Subscribe Now Subscribe Today
Research Article
 

Image Segmentation with Partial Differential Equations



Bin Zhou, Xiao-Lin Yang, Rui Liu and Wei Wei
 
Facebook Twitter Digg Reddit Linkedin StumbleUpon E-mail
ABSTRACT

In many practical fields, image segmentation is the most important procedure. The purpose of image segmentation is to detect the objects in images. The methods based on statistic theory can work well on images with no noise or little noise. But the procedure of segmentation is difficult to be obtained and the accuracy of result often depends on some artificial parameters. A lot of physical phenomenon can be described by Partial Differential Equations (PDEs) and related procedure is easy to be displayed. With the applications of PDEs, it is convenient to accomplish segmentation and represent the procedure.

Services
Related Articles in ASCI
Search in Google Scholar
View Citation
Report Citation

 
  How to cite this article:

Bin Zhou, Xiao-Lin Yang, Rui Liu and Wei Wei, 2010. Image Segmentation with Partial Differential Equations. Information Technology Journal, 9: 1049-1052.

DOI: 10.3923/itj.2010.1049.1052

URL: https://scialert.net/abstract/?doi=itj.2010.1049.1052
 
Received: January 26, 2010; Accepted: March 28, 2010; Published: May 12, 2010



INTRODUCTION

Image segmentation is an important activity in the fields of image analysis, pattern recognition and computer vision and there are many approaches about it (Caselles et al., 1997; Chan and Vese, 2001; Renka, 2009; Sleigh, 1986). Segmentation is the important technique for detecting objects and analyzing images. Segmentation methods fall into several categories such as histogram analysis, region growing, edge detection and partial differential equations (PDE)-based methods, etc.

An image can be segmented by the histogram analysis method (Russ, 2003) based on the distribution of its intensity and some pre-defined thresholds. This method does not make use of spatial structural information and it is effective for simple images with small amounts of structure.

Edge-based segmentation methods (Jahne, 2002) first search the edges of objects in the image and then use this edge information to reconstruct complete boundaries for the principal objects in the image. It is based on the fact that the position of an edge given by an extreme of the first-order derivative or a zero crossing in the second-order derivative. The main shortcoming is that the true edges are often fragmented by noise in the image and it will lead to error.

Region growing (Gonzalez and Woods, 2002) is one of the most popular segmentation methods. It segments an image by splitting the image into smaller regions and merging them into larger ones with k-means, k-nearest neighborhood or some other clustering method. Merging or grouping criteria may be based on criteria such as homogeneity, proximity, color, gray level or texture.

The PDE based methods (Morel and Solimini, 1995) are the most convenient and effective methods for image segmentation. The main advantage is that the theory behind the concept and the solution techniques are well established in other fields such as physics and mechanics. The snake (Kass et al., 1987), the gradient vector flow (Xu and Prince, 1998) and the level set method (Osher and Fedkiw, 2003), are typical of the methods. Recently, a level set model without re-initialization (Li et al., 2005), Sobolev gradient method (Renka, 2009), p-Laplace model (Zhou and Mu, 2010) are proposed.

The level set method is first introduced by Osher and Sethian (1988). It is simple and adaptable for computing and analyzing the motion of an interface in two or three dimensions. The moving interface or fronts are described by the zero level set in conventional level set methods. The level set method has been applied in a wide range of successful applications, including problems in fluid mechanics, combustion, solids modeling, computer animation, material science and image processing over the years (Han et al., 2003). These interfaces may easily develop sharp corners, break apart and merge together in a robust and stable way.

FUNDAMENTAL MATHEMATICAL FORMULATIONS

Many functional and PDEs have been applied in image segmentation.

Image for - Image Segmentation with Partial Differential Equations
Fig. 1: (a-h) Mean curvature motion and a topology change

One general image segmentation model was proposed by Mumford and Shah (1989). The related functional is denoted as following:

Image for - Image Segmentation with Partial Differential Equations
(1)

where, β and v are pre-defined.

A two-phase segmentation is proposed by Chan and Vese (2001) and the evolution equation can be denoted by

Image for - Image Segmentation with Partial Differential Equations
(2)

where, c1, c2 and β are pre-defined.

As follows, a p-Laplace equation (Zhou and Mu, 2010) is applied to image segmentation.

Image for - Image Segmentation with Partial Differential Equations
(3)

where, p>1 and λ is fixed.

Besides these mentioned above, there are also many other PDEs have been used in related applications.

In traditional level set method, curves or surfaces (active contours) are represented in implicit form as the zero level set of a high dimensional continuous function which called level set function. The evolution of the function φ can be governed by a Hamilton-Jacobi equation (Tian and Mu, 2009):

Image for - Image Segmentation with Partial Differential Equations
(4)

where, the function F is called the speed function. For image segmentation, the function F depends on the image data I and the level set function φ. The well-known mean curvature motion is shown in the first row of Fig. 1 a-h. Another example describes a topology change in the evolution is shown in the second row.

Recently, a new variational formulation of level set method without re-initialization is proposed by Li et al. (2005). In fact, the standard re-initialization method is to solve the re-initialization equation

Image for - Image Segmentation with Partial Differential Equations

where, φ0 is the function to be re-initialization and sign (φ) is the sign function. It is a simple version of Eq. 4.

LEVEL SET EVOLUTION BASED ON PARTIAL DIFFERENTIAL EQUATION

In order to get the level set equation, variational level set methods are traditional to be used. Based on this idea, the segmentation governed by a partial differential equation can be implemented.

Figure 2 a-d display the level set evolution for image segmentation of the first example, the image is 134x161 and the zero-contour of the initial level set function is shown in the first figure. It is encloses the object. The typical zero-contours and final zero-contours are shown in Fig. 3a-d.

As for the second numerical example, the evolution of the level sets and the two separately objects that we want to segment are shown in Fig. 3a-d. The image is 84x84 and it has been polluted seriously.

Image for - Image Segmentation with Partial Differential Equations
Fig. 2: (a-d) Extraction of one object with different initial contours

Image for - Image Segmentation with Partial Differential Equations
Fig. 3: (a-d) Extraction of two objects with serious pollution

The figure on the left shows the initial contour. Due to the level set evolution, the algorithm allows the automatical change of topology. This example shows the PDE-based method can deal with the image with pollution and complex initial contour.

CONCLUSION

This study discarded image segmentation and the partial differential equation. Many PDEs can be applied to image segmentation and a lot of PDE-based methods or models have been proposed. Those are convenient and effective for image segmentation.

ACKNOWLEDGMENTS

This study is supported by National Natural Science Foundation of China (No. 10926055). The authors would like to thank the anonymous reviewers for their valuable comments.

REFERENCES

1:  Caselles, V., R. Kimmel and G. Sapiro, 1997. Geodesic active contours. Int. J. Comput. Vision, 22: 61-79.
CrossRef  |  Direct Link  |  

2:  Chan, T.F. and L.A. Vese, 2001. Active contours without edges. IEEE Trans. Image Process., 10: 266-277.
CrossRef  |  Direct Link  |  

3:  Gonzalez, R.C. and R.E. Woods, 2002. Digital Image Processing. Pearson Education Inc., Upper Saddle River, NJ., USA

4:  Han, X., C. Xu and J.L. Prince, 2003. A topology preserving level set method for geometric deformable models. IEEE Trans. Pattern Anal. Machine Intell., 25: 755-768.
Direct Link  |  

5:  Jahne, B., 2002. Digital Image Processing. Springer, Heidelberg

6:  Kass, M., A. Witkin and D. Terzopoulos, 1987. Snakes active contour models. Int. J. Comp. Vision 1: 321-331.

7:  Li, C., C. Xu, Y, C. Gui and M.D. Fox, 2005. Level set evolution without re-initialization: A new variational formulation. Proceedings of the International Conference on Computer Vision and Pattern Recognition, June 20-26, 2005, Washington, DC., USA., pp: 430-436
CrossRef  |  

8:  Morel, J.M. and S. Solimini, 1995. Variational methods in image segmentation. Prog. Nonlinear Differential Equations Appl., 14: 245-245.
Direct Link  |  

9:  Mumford, D. and J. Shah, 1989. Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Applied Math., 42: 577-685.
CrossRef  |  Direct Link  |  

10:  Osher, S. and R. Fedkiw, 2003. Level Set Methods and Dynamic Implicit Surfaces. Springer, New York

11:  Osher, S. and J.A. Sethian, 1988. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys., 79: 12-49.
CrossRef  |  Direct Link  |  

12:  Renka, R.J., 2009. Image segmentation with a Sobolev gradient method. Nonlinear Anal. TMA., 71: 774-780.
Direct Link  |  

13:  Russ, J.C., 2003. The Image Processing Handbook. CRC Press, USA

14:  Sleigh, A.C., 1986. The extraction of boundaries using local measures driven by rules. Pattern Recognition Lett., 4: 247-258.
CrossRef  |  

15:  Tian, Y. and C.L. Mu, 2009. Extiction of viscous Hamiltion-Jacobi equations. J. Sichuan Univ., 46: 15-20.

16:  Xu, C. and J.L. Prince, 1998. Snakes, shapes and gradient vector flow. IEEE Trans. Image Process., 7: 359-369.
CrossRef  |  Direct Link  |  

17:  Zhou, B. and C.L. Mu, 2010. Level set evolution for boundary extraction based on a p-Laplace equation. Applied Math. Modelling,
CrossRef  |  

©  2022 Science Alert. All Rights Reserved