On the Instantaneous Coefficient of Variation for ... -...
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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, [email protected]. †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 [email protected].
Permission to publish this abstract separately is granted.
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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
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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
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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
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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
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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
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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)
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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)
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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
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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
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(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.
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(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].
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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