A fast and flexible multiresolution snake with a definite termination criterion

doi:10.1016/S0031-3203(00)00077-7

Pattern Recognition

Volume 34, Issue 7, 2001, Pages 1483-1490

https://doi.org/10.1016/S0031-3203(00)00077-7 Get rights and content

Abstract

This paper describes a fast process of parametric snake evolution with a multiresolution strategy. Conventional parametric evolution method relies on matrix inversion throughout the iteration intermittently, in contrast the proposed method relaxes the matrix inversion which is costly and time consuming in cases where the resulting snake is flexible. The proposed method also eliminates the input of snake rigidity parameters when the snake is flexible. Also, a robust and definite termination criterion for both conventional and proposed methods is demonstrated in this paper.

Introduction

Snakes [1], or active contours, are curves defined within an image domain that can move under the influence of internal forces coming from within the curve itself and external forces computed from the image data. The internal and external forces are defined so that the snake will conform to an object boundary or other desired features within an image. Snakes are widely used in many applications, including edge detection [1], shape modeling [2], [3], segmentation [4], [5], and motion tracking [4], [6].

There are, in general, two types of active contours presently: parametric [1] and geometric [7], [8], [9]. There have been several attempts to increase the capture range of parametric snakes through balloon model [10], through multiresolution scheme [11] and more recently through gradient vector flow method [12]. Comparatively only a few significant publications are there on the improvement of parametric snake evolution. Originally, Kass et al. introduced the method of finite difference [1] and subsequently Cohen et al. introduced finite element method [13]. This paper presents a fast snake evolution method coupled with a definite termination criterion for parametric snakes. A conventional evolution method of parametric snakes relies on inverting the rigidity matrix of the snake where one can apply any o(n) inversion method [4]. But during the whole evolution process this rigidity matrix changes as the number of snake points (snaxels) or the rigidity parameter changes [1]. So there is a need to invoke the matrix inversion routine quite frequently during the process [4]. This requirement naturally makes the process slow. Our proposed method bypasses this matrix inversion method in case of flexible snakes. In addition, we have proposed a multiresolution approach to speed up our algorithm further. Though multiresolution method as proposed by Leroy et al. [11] addresses the problem of initialization, it lacks the movement of snakes through different resolutions. In comparison, our proposed scheme explains the movement of snake through different resolutions successfuly.

Termination of the parametric snake evolution is a complicated process as the snake begins to oscillate at or near the solution [4]. We have also proposed a termination criterion that is based on the position of the snake and the external force at that position. This criterion looks for change of direction of the external force on the snaxels. The uniqueness of this termination condition is that it is well suited for the general snake equation (Euler equation) and also to the cases where external forces may not be specified in terms of gradient of a potential field [12].

The paper is organised as follows. We describe the necessary background briefly in Section 2. In Section 3, proposed multiresolution fast snake evolution process and the termination criterion are presented. Implementation of the algorithm and the experimental results are discussed in Section 4. Finally, concluding remarks are given in Section 5.

Section snippets

Background

A conventional snake is a parametric curve $C(s)=[x(s), y(s)]$ , $s∈[0, 1]$ that minimizes the energy functional [12] $E_{snake} = ∫ 01 12 {α|C′(s)|^{2} +β|C″(s)|^{2}}+E_{ext} (C(s)) d s,$ where the curly bracketed term of the integral represents internal energy of the snake and E_ext represents the external or image energy term. α and β of internal energy term represent, respectively, resistance to tension and that to the bending of the snake. Both of them are non-negative quantities. C′(s) and C″(s) denote the first and

Proposed method

From the foregoing discussion, it is evident that the existing conventional snake evolution method is inherently slow. Here we propose a fast method for flexible snake evolution coupled with a definite criterion for its termination.

A closer look at , reveals that they are force balance equations. The terms $−A x^{t}$ and $−A y^{t}$ are the x and y components of internal or snake force and the terms $u(x^{t}, y^{t})$ and $v(x^{t}, y^{t})$ are the corresponding components of external force at iteration t. They are added

Results and discussion

We have tested our proposed algorithm on a large number of synthetic and real images. Here we present only a few results obtained through the proposed multiresolution method. We also present the results of conventional evolution method as a basis of comparison with our method.

In all these experiments we have used GGVF as the guiding force for snake, with K=0.0005 [14] and 500 iterations. For the conventional snake evolution we have used α=0.6 and β=0.2 in all cases. We have implemented all our

Conclusion

This paper presents a fast implementation of parametric snake evolution coupled with a robust termination criterion. Our principal effort is to eliminate the o(n) matrix inversion and o(n) multiplications in flexible parametric snake evolution. This fast process may help in time critical processes like tracking cyclones from satellite images, tracking movement and change in cells in biomedical images, where flexibility of the active contour is a desirable property.

Termination criterion is based

Acknowledgements

The authors acknowledge the Indian Space Research Organisation (ISRO) for providing funds for carrying out the research and the Director, Indian Statistical Institute for providing the necessary infrastructure. The authors also acknowledge Prof. N.R. Pal and Dr. D.P. Muhkerjee for their valuable comments and suggestions.

About the Author—NILANJAN RAY received the B.E. degree in mechanical engineering in 1995 from Jadavpur University, Calcutta, India and the M.Tech. degree in computer science in 1997 from Indian Statistical Institute, Calcutta, India.

Currently, he is working as a research assistant in Oklahoma Imaging Laboratory, Oklahoma State University, USA, under the supervision of Dr. S.T. Acton. His research interests include image processing through anisotropic diffusion, level set methods, snake

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About the Author—BHABATOSH CHANDA Born in 1957. Received B.E. in Electronics and Telecommunication Engineering and Ph.D. in Electrical Engineering from University of Calcutta in 1979 and 1988, respectively. Received “Young Scientist Medal” of Indian National Science Academy in 1989 and “Computer Engineering Division Medal” of the Institution of Engineers (India) in 1998. He is also the recepient of UN fellowship, UNESCO-INRIA fellowship and fellowship of National Academy of Science, India during his career. He worked at Intelligent System lab, University of Washington, Seattle, USA as a visiting faculty from 1995 to 1996. He has published more than 50 technical articles. His research interest includes image processing, pattern recognition, computer vision and mathematical morphology. Currently he is working as a professor in Indian Statistical Institute, Calcutta, India.

About the Author—JYOTIRMAY DAS received the M. Tech. degree from the Institute of Radio Physics and Electronics and Ph.D. degree both from the University of Calcutta.

His current research interests include Atmospheric pattern analysis and remote sensing applications. He is currently the professor and head of the Electronics and Communication Sciences Unit of Indian Statistical Institute, Calcutta.

View full text

A fast and flexible multiresolution snake with a definite termination criterion

Abstract

Introduction

Section snippets

Background

Proposed method

Results and discussion

Conclusion

Acknowledgements

Comput. Med. Imag. Graph.

CVGIP: Image Understanding

Signal Processing

Snakes: Active contour models

Int. J. Comput. Vis.

Deformable models

Vis. Comput.

Tracking deformable objects in the plane using an active contour model

IEEE Trans. Pattern Anal. Machine Intell.

Dynamic contour: A texture approach and contour operations

Vis. Comput.