8. Bilateral Smoothing of Gradient Vector Field and Application to Image Segmentation

7/30/2019 8. Bilateral Smoothing of Gradient Vector Field and Application to Image Segmentation

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BILATERAL SMOOTHING OF GRADIENT VECTOR FIELD AND APPLICATION TOIMAGE SEGMENTATION

Ravindra S. Hegadi 1, Adithya Kumar Pediredla 2, and Chandra Sekhar Seelamantula 3

1 School of Computational Sciences, Solapur University, Solapur - 413 255, India

2,3Department of Electrical Engineering, Indian Institute of Science, Bangalore - 560 012, India

Emails: [email protected], [email protected], [email protected]

ABSTRACT toas snake energies and the snake outlining is formulated as

a minimization problem. Some early contributions in activeMedical image segmentation nds application in computer-

contour design date back to the work of Kass et al. [1]. Someaided diagnosis, computer-guided surgery, measuring tissue

important recent developments in snakes are the gradient vec-volumes, locating tumors, and pathologies. One approach

torow (GVF), which drives a pixel-based snake [3, 4]; theto segmentation is to use active contours or snakes. Active

parametric B-spline snake formulation of Jacob et al. [5], andcontours start from an initialization (often manually speci-

template-specic snakes [68].ed) and are guided by image-dependent forces to the ob-

We pursue active contour optimization along the lines ofject boundary. Snakes may also be guided by gradient vec-the

GVF concept of Xu and Prince. GVF is a dense vectortor elds associated with an image. The rst main result in

eldderived by minimizing a certain energy functional in athis direction is that of Xu and Prince, who proposed the no-

variational framework. The minimization is accomplished bytion of gradient vector ow (GVF), which is computed iter-

iteratively solving a pair of decoupled linear partial differen-

atively. We propose a new formalism to compute the vectortia

l equations that diffuses the gradient vectors of a gray-levelow based on the notion of bilateral ltering of the gradient

image or a binary edge map computed from an image. Theeld associated with the edge map we refer to it as the

GVFprovides local gradient information to the active con-bilateral vector ow (BVF). The range kernel denition that


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tour and drives the evolution process suitably. The methodwe employ is different from the one employed in the standard

requires that the initialization be provided close to the desiredGaussian bilateral lter. The advantage of the BVF formal-

contour so that the snake can navigate properly in the vectorism is that smooth gradient vector ow elds with enhanced

ow.The GVF has the ability to guide a snake into boundaryedge information can be computed noniteratively. The quality

concavities despite initializing inside, outside, or even acrossof image segmentation turned out to be on par with that ob-

theobject boundary. The standard GVF implementation is

tained using the GVF and in some cases better than the GVF.com

putationally expensive. Han et al. [9] proposed a numer-Index Terms Bilateral lter, Active contour, Snake, ical al

gorithm based on a multigrid method to speed up theGradient vector ow (GVF), Bilateral vector ow (BVF). GVF computations. Recently Li et al. [10] proposed a fast

algorithm for computing GVF. They posed GVF computation1. INTRODUCTION as

a convex optimization problem with an equality constraintand

solved it using an inexact augmented Lagrangian method.Active contours may be broadly classied into two types: In this paper, we propose a non-iterative vector ow con-parametric active contours [1] and geometric active construction by applying a bilateral lter to the gradient vectortours [2]. Parametric active contours are dened over the eld associated with an image edge map. Bilateral lters wereimage domain and move under the inuence of two types of rst in

troduced by Tomasi and Manduchi [11] and are nonlin-forces: internal force dened within the contour and external ear,edge-preserving smoothing lters. Contrary to classicalforce computed from the image data. The internal force en- linear lters that perform smoothing using spatially-invariantsures smoothness of the contour during deformation and the kernels dened on the image domain (domain kernel), bilat-external force pulls the contour points towards the location eral lters employ an additional kernel called the range kernelwith high image gradients (edges). These forces are referred that acts on the intensities associated with the pixels of inter-

est. Typically, both domain and range kernels are chosen as

R. S. Hegadi would like to thank the Indian Academy of Sciences for the Gaussian functions. The bilateral lter restricts the smoothingfellowship opportunity at the Indian Institute of Science, where this work was

process to photometrically similar regions. Bilateral lterscarried out. This work is partly supported by the Department of Science andTechnology (Government of India) Intensive Research in High-Priority havebeen shown to emerge from a Bayesian setting [12]Areas (IRHPA) grant.


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978-1-4673-2533-2/12/$26.00 2012 IEEE 317ICIP 2012

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using a suitably dened penalty function. Direct implemen- of the Gaussian and then computing the gradients along x andtation of the bilateral lters is computationally intensive as y. The vector ow concept also implicitly leads to the notionthe nonlinear ltering process cannot be accelerated using of a basinof attraction as the area around an object of interestfast Fourier transform algorithms. However, thanks to some that attracts the snake towards its boundary. Choosing gradi-ingenious recent developments, accurate approximations of ent map as the vector eld typically results in a small basin ofthe bilateral lter can now be implemented using fast algo- attractionaround the boundary.rithms [13, 14].

The basic idea in BVF construction is to smooth the 2.1. GVFapproachhorizontal and vertical gradients using the bilateral lter.The resulting smoothed eld highlights edge information Xu and Prince [3] proposed to obtain the vector eld v (x, y) =and suppresses the effect of noise in smooth regions. As a [u(x, y)

, v(x, y)], starting with an edge map f (x, y), as a so-consequence, the active contours initialized on the BVF get lution tothe optimization problem given by

attracted faster to the boundaries. Experimental results showthat the quality of outlining obtained is comparable with that

2 2 2 2 2 2 =

(ux +uy +vx +vy )+ |f | |v f | dx dy. (2)obtained with GVF and in some cases better. The key advan-tage of the BVF is that it is obtained noniteratively, whereas The vector eld is obtained by the following iterations.

the GVF is computed iteratively and it takes several tens ofiterations.2

ut(x, y, t) = u(x, y, t) [u(x, y, t) fx (x, y)]

This paper is organized as follows. In Section 2, we2 2

[fx (x, y) + fy (x, y)], andpresent an overview of vector elds, gradient vector ow, and

2then present the bilateral vector ow. In Section 3, we show vt (x

, y, t) = v (x, y, t) [v (x, y, t) fy (x, y)]how snakes perform on the bilateral vector ow. We also

[f 2 (x, y) + f 2 (x, y)],

x yprovide a comparative assessment of segmentation perfor-mance on some synthetic binary images as well as magnetic where thesubscripts denote the corresponding partial deriva-resonance images. We report results on the same images as tives.used by Xu and Prince in [3]. The


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key idea behind solving (2) to create the vector elds,instead o

f using the gradient eld f , is as follows: the rst2. GRADIENT VECTOR FLOW FIELD term of (

2) penalizes the vector eld for not being smooth andthe secon

d term makes it close to its gradient value. TheseA snake is a curve x(s) = [x(s), y(s)], s [0, 1] that moves two terms are optimally combined with the parameter . Theover an image to minimize the energy functional secondterm is weighted with |f |2 to enhance its impor-

tance where the edges are strong and more importance to

E = 1 1 |x (s)|2 + |x (s)|2 + Eext (x(s)) ds, the smoothingterms when it is small. If the image is noisy,

0 2 the valueof should be set high to give more importance to

smoothing this is the statistical signicance of .where and are parameters that control the snakes tensionand rigidity, respectively, and x (s) and x (s) denote the rst 2.2. Bilateral lter approach to compute v (x, y)

and second derivatives with respect to s. A snake that mini-mizes E must satisfy the Euler equation: A gradient vector ow together with associated smoothing

properties can be alternatively obtained by smoothing the vec-

x (s) x (s) Eext = 0, tor eld f (x) y using a bilateral lter. We propose

the use of a bilateral lter with a domain kernel as in the stan-which can be viewed as a force balance equation of the type dard case(Gaussian kernel with standard deviation d) and a

Fint + Fext = 0, where Fint = x (s) x (s) and specic range krnel specied as follows:

Fext = E xt . In the GVF formulation, the potential forceE is replaced with v (x, y), which is the GVF, to yield

2 2ext

f (x) f (x) f (y)the balance equation: h(x, y) = exp 22 , (3)

r

xt (s, t) = Fint + Fext = Eint Eext , where denotes he gradient operator; x is the point of in-

= x (s, t) x (s, t) + v. (1) terest and y isa point in its neighborhood. The same bilat-

eral lteris used to smooth the horizontal and vertical com-The curve solving the above dynamic equation is called a ponents of f . Note that the range lter denition is mod-GVF snake. The solution is computed iteratively beginning ied fromthe standard range lter denition (in particular,


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with an edge map derived from the image. Typically, this is note thepresence of the additional term f (x) in the deni-obtained by Gaussian smoothing rst with a certain parameter tion). Theintuition behind obtaining vector eld by ltering

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the gradient eld using the bilateral lter is as follows. Thespecic range kernel employed in the bilateral lter highlightsthe edge information. In regions where the edge informationis not strong, that is, where the regions are relatively homo-geneous, and corrupted by noise in some cases, the bilaterallter performs smoothing and suppresses noise in the gradi-ents. If the value of r is increased, that is, the spread ofthe range kernel is increased, smoothing occurs over regionshaving large intensity differences. This is preferable in high

(a) (b)noise situations. Thus, the parameter r effectively plays therole of in (2). The parameter d controls the domain of thebilateral lter, which determines the spread of the gradient in-formation. The counterpart of this effect in the standard GVFformulation is the role of the iterations in smoothing the vec-

tor ow eld and ensuring that it creates a basin of attractionaround the edges. Hence, the proposed methodology is analo-gous to the GVF, with the main advantage that it can be com-puted noniteratively. Thanks to some recent developments,

(c) (d)fast implementation of the bilateral lter is possible [13, 14].

2.3. Discrete BVF computation

The BVF on edge-map f (x) is obtained as

BVF(x) = kd 1 (x) R2 f ()c(, x) d, where

(e) (f )

Fig. 1.Segmentation of frontal horn from brain MRI image: (a)

2 2 2 f (x) f (x) f () x original image

alization, (b) edge map obtained after Gaus-22 22 sian smooth

ing with =5, (c) BVF generated on original image, (d)c(, x) = e r e r ,

resultof segmentation using BVF, (e) GVF generated on original

kd (x) = 2 c(, x) d. image, and(f) result of segmentation using GVF with =2 (Data

R courtesy http://www.mr-tip.com).

The choice of kernel is based on the noise distribution. Wechose the standard Gaussian kernel. For discrete images and

3. ACTIVE CONTOURS USING BVFnite support kernels, which are more practical, replacing the


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integrals by sums, we have In Figure 1, we show the segmentation of frontal horn from

brain MRI image. The scale parameter is chosen empiri-

x+m n+m

cally based on basin of attraction. From Figure 2, we observeBVF(x, y) = k1 (x, y) f (k, l)c(x, y, k, l), that the BVF performs segmentation quite effectively and as

dk=xm l=nm accurately

as the GVF. The GVF is computed iteratively usingthe MAT

LAB toolbox of Xu and Prince, available at authorswhere

website: http://www.iacl.ece.jhu.edu/static/gvf/.

2 2 Inorder to verify that BVF is also capable of driving ac-

f (x, y) f (x, y) f (k, l) 22 tive contou

rs into boundary concavities, we consider the samec(x, y, k, l) = e r example

s as those of Xu and Prince. The test images are(x k)2 + (y l)2 shown inFigure 2(a). The initial contours are also shown

22 on the images. The corresponding BVF and GVF and the

e d , segmentation results obtained are also shown. We observe

x+m n+m

that the BVF-based segmentation is comparable to the GVF-

kd (x, y) = c(x, y, k, l). based s

egmentation in the rst two cases (rst two rows ofk=xm l=nm Figure 2),and in case of the cardiac MRI image, it is better

than the GVF-based segmentation.

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

Fig. 2. Comparison of results obtained using BVF and GVF: (a) Original image with initialization, (b) bilateral vector eld with d =3 and r = 2, (c) segmentation using BVF, (d) gradient vector eld, and (e) segmentation using GVF (Data courtesy: Xu et al. [3]).

4. CONCLUSIONS[6] A. K. Pediredla and C. S .Seelamantula, Active-contour-

based automated image quantitation techniques for WesternWe showed that by applying a bilateral lter to the gradient of


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Blot Analysis, 7th Intl. Symp. on Image and Signal Process.an edge map, a smooth vector ow, which we referred to as

and Anal., pp. 331-336, September 2011.the bilateral vector ow, can be obtained. The BVF enhances

[7] R. Delgado-Gonzalo, P. Thevenaz, C. S. Seelamantula, andthe edge information and suppresses noisy gradients. This is

M. Unser, Snakes with an ellipse-reproducing property,a noniterative alternative to the gradient vector ow. Experi-

IEEE Trans. Image Process., vol. 21, no. 3, pp. 12581271,mental results show that the BVF is capable of providing good

March 2012.segmentation performance even in the presence of boundary

[8] A. K. Pediredla and C. S. Seelamantula, A unied approachconcavities. The accuracy of segmentation is comparable to

for optimization of snakuscules and ovuscules, Proc. IEEEthat of GVF and in some cases better. The proposed BVF for-

Intl. Conf. on Acoust. Speech and Signal Process., pp. 681-malism is extendable to 3-D to perform volume segmentation.

684, March 2012.

[9] X. Han, C. Xu, and J. L. Prince, Fast numerical scheme for5. REFERENCES

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