0

)

( ) ( ) ( )[ ] ( )[ ]2222S /2exp/2exp/2,

0 cyycxxcycxyx ffffffff πππ −−=W . (AI.2)

L-look speckle intensity S(x, y) can be obtained by convolving S0(x, y) with a 2D Gaussian kernel

having standard deviation t2 (where t is to be determined); and the autocovariance function of the L-

look intensity fluctuation is then given by

( ) ( ) ),(),(,,220S yxgyxgyxCyx

ttSC ∆∆⊗∆∆⊗∆∆=∆∆

∆Ω−

∆Ω−

ΩΩ=

2

)(

2)(

exp22 yx

ffyx

cycx

yx , (AI.3)

where tff cxcxx241+=Ω and tff cycyy

241+=Ω . Taking the Fourier transform of

yields the PSD for the L-look speckle intensity ( yxCS ∆∆ , )

( ) [ ] [ ]222222S /2exp/2exp)(2, yyxxcycxyx ffffff Ω−Ω−= πππW . (AI.4)

Noting that the variance of the multi-look speckle , we get the

relationhsip between L and t :

∫ ∫∞

∞−

∞

∞−== LdudvvuWSS /1),(2σ

tffL cycx41+= . (AI.5)

Therefore, (AI.3) can be reduced to

39

( )

∆−

∆−=∆∆

L

yf

Lxf

LyxC cycx

2

)(

2)(

exp1,22

S . (AI.6)

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

. If S),(),(ˆ 3/1 yxSyxS = 1 and S2 are the L-look intensities of speckle at two points (x1, y1) and (x2, y2),

then the second order probability density function is

( ) ( )

−

−

+−

−Γ= −

−+

CsCs

LICss

LCL

CssLss L

LL

yxSyxS1

21

exp)1)((

/, 21

121

2/)1(21

1

21),(),,( 2211ρ , (AI.7)

where In(x) is the modified Bessel function of nth order. C is the (correlation coefficient) modulus of the

autocorrelation function of the S(x, y) normalized to unity at lag values ∆ and 0=x 0=∆y , i,e.,

( ) )0,0(),(, SSS CyxCyxC ∆∆=∆∆

∆Ω+∆Ω−=

2

)()(exp

22 yx yx .

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

function [ ]

( ),;;3/1,3/1)()3/1(

12

2

3/1

3/12

3/11 SCLF

LLLSS −−

Γ

+Γ= where is a hypergeometric

function. Since

);;,(12 zcbaF

( ))(

3/13/1

3/1

LLLS

Γ

+Γ= ( ), we have ( )[ ]1;;3/1,3

(1(,

3/1ˆ −−

Γ

+Γ=∆∆ SS CL

LLyxC /1

))3/

12

2

−

F

L.

Now ⋅⋅⋅++

++=−− 212

)1(811

911);;3/1,3/1( C

LLC

LCLF . Retaining terms up to the first order in C,

we get

( ) SS CLLL

LyxC 1)(3)3/1(,

2

3/1ˆ

Γ

+Γ≅∆∆ . (AI.8)

Taking the Fourier transform of (AI.8) yields

),()(3)3/1(),(

2

3/1 yxSyxT ffWLL

Lff

Γ

+Γ≅=W . (AI.9)

Appendix II

40

For a real-valued Gaussian random process X(t) with autocorrelation function R(τ), the cross

correlation of the product of X(t) and its first derivative is given by )(tX&

∆−∆+

=

∆−∆+

=→∆→∆ t

tXttXtXEt

tXttXtXEtXtXEtt

)()()()()()()()(2

00limlim&

( ) ( ) 0)0(0)()()( limlim0

2

0==

∆−∆

=∆

−∆+=

→∆→∆

Rt

RtRt

tXEttXtXEtt

& .

Denoting the expectation of X(t) by Xµ , we find that

0)()()()(])([ =−=− tXEtXtXEtXtXE XX&&& µµ

)(tX&

, since X is zero mean, indicating that X(t) and

are uncorrelated at any time t. Since at a given time, both X(t) and X are Gaussian random

variables, we know that they must be mutually independent.

)(t&

)(t&

Appendix III

Suppose that three Gaussian random variables ( )),0(~ 1σNX , ( )),0(~ 1σNY and

( ),(~ 22 )σµNZ are mutually independent. The mean of Z, µ2, is non-zero. The joint PDF for X, Y and

Z can be expressed by )()()(),(,, zyxyx ZYXZYX ρρρρ = .

We now define two ratio random variables ZXU = and ZYV = . Substituting U for X and V

for Y, we find that the joint PDF for U, V and Z is given by . The

marginal PDF for U and V can be obtained by doing the following integral [13]

2,, )()()( zzzvzu ZYXVU ρρρρ ),,( zvuZ =

, dzzzzvzuvu ZYXVU ∫∞

∞−= 2

, )()()(),( ρρρρ

in which term z2 is the Jacobian of the variable transformation. Inserting the PDFs of the random variables

X, Y and Z into above integral, after simplification, yields

++

+

+−=

2/522/32

222

221

222

21

22

,)1(

1)1()(

)1(2exp

2),(

wwww

vuVUγγ

µσ

γσ

µ

πσ

µρ ,

41

where and 222 vuw += 212 )( σσγ = .

Similarly, for four mutually independent Gaussian random variables ( )),0(~ 1σNX ,

( )),0(~ 1σNY , ( )),0(~ 3σNU and ( )),1(~ 4σNV , if defining that VX=1Z , VYZ =2 , and

VUZ =3 , we can find that the joint pdf function of Z1 , Z2 and Z3 is given by

( )

++⋅

+−=

)~1(23

)~1(1

~12exp

41),,(

2

24

2/32221

2

321

2321,, 321 rrrrzzzZZZ

γ

σπ

γγσσσπρ

( ) ( )[ ]uerfcuueurr

u 22

2/12

34

2223)1(

)~1(4

)~1(2 2

+−++

++

+ − πγ

σ

γπ

,

where ( ) 23

231

22

21

2 zzzr σσ++= , and [ ] 2/1224 )~1(2 −

+= ru γσ 214 )(~ σσγ = .

42