Region-Based PDEs for Cells Counting and Segmentation in 3D+Time Images of Vertebrate Early Embryogenesis

This paper is devoted to the segmentation of cell nuclei from time lapse confocal microscopy images, taken throughout early Zebrafish embryogenesis. The segmentation allows to identify and quantify the number of cells in the animal model. This kind of information is relevant to estimate important biological parameters such as the cell proliferation rate in time and space. Our approach is based on the active contour model without edges. We compare two different formulations of the model equation and evaluate their performances in segmenting nuclei of different shapes and sizes. Qualitative and quantitative comparisons are performed on both synthetic and real data, by means of suitable gold standard. The best approach is then applied on a number of time lapses for the segmentation and counting of cells during the development of a zebrafish embryo between the sphere and the shield stage

A Variational Shape Optimization Approach for Image Segmentation with a Mumford-Shah Functional

We introduce a novel computational method for a Mumford–Shah functional, which decomposes a given image into smooth regions separated by closed curves. Casting this as a shape optimization problem, we develop a gradient descent approach at the continuous level that yields nonlinear PDE flows. We propose time discretizations that linearize the problem and space discretization by continuous piecewise linear finite elements. The method incorporates topological changes, such as splitting and merging for detection of multiple objects, space–time adaptivity, and a coarse-to-fine approach to process large images efficiently. We present several simulations that illustrate the performance of the method and investigate the model sensitivity to various parameters

Segmentation and Restoration of Images on Surfaces by Parametric Active Contours with Topology Changes

In this article, a new method for segmentation and restoration of images on two-dimensional surfaces is given. Active contour models for image segmentation are extended to images on surfaces. The evolving curves on the surfaces are mathematically described using a parametric approach. For image restoration, a diffusion equation with Neumann boundary conditions is solved in a postprocessing step in the individual regions. Numerical schemes are presented which allow to efficiently compute segmentations and denoised versions of images on surfaces. Also topology changes of the evolving curves are detected and performed using a fast sub-routine. Finally, several experiments are presented where the developed methods are applied on different artificial and real images defined on different surfaces

Anisotropic Chan-Vese segmentation

In this paper we study a variant to Chan-Vese image segmentation model with rectilinear anisotropy. We show existence of minimizers in the $2$-phases case and how they are related to the (anisotropic) Rudin-Osher-Fatemi denoising model (ROF). Our analysis shows that in the natural case of a piecewise constant on rectangles image (PCR function in short), there exists a minimizer of the Chan-Vese functional which is also piecewise constant on rectangles over the same grid that the one defined by the original image. In the multiphase case, we show that minimizers of the Chan-Vese multiphase functional also share this property in the case that the initial image is a PCR function. We also investigate a multiphase and anisotropic version of the Truncated ROF algorithm, and we compare the solutions given by this algorithm with minimizers of the multiphase anisotropic Chan-Vese functional.Comment: Revised version. 29 pages, 3 figure

Variational Image Segmentation with Constraints

The research of Huizhu Pan addresses the problem of image segmentation with constraints though designing and solving various variational models. A novel constraint term is designed for the use of landmarks in image segmentation. Two region-based segmentation models were proposed where the segmentation contour passes through landmark points. A more stable and memory efficient solution to the self-repelling snakes model, a variational model with the topology preservation constraint, was also designed

Joint methods in imaging based on diffuse image representations

This thesis deals with the application and the analysis of different variants of the Mumford-Shah model in the context of image processing. In this kind of models, a given function is approximated in a piecewise smooth or piecewise constant manner. Especially the numerical treatment of the discontinuities requires additional models that are also outlined in this work. The main part of this thesis is concerned with four different topics. Simultaneous edge detection and registration of two images: The image edges are detected with the Ambrosio-Tortorelli model, an approximation of the Mumford-Shah model that approximates the discontinuity set with a phase field, and the registration is based on these edges. The registration obtained by this model is fully symmetric in the sense that the same matching is obtained if the roles of the two input images are swapped. Detection of grain boundaries from atomic scale images of metals or metal alloys: This is an image processing problem from materials science where atomic scale images are obtained either experimentally for instance by transmission electron microscopy or by numerical simulation tools. Grains are homogenous material regions whose atomic lattice orientation differs from their surroundings. Based on a Mumford-Shah type functional, the grain boundaries are modeled as the discontinuity set of the lattice orientation. In addition to the grain boundaries, the model incorporates the extraction of a global elastic deformation of the atomic lattice. Numerically, the discontinuity set is modeled by a level set function following the approach by Chan and Vese. Joint motion estimation and restoration of motion-blurred video: A variational model for joint object detection, motion estimation and deblurring of consecutive video frames is proposed. For this purpose, a new motion blur model is developed that accurately describes the blur also close to the boundary of a moving object. Here, the video is assumed to consist of an object moving in front of a static background. The segmentation into object and background is handled by a Mumford-Shah type aspect of the proposed model. Convexification of the binary Mumford-Shah segmentation model: After considering the application of Mumford-Shah type models to tackle specific image processing problems in the previous topics, the Mumford-Shah model itself is studied more closely. Inspired by the work of Nikolova, Esedoglu and Chan, a method is developed that allows global minimization of the binary Mumford-Shah segmentation model by solving a convex, unconstrained optimization problem. In an outlook, segmentation of flowfields into piecewise affine regions using this convexification method is briefly discussed