Active contours without edges

We propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by the gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We give a numerical algorithm using finite differences. Finally, we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.     

Image segmentation and selective smoothing by using Mumford-Shah model.

Recently, Chan and Vese developed an active contour model for image segmentation and smoothing by using piecewise constant and smooth representation of an image. Tsai et al. also independently developed a segmentation and smoothing method similar to the Chan and Vese piecewise smooth approach. These models are active contours based on the Mumford-Shah variational approach and the level-set method. In this paper, we develop a new hierarchical method which has many advantages compared to the Chan and Vese multiphase active contour models. First, unlike previous works, the curve evolution partial differential equations (PDEs) for different level-set functions are decoupled. Each curve evolution PDE is the equation of motion of just one level-set function, and different level-set equations of motion are solved in a hierarchy. This decoupling of the motion equations of the level-set functions speeds up the segmentation process significantly. Second, because of the coupling of the curve evolution equations associated with different level-set functions, the initialization of the level sets in Chan and Vese's method is difficult to handle. In fact, different initial conditions may produce completely different results. The hierarchical method proposed in this paper can avoid the problem due to the choice of initial conditions. Third, in this paper, we use the diffusion equation for denoising. This method, therefore, can deal with very noisy images. In general, our method is fast, flexible, not sensitive to the choice of initial conditions, and produces very good results.     

一种新的曲线演化混合模型图像分割算法

本文在Mumford-Shah模型的基础上，将传统几何曲线演化的驱动力（图像梯度局部信息）、Mumford-Shah模型的全局信息以及水平集的符号距离函数统一在一个变分框架之下，完成曲线演化过程的数值计算。本混合模型无需重新计算演化曲线的初始位置，可选择较大时间步长。实验结果表明，新的混合模型既保留了原有曲线演化模型的优势，又能高效稳健、快速地完成曲线演化过程。     

The Piecewise Smooth Mumford–Shah Functional on an Arbitrary Graph

The Mumford-Shah functional has had a major impact on a variety of image analysis problems, including image segmentation and filtering, and, despite being introduced over two decades ago, it is still in widespread use. Present day optimization of the Mumford-Shah functional is predominated by active contour methods. Until recently, these formulations necessitated optimization of the contour by evolving via gradient descent, which is known for its overdependence on initialization and the tendency to produce undesirable local minima. In order to reduce these problems, we reformulate the corresponding Mumford-Shah functional on an arbitrary graph and apply the techniques of combinatorial optimization to produce a fast, low-energy solution. In contrast to traditional optimization methods, use of these combinatorial techniques necessitates consideration of the reconstructed image outside of its usual boundary, requiring additionally the inclusion of regularization for generating these values. The energy of the solution provided by this graph formulation is compared with the energy of the solution computed via traditional gradient descent-based narrow-band level set methods. This comparison demonstrates that our graph formulation and optimization produces lower energy solutions than the traditional gradient descent based contour evolution methods in significantly less time. Finally, we demonstrate the usefulness of the graph formulation to apply the Mumford-Shah functional to new applications such as point clustering and filtering of nonuniformly sampled images.     

Unsupervised Multiphase Segmentation: A Phase Balancing Model

Variational models have been studied for image segmentation application since the Mumford-Shah functional was introduced in the late 1980s. In this paper, we focus on multiphase segmentation with a new regularization term that yields an unsupervised segmentation model. We propose a functional that automatically chooses a favorable number of phases as it segments the image. The primary driving force of the segmentation is the intensity fitting term while a phase scale measure complements the regularization term. We propose a fast, yet simple, brute-force numerical algorithm and present experimental results showing the robustness and stability of the proposed model.     

Constraining active contour evolution via Lie Groups of transformation

We present a novel approach to constraining the evolution of active contours used in image analysis. The proposed approach constrains the final curve obtained at convergence of curve evolution to be related to the initial curve from which evolution begins through an element of a desired Lie group of plane transformations. Constraining curve evolution in such a way is important in numerous tracking applications where the contour being tracked in a certain frame is known to be related to the contour in the previous frame through a geometric transformation such as translation, rotation, or affine transformation, for example. It is also of importance in segmentation applications where the region to be segmented is known up to a geometric transformation. Our approach is based on suitably modifying the Euler-Lagrange descent equations by using the correspondence between Lie groups of plane actions and their Lie algebras of infinitesimal generators, and thereby ensures that curve evolution takes place on an orbit of the chosen transformation group while remaining a descent equation of the original functional. The main advantage of our approach is that it does not necessitate any knowledge of nor any modification to the original curve functional and is extremely straightforward to implement. Our approach therefore stands in sharp contrast to other approaches where the curve functional is modified by the addition of geometric penalty terms. We illustrate our algorithm on numerous real and synthetic examples.     

A real-time algorithm for the approximation of level-set-based curve evolution.

In this paper, we present a complete and practical algorithm for the approximation of level-set-based curve evolution suitable for real-time implementation. In particular, we propose a two-cycle algorithm to approximate level-set-based curve evolution without the need of solving partial differential equations (PDEs). Our algorithm is applicable to a broad class of evolution speeds that can be viewed as composed of a data-dependent term and a curve smoothness regularization term. We achieve curve evolution corresponding to such evolution speeds by separating the evolution process into two different cycles: one cycle for the data-dependent term and a second cycle for the smoothness regularization. The smoothing term is derived from a Gaussian filtering process. In both cycles, the evolution is realized through a simple element switching mechanism between two linked lists, that implicitly represents the curve using an integer valued level-set function. By careful construction, all the key evolution steps require only integer operations. A consequence is that we obtain significant computation speedups compared to exact PDE-based approaches while obtaining excellent agreement with these methods for problems of practical engineering interest. In particular, the resulting algorithm is fast enough for use in real-time video processing applications, which we demonstrate through several image segmentation and video tracking experiments.     

基于形态学尺度空间和梯度修正的分水岭分割

分水岭是一种有效的图像分割方法,但存在过分割现象,为此提出了一种基于形态学尺度空间和梯度修正的分水岭图像分割方法,该方法利用形态学混合开闭重建尺度空间和梯度修正技术,在平滑原始图像的同时保留了重要的区域轮廓而去除了易造成过分割的区域细节和噪声,克服了传统的形态学开闭尺度空间在平滑细节和噪声时,部分重要区域轮廓也被平滑及不满足尺度因果性的问题。对平滑后的图像采用梯度修正分水岭变换,保持了尺度和分割区域数目间的因果性,进一步消除了标准分水岭的过分割现象。仿真实验表明,该方法能有效地消除过分割现象,分割的区域数目满足尺度因果性,且具有较高的区域轮廓定位能力。     

Stochastic differential equations and geometric flows

In previous years, curve evolution, applied to a single contour or to the level sets of an image via partial differential equations, has emerged as an important tool in image processing and computer vision. Curve evolution techniques have been utilized in problems such as image smoothing, segmentation, and shape analysis. We give a local stochastic interpretation of the basic curve smoothing equation, the so called geometric heat equation, and show that this evolution amounts to a tangential diffusion movement of the particles along the contour. Moreover, assuming that a priori information about the shapes of objects in an image is known, we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise. We also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features, in contrast to the original geometric heat flow which tends to circularize any closed curve.     

Low bit-rate image coding using adaptive geometric piecewise polynomial approximation.

We present a new image coding algorithm, the geometric piecewise polynomials (GPP) method, that draws on recent developments in the theory of adaptive multivariate piecewise polynomials approximation. The algorithm relies on a segmentation stage whose goal is to minimize a functional that is conceptually similar to the Mumford-Shah functional except that it measures the smoothness of the segmentation instead of the length. The initial segmentation is "pruned" and the remaining curve portions are lossy encoded. The image is then further partitioned and approximated by low order polynomials on the subdomains. We show examples where our algorithm outperforms state-of-the-art wavelet coding in the low bit-rate range. The GPP algorithm significantly outperforms wavelet based coding methods on graphic and cartoon images. Also, at the bit rate 0.05 bits per pixel, the GPP algorithm achieves on the test image Cameraman, which has a geometric structure, a PSNR of 21.5 dB, while the JPEG2000 Kakadu software obtains PSNR of 20 dB. For the test image Lena, the GPP algorithm obtains the same PSNR as JPEG2000, but with better visual quality at 0.03 bpp.     

于形态学梯度重建的分水岭分割

提出一种基于形态学梯度重建的分水岭图像分割方法.该方法在形态学梯度图像的基础上,利用形态学开闭重建运算对梯度图像进行重建,在保留重要区域轮廓的同时去除了细节和噪声.避免了标准分水岭存在的过分割现象及传统形态学开闭运算先平滑原始图像,后进行分水岭变换而造成的区域轮廓位置偏移.仿真实验证明,无论从消除过分割还是区域轮廓定位等性能方面,该方法均具有较好的分割效果.整个分割过程无需进行分割后的区域合并处理,降低了分割的复杂性;且分割过程只需选择合适的结构元素大小,增强了算法的灵活性.     

Unsupervised variational image segmentation/classification using a Weibull observation model.

Studies have shown that the Weibull distribution can model accurately a wide variety of images. Its parameters index a family of distributions which includes the exponential and approximations of the Gaussian and the Raleigh models widely used in image segmentation. This study investigates the Weibull distribution in unsupervised image segmentation and classification by a variational method. The data term of the segmentation functional measures the conformity of the image intensity in each region to a Weibull distribution whose parameters are determined jointly with the segmentation. Minimization of the functional is implemented by active curves via level sets and consists of iterations of two consecutive steps: curve evolution via Euler-Lagrange descent equations and evaluation of the Weibull distribution parameters. Experiments with synthetic and real images are described which verify the validity of method and its implementation.     

融合模糊聚类的Mumford-Shah模型

Mumford-Shah模型和模糊聚类技术是图像分割的两类重要方法,前者着重于控制图像分割区域的连通性和边界的光滑性,而后者更多地分析了图像色彩的统计特征.受此启发,文中通过在第一种方法中融入模糊聚类技术,提出了融合模糊聚类的Mumford-Shah模型(简称FCMS模型),它能很好地结合两类方法各自的优点.在FCMS中,通过引入三个策略实现两类方法的融合,理论分析可知,现有的多类模糊聚类技术与许多Mumford-Shah模型的变形方法都能在此框架下很好地融合.文中以FCM和基本Mumford-Shah模型为例,给出了FCMS的一个具体实现,并对其做了理论和实验上的分析研究,所得结果证明了这一新模型的合理性与有效性.     

Connectivity of projected high dimensional data charts on one-dimensional curves

We propose a principal curve tracing algorithm, which uses the gradient and the Hessian of a given density estimate. Curve definition requires the local smoothness of data density and is based on the concept of subspace local maxima. Tracing of the curve is handled through the leading eigenvector where fixed step updates are used. We also propose an image segmentation algorithm based on the original idea and show the effectiveness of the proposed algorithm on a Brainbow dataset. Lastly, we showed a simple approach to define connectivity in complex topologies, by providing a tree representation for the bifurcating synthetic data.     

基于图切割与C-V模型的运动目标分割

将一种基于图切割与简化Mumford-Shah模型Chan-vese模型(G-V模型)相结合的方法应用于运动目标分割中.在此方法中,利用图切割技术求解能量最优化,利用C-V模型自适应处理目标几何的拓扑变化.通过实验对此方法在图像序列中的运动目标进行了检测与分割研究.实验结果表明,图切割能量优化加速了曲线进化进程,迭代次数大大减少,同时避免了常规水平集方法中符号函数的初始化和迭代更新.对图像序列中的运动目标进行分割的仿真实验验证了该方法的有效性.     

A Fully Global Approach to Image Segmentation via Coupled Curve Evolution Equations

In this paper, we develop a novel region-based approach to snakes designed to optimally separate the values of certain image statistics over a known number of region types. Multiple sets of contours deform according to a coupled set of curve evolution equations derived from a single global cost functional. The resulting active contour model, in contrast to many other edge and region based models, is fully global in that the evolution of each curve depends at all times upon every pixel in the image and is directly coupled to the evolution of every other curve regardless of their mutual proximity. As such evolving contours enjoy a very wide “field of view,” endowing the algorithm with a robustness to initial contour placement above and beyond the significant improvement exhibited by other region based snakes over earlier edge based snakes.     

基于Mumford-Shah模型的水上桥梁目标分割与识别

在桥梁知识的基础上,从河流和陆地的区域特性出发,提出了一种基于Mumford-Shah（MS）模型的水上桥梁分割算法.首先对Mumford-Shah模型中的区域权重进行相关性定义,实现对水域的分割,然后根据桥梁与河流区域边界的几何位置关系实现对桥梁目标的提取和识别.实验表明,该算法能够实现对水上桥梁目标的提取,尤其对远距离小目标水上桥梁及灰度梯度较弱图像的桥梁分割更有效.     

Image segmentation using active contours driven by the Bhattacharyya gradient flow.

This paper addresses the problem of image segmentation by means of active contours, whose evolution is driven by the gradient flow derived from an energy functional that is based on the Bhattacharyya distance. In particular, given the values of a photometric variable (or of a set thereof), which is to be used for classifying the image pixels, the active contours are designed to converge to the shape that results in maximal discrepancy between the empirical distributions of the photometric variable inside and outside of the contours. The above discrepancy is measured by means of the Bhattacharyya distance that proves to be an extremely useful tool for solving the problem at hand. The proposed methodology can be viewed as a generalization of the segmentation methods, in which active contours maximize the difference between a finite number of empirical moments of the "inside" and "outside" distributions. Furthermore, it is shown that the proposed methodology is very versatile and flexible in the sense that it allows one to easily accommodate a diversity of the image features based on which the segmentation should be performed. As an additional contribution, a method for automatically adjusting the smoothness properties of the empirical distributions is proposed. Such a procedure is crucial in situations when the number of data samples (supporting a certain segmentation class) varies considerably in the course of the evolution of the active contour. In this case, the smoothness properties of the empirical distributions have to be properly adjusted to avoid either over- or underestimation artifacts. Finally, a number of relevant segmentation results are demonstrated and some further research directions are discussed.     

Effective Level Set Image Segmentation With a Kernel Induced Data Term

This study investigates level set multiphase image segmentation by kernel mapping and piecewise constant modeling of the image data thereof. A kernel function maps implicitly the original data into data of a higher dimension so that the piecewise constant model becomes applicable. This leads to a flexible and effective alternative to complex modeling of the image data. The method uses an active curve objective functional with two terms: an original term which evaluates the deviation of the mapped image data within each segmentation region from the piecewise constant model and a classic length regularization term for smooth region boundaries. Functional minimization is carried out by iterations of two consecutive steps: 1) minimization with respect to the segmentation by curve evolution via Euler-Lagrange descent equations and 2) minimization with respect to the regions parameters via fixed point iterations. Using a common kernel function, this step amounts to a mean shift parameter update. We verified the effectiveness of the method by a quantitative and comparative performance evaluation over a large number of experiments on synthetic images, as well as experiments with a variety of real images such as medical, satellite, and natural images, as well as motion maps.     

Shape statistics variational approach for the outer contour segmentation of left ventricle MR images.

Segmentation of left ventricles is one of the important research topics in cardiac magnetic resonance (MR) imaging. The segmentation precision influences the authenticity of ventricular motion reconstruction. In left ventricle MR images, the weak and broken boundary increases the difficulty of segmenting the outer contour precisely. In this paper, we present an improved shape statistics variational approach for the outer contour segmentation of left ventricle MR images. We use the Mumford-Shah model in an object feature space and incorporate the shape statistics and an edge image to the variational framework. The introduction of shape statistics can improve the segmentation with broken boundaries. The edge image can enhance the weak boundary and thus improve the segmentation precision. The generation of the object feature image, which has homogenous "intensities" in the left ventricle, facilitates the application of the Mumford-Shah model. A comparison of mean absolute distance analysis between different contours generated with our algorithm and that generated by hand demonstrated that our method can achieve a higher segmentation precision and a better stability than various approaches. It is a semiautomatic way for the segmentation of the outer contour of the left ventricle in clinical applications.