Image Segmentation of Homogeneous Intensity Regions using Wavelets based Level Set

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 3, Issue 10, October 2013)

515

Image Segmentation of Homogeneous Intensity Regions

using Wavelets based Level Set

K. Karthik

1

, P. Hrushikesh

2 1,2

B. Tech Student, Dept. of ECE, Kakatiya Institute of Technology & Science, Warangal, AP, INDIA.

Abstract — This paper proposes an image segmentation

method that integrates a wavelets feature, which is able to enhance the dissimilarity between regions with low variations in intensity. This feature is integrated to formulate a new level set based active contour model that addresses the segmentation of regions with highly similar intensities, which do not have clear boundaries between them. In the first phase, the strength of wavelet transform will be adapted to formulate wavelet energies. The second phase will be dedicated to regularize a new wavelets based level set formulation. This formulation is composed of two terms that guide the contour, the wavelet energy incorporated region term and the contour smoothness term. This approach is useful and suitable for segmentation of regions within an image with improper boundaries.

Keywords—Wavelets, level sets, image segmentation. I. INTRODUCTION

Image Segmentation refers to the process of partitioning a digital image into multiple segments. Image segmentation is typically used to locate objects and boundaries (lines, curves, etc.) in images. More precisely, it is the process of assigning a label to every pixel in an image such that pixels with the same label share certain visual characteristics. The result of image segmentation is a set of segments that collectively cover the entire image, or a set of contours extracted from the image. Image segmentation is one of the most important preliminary stages in computer-aided diagnosis [3] system that facilitates further object identification, recognition, and quantification.

In past researches, various image segmentation methods have been employed depending upon the image domain, structure of interest and imaging modality. Although there are vast number of gray scale image segmentation methods available, one of the main challenges still faced is the segmentation of regions with high similarity in intensity values with their background.

The level set method is a numerical technique for tracking interfaces and shapes. The advantage of the level set method is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects.

In two dimensions, the level set method representing a closed curve 𝛤 using an auxiliary function 𝜑 , called the level set function. 𝛤 is represented as the zero level set 𝜑 of by Eq. (1) and the level set method manipulates 𝛤 implicitly, through the function 𝜑 . 𝜑 is assumed to take positive values inside the region delimited by the curve and negative values outside.



 



{ x y, |



x y, } 0

  

(1) The basic idea is to represent contours as the zero level set of an implicit function defined in a higher dimension, usually referred as the level set function, and to evolve the level set function according to a partial differential equation (PDE). This approach presents several advantages over the traditional parametric active contours. First, the contours represented by the level set function may break or merge naturally during the evolution, and the topological changes are thus automatically handled. Second, the level set function always remains a function on a fixed grid, which allows efficient numerical schemes. Re-initialization, a technique for periodically re-initializing the level set function to a signed distance function during the evolution [1]. The level set function is obviously a disagreement between the theory of the level set method and its implementation using re-initialization. It still remains a serious problem as when and how to apply the re-initialization. Our variational energy functional consists of an internal energy term and an external energy term, respectively. The resulting evolution of the level set function is the gradient flow that minimizes the overall energy functional. Due to the internal energy [2], the level set function is naturally and automatically kept as an approximate signed distance function during the evolution. Therefore the re-initialization procedure is completely eliminated.

II. BACKGROUND WORK

If the curve moves with a velocity v, then the level set satisfies the level set equation [1]

v t



_

 _{ }

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Here | | is the euclidean norm and t is the time. This is a partial differential equation, in particular a Hamilton-jacobi equation [9], and can be solved numerically.

a. Variational Level Set Formulation of Curve Evolution without Re-initialization

The level-set are dynamic curves that move toward the object boundaries. Therefore we define an external energy[1] that can move towards the edges. If I be the image, then edge indicator function (g) is defined by

2

1 (| |)

1 | * |

g I

G_ I

 

  (3)

Where G is Gaussian kernel with standard deviation



we define an external energy for a function





x, y



as below

Eg ( )_ext   Lg ( )  v Ag( ) (4)

Here





0

and v are constants, and the length and area terms are defined respectively by

Lg( ) g ( ) | |

Ag( ) gH ( ) dxdy

dxdy

   

 



  

   (5)

Where,



is the Dirac function, and H is the Heaviside function.

Now, the following is the total energy functional.

 

int

 

Etotal   Eg   Egext  (6)

The external energy drives the zero level set towards the object boundaries, while the internal energy penalizes the deviation of



from a distance function during its evolution. The variational formula derives from the penalize energy Eq. 6

Where 0 is a parameter controlling the effect of

penalizing the deviation of



from a distance function, and The energy functional Ag (∅) [5]_{introduced are used}

to speed up curve evolution. The coefficient v of Ag can be positive or negative, depending on the relative position of the initial level-set to the object of interest. If the initial level-sets are placed inside the object, the coefficient v

should take negative value to speed up the expansion of the level-sets. By calculus of variations, the Gateaux derivative of the functional E in [6] can be written as

μ[ Δ div ])] λδ ( )div g g ( )

total

E

v t

 _  _{ }

 

   

 _{ } _  _  _

   

 _ _ _  _

(7)

Where Δ is the Laplacian operator, Therefore, the function ∅ that minimizes this functional satisfies the Euler-Lagrange equation .

The gradient flows of the energy function λ, Lg(∅) and

v Ag(∅ ), are responsible of driving the zero level curve towards the object boundaries.

b. Wavelet Decomposition

Wavelet packet decomposition (WPD) is a wavelet transform where the signal is passed through more filters than the discrete wavelet transform (DWT). In the DWT, each level is calculated by passing only the previous approximation coefficients (cAj) through low

and high pass quadrature mirror filters. However in the WPD, both the detail (cDj (in the 1-D case), cHj, cVj,

cDj (in the 2-D case)) and approximation coefficients are

decomposed to create the full binary tree as shown in figure.1 where x[n] signal is decomposed into three levels for which g[n] is the low pass approximation coefficient, h[n] is high pass detail coefficient. For n levels of decomposition [8] the WPD produces 2n different sets of coefficients (or nodes) as opposed to (3n + 1) sets for the DWT. However, due to the down sampling process the overall number of coefficients is still the same and there is no redundancy.

c. Wavelet Energy

Two-dimensional wavelet decomposition is applied on the input image returns EA, which is the percentage of

energy corresponding to the approximation, and vectors EH, EV, ED, which contain the percentages of energy

corresponding to the horizontal, vertical, and diagonal details [7], respectively.

2 ,

( , ) ( ) ( , )

H r s H

V r s

D r D

V

s

E x y D K x r y s

  



(8)

Where K(x, y) is a Gaussian kernel function. Generally, the computation of local wavelet energy of a pixel involves squaring in combination with a logarithmic normalizing nonlinearity of the wavelet coefficients at each pixel.

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This formulation results in one wavelet energy feature per pixel. Each of the wavelet energy value represents the respective pixel’s structures information. The final wavelet energies of all of the pixels in the image produce a wavelet energy map that shows the various structures that are present in the image. It is assumed that each structure in the image will demonstrates a distinctive range of wavelet energy values.

III. LEVEL SET AIDED WITH WAVELETS

Ourproposed work consists of two phases. In the first phase, wavelet energies of an image is extracted from the wavelet transform and in the second phase, wave-energy for the image in phase one is calculated and level set is formulated.

A.Image Feature Extraction via Wavelet Transform

In this phase, discrete wavelet transformation is performed on the image. The wavelet transformation process is iterated until a predefined number of levels to obtain the multi-resolution representation of the image. The levels refer to the inverse of the size aperture of the wavelet filter to be applied on the image the highest level contains the coarsest intensity variation of the image, while the lowest level contains the finest intensity variation.

The wavelet transformation process decomposes the original image into four sub bands: approximation and the three detail sub bands in the horizontal, vertical, and diagonal directions. The approximation sub band preserves the approximate information of the original image, and the detail sub bands capture the intensity variations in the mentioned three directions. Wavelet frame approach is used to scale up the size of each sub band to the size of the original image to produce translation invariant sub bands. This approach results in sub bands with the same size as the original image. After wave let transformation is applied on the image, wavelet coefficients from the detail sub bands of all the decomposition levels are used to formulate the wavelet-based feature [12][4]. This wavelet-based feature is called wavelet energy, which is computed in pixel wise manner. This pixel wise approach assigns one feature for each pixel in the image. Hence, each of the pixels in the image is represented with a wavelet energy value. The formulation of the wavelet energy feature is carried out using Eqn. (8)

B.Wavelet Guided Level Set Evolution

In this second phase of the proposed work, wavelet energy computed from the first phase will be used to infer the statistical information of the pixels in the image. These wavelet energy values are then formulated into the region-based LSAC model.

Region- based LSAC is adapted in this work due to the properties of the regions with low intensity variations demonstrating weak edge information.

Assume that an image I contains two structures: The object 𝛺1, within the background 𝛺2. The proposed model segments 𝛺1 from the 𝛺2, with contour C separating these two structures. Based on the principles stated above, the energy function [7] of the proposed model is formulated as

 

0 1 1 2 2

F







F







F



(9)

The objective of the proposed model is to segment a given image domain,𝛺, into two regions: 𝛺1and, 𝛺2 with the contour, 𝜙 as the zero of the level set function of the contour that delineates these two regions. The maximum likelihood function that maximizes the probability of a given pixel belonging to each corresponding region is used to model the region information. The probability distribution, M1, represents the distribution of 𝛺1’s

wavelet energy and M2 represents the distribution of 𝛺2’s

wavelet energy.

With the image pixels {(i, j) ∈ 𝛺}, the pixels in the 𝛺1

domain described by M1 have the wavelet energy values,









1 i, j 1

W  W : i , j  (10)

The pixels in the 𝛺2 domain described by M2 have the

wavelet energy values,









2 i, j 2

W  W : i , j  (11)

The level set contour 𝜙 is updated to new contour points by maximizing the likelihood that the probability W1 and W2 are random samples drawn from the models

M1 and M2, respectively. This maximum likelihood is

F1(c), which is expressed as F1 (𝜙)

1 1, 2

F ( ) P(W | , M M ) (12)

Where P (W | 𝜙, M1, M2) is the joint probability density

of the wavelet energy W with contour 𝜙, models M1 and

M2.

Let P1 and P2 be the probability density function (PDF)

of model M1 andM2 respectively. The wavelet energy of

all pixels in each region is assumed to be statistically independent. Thus Eqn. (11) is equivalent to the product of two PDF’s, P1 and P2 and with the assumption that the

wavelet energy of all pixels in each region is statistically independent, the PDF’s, P1 and P2 are given as





 

k k i , j k ij

P W |   _kP W for k1, 2 (13)

Thus, Eqn. 10 can be re written as

   

 

1 i, j 1 1 ij i, j 2 2 ij

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International Journal of Emerging Technology and Advanced Engineering

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C.Contour Smoothness Term F2(𝝓)

In order to produce a smooth contour of the segmented region, the length of contour C [5] [6] [10] represented as zero in the level set function 𝜙 is minimized by minimizing the total Euclidean arc length of the contour. Using the regularized Heaviside function, the gradient of Heaviside function gives the contour. Thus the integral of the gradient of the Heaviside function over the image domain gives the length of the contour. The length of the contour is given by

 





2

F   



H  x y, dxdy (15)

The minimization of Eqn. 14 gives the contour smoothness term that will be integrated in the energy function. The following section will present the integration of the region and contour smoothness terms into the proposed energy function.

D. Energy Function F(𝝓)

The region term in Eqn. (14) and contour smoothness term in Eqn. (15) are integrated into the energy function given in Eqn.(9)

 



 





 



   

1 1 1 2

2

, 1 ,

, , (16)

F M H x y M H x y

x y x y dxdy

 



    

   

  

  _  _

 



Where Mk is assumed to have a Gaussian distribution is

given by



2





 



2

, 1

ln 2 (17)

2 2

k

k k

k

u x y M

M 

 

 

The means, m1 and m2, and variances, 2 1

 and 2 2  , will be calculated from Ω1 and Ω2 , respectively.

The final Euler–Lagrange equation corresponding to the energy function, F, with respect to level set function ∅. The minimization of the energy function is computed by making ∅ as a function of time and replacing the zero with the time derivative of ∅.

IV. EXPERIMENTAL RESULTS

The proposed algorithm is been applied on to four different images which are shown in the following figures below. The simulation is carried using MATLAB 7.8 at the processor speed of 2.27 GHz and 3 GB RAM. With initial contour taken as square contour with different size, the parameters controlling the contour evolution are selected accordingly. In Figure 2 the images [a, c, e, g] are the input images for which the proposed model is applied the final segmented images shown by the final contour formed is for just 60 iterations which is very less as compared with that well known methods such as chan-vese, chumming li, and others. Final contour is obtained within 6-9 seconds for all different images.

V. CONCLUSIONS

In this paper, we proposed a new formulation of Level set that utilizes the strength of wavelet transform to segment regions exhibiting homogenous intensity values with the background. The integration of the wavelet energy map to formulate the proposed model has been proven to produce segmented results with high accuracy in perception in comparison with methods that uses intensity values which is hard to segment using conventional. The proposed model of image segmentation outstands in term speed and accuracy.

(a) (b) Final Result

(c) (d)

(e) (f)

(g) (h)

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International Journal of Emerging Technology and Advanced Engineering

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REFERENCES

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[2] Shaojun Liu and Jia Li “Automatic Medical Image Segmentation Using Gradient and Intensity Combined Level Method”, IEEE Annual International Conference on Medical Imaging, pp: 78-82, 2006.

[3] Gang Chen, Lixu Gu , Lijun Qian and Jianrong Xu “An Improved Level Set For Liver Segmentation and Perfusion Analysis in MRIs”, IEEE Transactions on Information Technology in Biomedicine, Vol: 13, pp: 94-103, Jan. 2009.

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[6] T. F. Chan and L. A. Vese, “Active Contours Without Edges”, IEEE Transactions on Image Processing, vol.10(2), 2001, pp.266– 277.

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[9] S. Osher and J. A. Sethian, “Fronts Propagating With Curvature Dependent Speed: Algorithms based on Hamilton-Jacobi Formulation”, Journal of Computational Physics, vol.79, 1988, pp.12–49.

[10] T. Randen and J. H. Husoy, “Filtering for Texture Classification: A Comparative Study”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1999, vol.21 (4), pp.291–310. [11] B. Y. Sandberg, T. F. Chan and L.A.Vese, “A Level-Set and

Gabor Based Active Contour Algorithm for Segmenting Textured Images”, Technical report, Mathematics Department, UCLA, 2002.

[12] M. Unser, “Texture Classification and Segmentation using Wavelet Frames”, IEEE Transactions on Image Processing, vol.4 (11), 19993, pp.1549–1560.