Curvelet-based geodesic snakes for image segmentation with multiple objects

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Abstract

Curvelet transform is a multiscale and multidirectional geometric wavelet transform, which is an optimal sparse representation of edges and contours of objects. In this paper, a curvelet-based geodesic snake (CGS) is proposed for image segmentation of multiple objects. By producing the edge map of objects by curvelet thresholding instead of simple gradient methods, the proposed method shows great promises to recognize edges of multiple objects with weak edges and strong noises. In addition, we design several rules to quantitatively compare the segmentation accuracy.

Introduction

Image segmentation is one of the focused topics in image processing and computer vision. Snake-based image segmentation has been widely applied in medical images (Yezzi et al., 1997), texture segmentation (Sagiv et al., 2006) and visual tracking (Paragios and Deriche, 2000, Niethammer et al., 2006).

Traditional snake mainly refers to the parametric snake (Kass et al., 1987, Xu and Prince, 1998a, Xu and Prince, 1998b). The main limitations of the traditional snake are that it is dependent on parameters and cannot deal with topology changes. To solve the problems, geometric snake was proposed based on the mean curvature motion equation (Caselles et al., 1993, Malladi et al., 1994). However, the geometric snake uses heuristic stopping procedures when an edge is reached, and the image force cannot realize a complete stop. An improved version named geodesic active contour (GAC) was then proposed (Caselles et al., 1997). These two versions are both implemented by level sets (Osher and Sethian, 1988). The GAC is a particular case of the classical energy-based geometric snake, and equivalent to finding a curve of minimal weighted length in Riemannian space based on geodesic or local minimal distance computations. The GAC uses the negative gradient of the edge detector as an attraction force to project on the normal of the curve. This force balances the other terms close to the boundaries and causes the curve to stop there. In this paper, we follow the GAC method.

Improvements of active contours have been presented in the last few years. Paragios and Deriche (2002) presented a curve-based variational framework that integrates boundary- and region-based information modules to deal with partition problems. Chan and Vese (2001) proposed a model of image region segmentation where the stopping term at object boundaries does not depend on the image gradient. Xie and Mirmehdi (2004) proposed a region-aided geometric snake that integrates gradient flow forces with region constraints, thus more tolerant towards weak edges and noise. Xie and Mirmehdi (2008) also introduced a magnetostatic active contour model where the external force field is based on magnetostatics and hypothesized magnetic interactions between contours and object boundaries.

Applying multiscale methods to snakes is one of the hot topics in image segmentation. Coarse-to-fine scale strategies make the segmentation algorithms more economical for computation. Wu et al. (2000) proposed a directional image force for active contours based on wavelet frames. The wavelet-based snakes are helpful for noise suppression. Bresson et al. (2006) applied linear scale space into the parametric snake model. Mignotte and Meunier (2001) presented a multiscale approach for deformable contour optimization relying on a multigrid coarse-to-fine relaxation algorithm. Mukherjee and Acton (2002) introduced an integrated approach to cloud tracking based on scale-space classification. Keserci and Yoshida (2002) proposed an edge-guided wavelet snake model to fit multiscale edges. Liu and Hwang (2003) proposed an integrated wavelet-based snake model for segmentation and tracking based on the coarse-to-fine strategy. Tang and Acton (2004) combined the B-spline model with multiscale gradient vector flow external force to elude clutter and to reliably localize the vessel boundaries. Recently, a curvelet-based parametric snake has been proposed by Ma et al. (2006) for multiscale tracking of geophysical fluids, which can deal with the segmentation of images with weak edges and noise in an efficient way. However, this curvelet-based snake is performed only for single-object detection and tracking.

In this paper, we extend the snake model proposed in (Ma et al., 2006) to multi-object detection and segmentation by combining the curvelets with the traditional GAC model.

Section snippets

Curvelets

Curvelet transform is a new multiscale geometric wavelet transform, which can represent edges and curve-singularities much more efficiently than traditional wavelet transforms. Curvelets can also capture intrinsic geometric structures of objects with fewer coefficients for a given accuracy of reconstruction in comparison with wavelets. The needle-shaped elements of curvelets are shown in Fig. 1. In this paper, we apply the second generation discrete curvelet transform (DCuT) (Candès and Donoho,

Curvelet-based geometric snakes

The traditional snake is defined by an evolution course of a closed curve C(s)=[x(s),y(s)]T, s  [0,1] obeying a specific energy formula (Kass et al., 1987)E(C,Cs,Css)=0112α(s)Cs2+12β(s)Css2+g(C)ds,where Cs and Css denote the first and second order partial derivatives of the curve C with respect to the parameter s, respectively. The α(s) and β(s) are dependent on design parameters (usually set constants) of an internal energy function controlling the elasticity and rigidity of the snake. The

Experimental results

We present numerical experiments by applying the CGS to various images with different shapes, topologies and noises. Firstly, we show the basic running process of the CGS and evaluate its performances in comparison with the standard GAC. Secondly, we show good ability of the CGS for segmentation of images with noise. Thirdly, a case with weak edges is tested.

Conclusion

In this paper we propose a curvelet-based geometric snake for image segmentation with multiple objects. The main contribution of this paper is to extend the existent curvelet-based parameter snake for single-object segmentation to the geometric snake for multi-object segmentation. The edge map of the objects contained in the given images is obtained by curvelet thresholding, which, contrary to simple gradient methods, overcomes the issue of strong noise and weak edges. Curvelets are indeed

Acknowledgments

The authors thank the anonymous reviewers for their constructive comments and fruitful suggestions, which have contributed in improving the quality of the paper. The authors also thank financial support by National Natural Science Foundation of China under Grant Nos. 40704019 and 40674061, Tsinghua Basic Research Fund (JC2007030).

References (28)

  • V. Caselles et al.

    Geodesic active contours

    Internat. J. Comput. Vision

    (1997)
  • T. Chan et al.

    Active contours without edges

    IEEE Trans. Image Process.

    (2001)
  • M. Kass et al.

    Snakes: Active contour models

    Internat. J. Comput. Vision

    (1987)
  • J. Liu et al.

    Active contour model using wavelet modulus for object segmentation and tracking in video sequences

    Internat. J. Wavelets, Multiresolution Infor. Process.

    (2003)
  • Cited by (0)

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