A new model for segmentation of gray-scale and color images
© Zhang et al.; licensee Springer. 2013
Received: 2 March 2013
Accepted: 22 October 2013
Published: 22 November 2013
In this paper, we propose a new model for segmentation of both gray-scale and color images. This model is inspired by the GAC model, the region-scalable fitting model, the weighted bounded variation model and the active contour model based on the Mumford-Shah model. Compared with other active contour models, our new model cannot only make full use of advantages of both edge-based and region-based models, but also maintain more accurate overall message of segmented objects. Moreover, we establish the existence of the global minimum of the new energy functional and analyze the property of it. Finally, numerical results show the effectiveness of our proposed model.
The image segmentation problem is fundamental in the field of computer vision, and the aim of it is to divide an image into a finite number of important regions. Recently, variational methods have been extensively studied for image segmentation because of their flexibility in modeling and advantages in numerical implementation.
The geometric active contour (GAC) model is one of the most well-known variational models [1–4] for image segmentation. It was proposed in  and has been widely used in practice . The main idea of the GAC model is to utilize the image gradient to stop evolving contours on object boundaries. However, this model has one major disadvantage, that is, giving an initial curve, during the evolution, the energy may evolve to its local minimizers.
Moreover, there are other variational models for image segmentation [6–8]. These models do not utilize the image gradient and are significantly less sensitive to the location of initial contours than other modes. Therefore, they have good performance for the image with weak object boundaries. Among these models, the Chan-Vese model  is very popular and useful. This model can be applied for the segmentation of images with two regions, each having a distinct mean of pixel intensities. In order to handle images with multiple regions, Vese and Chan proposed the piecewise constant (PC) models , in which multiple regions can be represented by multiple level set functions. And yet, these PC models are not very successful for images with intensity inhomogeneity. To deal with more general situation efficiently, Chunming Li proposed the region-scalable fitting model in . However, this model is not always valid when the whole object needs to be segmented.
In this paper, inspired by the GAC model, the region-scalable fitting model, the weighted bounded variation model [3–5] and the active contour model based on the Mumford-Shah model , we propose a new model which can be applied to segment both gray-scale and color images. In order to make our model apply to color images, we shall use the red-green-blue model . Although there are some other models which are similar to our new model, they have different essence and applications. Our new model cannot only make full use of advantages of both edge-based and region-based models, but also overcome the usual drawback in the level set approach. Moreover, compared with other active contours models, our model can keep more accurate overall message of segmented objects for simple and complex images of different modalities. Finally, we investigate the new model mathematically and establish the existence of the minimum to the new energy functional.
The remainder of the paper is organized as follows. In Section 2, we show some background. In Section 3, our new model is proposed. Theoretical results, iterating schemes and experimental results are also given in this section. Finally, we conclude our paper in Section 4.
2.1 GAC model
where β is an arbitrary positive constant.
where R, G and B represent the pixel values of red, green and blue after Gaussian convolution, respectively, i.e., , and .
2.2 The region-scalable fitting model
where is the Gaussian function and , are two functions that fit image intensities near the point x. Moreover, ϕ is the level set function embedding the evolving active contour and is the Heaviside function.
3 Our proposed model
where g is a diffusion coefficient defined in the same way as formula (2.2) or (2.3) and is the Gaussian function.
where takes larger values at the points near the center point y, and decreases to 0 as x goes away from y. Therefore, , are allowed to vary in space.
where C is a constant.
when approximating with spatially varying fitting functions , .
The above analysis shows that our new model uses not only the edge detector which contains information concerning the boundaries of objects, but also the spatially varying fitting functions , which are used to approximate the image intensities and avoid existence of the local minimizers to energy functional (3.3).
3.1 Mathematical results
Theorem 1 For any given (, where ), there exists a function minimizing the energy functional E 1 in (3.3).
If , from (2.2) we can obtain with c being a positive constant. Since , where C 1 is a positive constant.
where (). Since (), , where C 2 is a positive constant.
So, is bounded.
Since , the sequence has a bounded BV-norm.
Thus, there is a subsequence, also denoted by , and such that strongly in . Moreover, according to the formula in , we know that there is a subsequence, also denoted by , satisfying a.e. for . Since for any , a.e. for .
That is, strongly in and for every .
Therefore and is a minimum of the energy functional E 1. □
Theorem 2 Let (), . For any given , (), if is any minimum of the energy functional E 1 defined in (3.3), then for almost every , the characteristic function is a global minimum of the functional E 1 where c is the boundary of the set .
3.2 Numerical implementation
Considering u fixed, compute and by using formula (3.2);
Considering and fixed, update u by using the iterative schemes of minimization problem (3.3).
When a steady state is found, the final segmentation is obtained by thresholding u at any level in (in our experiments, we choose ).
where is an exact penalty function provided that the constant α is chosen large enough. The energy functional E 2 is convex, so E 2 does not possess local minimizers. Hence, any minimizer of E 2 is global.
- (1)v being fixed, we search for u as a solution of(3.8)
- (2)u being fixed, we search for v as a solution of(3.9)
3.3 Experimental results
This paper describes a new model for segmentation of gray-scale and color images. This model is based on the GAC model, the region-scalable fitting model, the weighted bounded variation model and the active contour model based on the Mumford-Shah model. Compared with other active contour models, our new model cannot only make full use of advantages of both edge-based and region-based models, but also keep more accurate overall message of the segmented objects. Our numerical results confirm the effectiveness of our algorithm. Moreover, we investigate the new model mathematically and establish the existence of the minimum to the new energy functional.
This work is partially supported by the National Science Foundation of China (11271100, 11126222, 11301113, 71303067), the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2012065, Grant No. HIT. HSS. 201201), China Postdoctoral Science Foundation funded project (Grant No. 2012M510933, Grant No. 2013M541400), the Heilongjiang Postdoctoral Fund (LBH-Z12102), the Aerospace Supported Fund of China (Contract No. 2011-HT-HGD-06), and also the Financial Support from Postdoctoral Science-Research Developmental Foundation of Heilongjiang Province (LBH-Q11111).
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