Image segmentation by iterative optimization of multiphase multiple piecewise constant model and Four-Color relabeling

In the paper an iteratively unsupervised image segmentation algorithm is developed, which is based on our proposed multiphase multiple piecewise constant (MMPC) model and its graph cuts optimization. The MMPC model use multiple constants to model each phase instead of one single constant used in Chan and Vese (CV) model and cartoon limit so that heterogeneous image object segmentation can be effectively dealt with. We show that the multiphase optimization problem based on our proposed model can be approximately solved by graph cuts methods. Four-Color theorem is used to relabel the regions of image after every iteration, which makes it possible to represent and segment an arbitrary number of regions in image with only four phases. Therefore, the computational cost and memory usage are greatly reduced. The comparison with some typical unsupervised image segmentation methods using a large number of images from the Berkeley Segmentation Dataset demonstrates the proposed algorithm can effectively segment natural images with a good performance and acceptable computational time.     

Unsupervised multiphase segmentation: A recursive approach

We propose an unsupervised multiphase segmentation algorithm based on Bresson et al.’s fast global minimization of Chan and Vese’s two-phase piecewise constant segmentation model. The proposed algorithm recursively partitions a region into two subregions, starting from the largest scale. The segmentation process automatically terminates and detects when all the regions cannot be partitioned further. The number of regions is not given and can be arbitrary. Furthermore, this method provides a full hierarchical representation that gives a structure of a given image.     

An enhanced multiphase Chan–Vese model for the remote sensing image segmentation

The level set method has been widely used in image segmentation; however, the complexity of the computation has restricted its application field. Also, it is a big challenge to segment remote sensing image mainly because of the complex terrain. In this paper, an enhanced multiphase phase level set method based on the Chan–Vese (C‐V) model is proposed for segmenting remote sensing images. Compared with the C‐V model, two main contributions of the proposed model mainly include the following: First, we introduce a new strategy of initialization in which the contours of the first k biggest connected regions are extracted as the initial curves (k is the number of level set functions); Second, to increase the accuracy, a morphological gradient component is added to the original intensity image. To investigate the effectiveness and efficiency of the proposed model, we have applied it to analyze different kinds of images, including synthetic, real, and remote sensing images. The experimental results have shown that our method is able to achieve better segmentation with less computational consumption compared with the traditional multiphase C‐V model and local and global intensity fitting model. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiple piecewise constant with geodesic active contours (MPC-GAC) framework for interactive image segmentation using graph cut optimization

This paper proposes an improved variational model, multiple piecewise constant with geodesic active contour (MPC-GAC) model, which generalizes the region-based active contour model by Chan and Vese, 2001 [11] and merges the edge-based active contour by Caselles et al., 1997 [7] to inherit the advantages of region-based and edge-based image segmentation models. We show that the new MPC-GAC energy functional can be iteratively minimized by graph cut algorithms with high computational efficiency compared with the level set framework. This iterative algorithm alternates between the piecewise constant functional learning and the foreground and background updating so that the energy value gradually decreases to the minimum of the energy functional. The k-means method is used to compute the piecewise constant values of the foreground and background of image. We use a graph cut method to detect and update the foreground and background. Numerical experiments show that the proposed interactive segmentation method based on the MPC-GAC model by graph cut optimization can effectively segment images with inhomogeneous objects and background.     

Combinatorial Optimization of the Discretized Multiphase Mumford–Shah Functional

The Mumford–Shah model has been one of the most influential models in image segmentation and denoising. The optimization of the multiphase Mumford–Shah energy functional has been performed using level sets methods that optimize the Mumford–Shah energy by evolving the level sets via the gradient descent. These methods are very slow and prone to getting stuck in local optima due to the use of gradient descent. After the reformulation of the 2-phase Mumford–Shah functional on a graph, several groups investigated the hierarchical extension of the graph representation to multi class. The discrete hierarchical approaches are more effective than hierarchical (or direct) multiphase formulation using level sets. However, they provide approximate solutions and can diverge away from the optimal solution. In this paper, we present a discrete alternating optimization for the discretized Vese–Chan approximation of the piecewise constant multiphase Mumford–Shah functional that directly minimizes the multiphase functional without recursive bisection on the labels. Our approach handles the nonsubmodularity of the multiphase energy function and provides a global optimum if the image estimation data term is known apriori.     

Multiphase Soft Segmentation with Total Variation and <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> Regularization

In this paper, we propose a variational soft segmentation framework inspired by the level set formulation of multiphase Chan-Vese model. We use soft membership functions valued in [0,1] to replace the Heaviside functions of level sets (or characteristic functions) such that we get a representation of regions by soft membership functions which automatically satisfies the sum to one constraint. We give general formulas for arbitrary N-phase segmentation, in contrast to Chan-Vese’s level set method only 2 ^m -phase are studied. To ensure smoothness on membership functions, both total variation (TV) regularization and H ¹ regularization used as two choices for the definition of regularization term. TV regularization has geometric meaning which requires that the segmentation curve length as short as possible, while H ¹ regularization has no explicit geometric meaning but is easier to implement with less parameters and has higher tolerance to noise. Fast numerical schemes are designed for both of the regularization methods. By changing the distance function, the proposed segmentation framework can be easily extended to the segmentation of other types of images. Numerical results on cartoon images, piecewise smooth images and texture images demonstrate that our methods are effective in multiphase image segmentation.     

C-V活动轮廓模型的一个注记

Chan-Vese提出的“无边活动轮廓”模型（C-V模型）是一个著名的基于区域的图像分割模型,它是基于Mumford-Shah泛函和二值PC函数（目标区域取一个值,背景区域取另一个值）解决图像分割问题的。在C-V模型中,定义能量泛函的面积项的系数被要求为非负值,这个要求限制了模型适用的范围。实验研究表明：面积项系数取负值时,C-V模型能够分割某些原来不适用的图像。     

A new active contour remote sensing river image segmentation algorithm inspired from the cross entropy

The CV (Chan–Vese) model is a piecewise constant approximation of the Mumford and Shah model. It assumes that the original image can be segmented into two regions such that each region can be represented as constant grayscale value. In fact, the objective functional of the CV model actually finds a segmentation of the image such that the within-class variance is minimized. This is equivalent to the Otsu image thresholding algorithm which also aims to minimize the within-class variance. Similarly to the Otsu image thresholding algorithm, cross entropy is another widely used image thresholding algorithm and it finds a segmentation such that the cross entropy of the segmented image and the original image is minimized. Inspired from the cross entropy, a new active contour image segmentation algorithm is proposed. The region term in the new objective functional is the integral of the logarithm of the ratio between the grayscale of the original image and the mean value computed from the segmented image weighted by the grayscale of the original image. The new objective functional can be solved by the level set evolution method. A distance regularized term is added to the level set evolution equation so the level set need not be reinitialized periodically. A fast global minimization algorithm of the objective functional is also proposed which incorporates the edge term originated from the geodesic active contour model. Experimental results show that, the algorithm proposed can segment images more accurately than the CV model and the implementation speed of the fast global minimization algorithm is fast.     

一种整合单通道各向异性扩散信息的水平集图像分割方法*

在对Chan-Vese提出的基于简化Mumford-Shah模型(C-V模型)改进的基础上,针对彩色图像、多光谱图像等多通道图像,提出了一种多通道C-V模型水平集图像分割方法.首先将多通道图像分解到各单通道,使用一种新的各向异性扩散方法对各通道进行平滑滤波,然后使用能够整合各通道各向异性扩散信息的多通道C-V模型进行分割.普通彩色图像与多光谱图像数据的实验结果表明,该方法分割质量明显优于传统的C-V模型分割.