Active contours driven by local image fitting energy
Introduction
The active contour models (ACM) [1], which are based on the theory of surface evolution and geometric flows, have been extensively studied and successfully used in image processing. The level set method proposed by Osher and Sethian [2] is widely used in solving the problems of surface evolution. Later, geometric flows were unified into the classic energy minimization formulations for image segmentation [3], [4], [5], [9], [10], [11], [16], [17], [23]. Generally speaking, the existing ACM methods can be classified into two types: edge-based models [1], [2], [3], [4], [6], [9], [14], [19], [20], [23] and region-based models [5], [7], [8], [10], [11], [16], [17], [18], [21]. Each of them has its own pros and cons.
The edge-based models utilize image gradient as an additional constraint to stop the contours on the boundaries of desired objects. Usually, a stopping function is used to attract the contours to the desired boundaries. In order to enlarge the capture range of the force, a balloon force term is often incorporated into the evolution function, which controls the contour to shrink or expand. However, it is difficult to choose a proper balloon force. Either a too large or too small balloon force will result in undesirable effects [10].
Region-based models utilize the image statistical information to construct constraints, and have more advantages over edge-based models. First, they do not use the image gradient, and can successfully segment objects with weak boundaries or even without boundaries. Second, the initial contour can start anywhere in the image, and the interior contours can be automatically detected. One of the most popular region-based models is the C–V model [5], which has been successfully used in binary phase segmentation with the assumption that each image region is statistically homogeneous. However, the C–V model does not work well for the images with intensity inhomogeneity. Vese and Chan extended their work in [17] to utilize multiphase level set functions to represent multiple regions. These models are called the piecewise constant (PC) models. Nonetheless, both the C–V and the PC models have the drawback described above.
In order to segment images with intensity inhomogeneities, Vese and Chan [17] and Tsai et al. [16] proposed two similar models, which are called piecewise smooth (PS) model. However, these methods are computationally inefficient. More discussions above the properties of PC model and PS model can be found in [10], [11]. Li et al. [10], [11] proposed the LBF (local binary fitting) model, which utilizes the local image information as constraints, can well segment objects with intensity inhomogeneities. Furthermore, LBF model has better performance than PC and PS models in segmentation accuracy and computational efficiency.
In this paper, we propose a novel ACM model that can be used to segment images with intensity inhomogeneities. We utilize the local image information to construct a local image fitting (LIF) energy functional, which can be viewed as a constraint of the differences between the fitting image [10], [11] and the original image. Furthermore, a novel method is used to regularize the level set function by using Gaussian kernel filtering after each iteration. In addition, re-initialization is not needed in the proposed method. The complexity analysis and experimental results show that the proposed method is more efficient than the LBF model, while yielding similar results.
The rest of the paper is organized as follows. In Section 2, we review some classic models and indicate their limitations. Section 3 describes our model and its variational formulation. In Section 4, we validate our method by various experimentations on synthetic and real images. Conclusion is made in Section 5.
Section snippets
The Mumford and Shah (MS) model
In [8], Mumford and Shah formulated the image segmentation problem as follows: find an optimal piecewise smooth approximation function u of image I, which varies smoothly within each sub-region of image domain , and rapidly or discontinuously goes across the boundaries of . They proposed the following energy functional:where is the length of the contour C, are fixed parameters.
The unknown set C and the non-convexity of the above energy
LIF model and its variational level set formulation
A local fitted image (LFI) formulation is defined as follows:where and are defined as follows:where is a rectangular window function, e.g. a truncated Gaussian window or a constant window. In our experiment, we choose a truncated Gaussian window with standard deviation and of size by , where k is the greatest integer smaller than . Similar segmentation results can be achieved if
Experimental results
Our algorithm is implemented in Matlab 7.0 on a 2.8-GHz Intel Pentium IV personal computer. In this section, we apply our method to synthetic images and real images of different modalities, and use the same parameters , , , and time-step . Parameter σ is chosen by experience according to the images The Matlab source code of the proposed algorithm can be downloaded at http://www.comp.polyu.edu.hk/∼cslzhang/code/LIF.zip.
Conclusions
In this paper, we proposed a novel active contour model driven by local image fitting (LIF) energy. The proposed LIF energy functional has less computational complexity than the local binary fitting (LBF) energy functional. A novel level set method based on Gaussian filtering was used to implement our variational formulation, and the experimental results revealed that it is not only robust to prevent the energy functional from being trapped into local minimum, but also capable of keeping the
About the Author—KAIHUA ZHANG was born in Rizhao, China, in 1983. He received his B.S. degree in science and technology of electronic information from Ocean University of China (OUC) in 2006 and master degree in signal and information processing from the University of Science and Technology of China (USTC) in 2009. Currently he is a research assistant in the department of computing, The Hong Kong Polytechnic University. His research interests include pattern recognition and image processing.
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About the Author—KAIHUA ZHANG was born in Rizhao, China, in 1983. He received his B.S. degree in science and technology of electronic information from Ocean University of China (OUC) in 2006 and master degree in signal and information processing from the University of Science and Technology of China (USTC) in 2009. Currently he is a research assistant in the department of computing, The Hong Kong Polytechnic University. His research interests include pattern recognition and image processing.
About the Author—HUIHUI SONG was born in Liao Cheng, China, in 1986. She received her B.S degree in science and technology of electronic information from Ocean University of China (OUC) in 2008. Currently she is a M.S. candidate of University of Science and Technology of China (USTC). Her research interests include pattern recognition and image processing.
About the Author—LEI ZHANG received the B.S. degree in 1995 from Shenyang Institute of Aeronautical Engineering, Shenyang, P.R. China, the M.S. and Ph.D degrees in Electrical and Engineering from Northwestern Polytechnical University, Xi’an, P.R. China, respectively, in 1998 and 2001. From 2001 to 2002, he was a research associate in the Department of Computing, The Hong Kong Polytechnic University. From January 2003 to January 2006 he worked as a Postdoctoral Fellow in the Department of Electrical and Computer Engineering, McMaster University, Canada. Since January 2006, he has been an Assistant Professor in the Department of Computing, The Hong Kong Polytechnic University. His research interests include image and video processing, biometrics, pattern recognition, computer vision, multisensor data fusion and optimal estimation theory, etc. Dr. Zhang is an associate editor of IEEE Transactions on Systems, Man and Cybernetics, Part C.