Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

/pdf/nonlinear-partial-differential-equations-with-applications-204hhsvanv.pdf

Nonlinear partial differential equations with applications

http://www4.comp.polyu.edu.hk/~cslzhang/paper/IVC_10.pdf

Active contours with selective local or global segmentation: A new formulation and level set method

Automatic segmentation of cell nuclei is an essential step in image cytometry and histometry. Despite substantial progress, there is a need to improve accuracy, speed, level of automation, and adaptability to new applications. This paper presents a robust and accurate novel method for segmenting cell nuclei using a combination of ideas. The image foreground is extracted automatically using a graph-cuts-based binarization. Next, nuclear seed points are detected by a novel method combining multiscale Laplacian-of-Gaussian filtering constrained by distance-map-based adaptive scale selection. These points are used to perform an initial segmentation that is refined using a second graph-cuts-based algorithm incorporating the method of alpha expansions and graph coloring to reduce computational complexity. Nuclear segmentation results were manually validated over 25 representative images (15 in vitro images and 10 in vivo images, containing more than 7400 nuclei) drawn from diverse cancer histopathology studies, and four types of segmentation errors were investigated. The overall accuracy of the proposed segmentation algorithm exceeded 86%. The accuracy was found to exceed 94% when only over- and undersegmentation errors were considered. The confounding image characteristics that led to most detection/segmentation errors were high cell density, high degree of clustering, poor image contrast and noisy background, damaged/irregular nuclei, and poor edge information. We present an efficient semiautomated approach to editing automated segmentation results that requires two mouse clicks per operation.

Improved Automatic Detection and Segmentation of Cell Nuclei in Histopathology Images

http://www4.comp.polyu.edu.hk/%7Ecslzhang/paper/PR_10_LIF.pdf

Active contours driven by local image fitting energy

Manufacturing-oriented topology optimization has been extensively studied the past two decades, in particular for the conventional manufacturing methods, for example, machining and injection molding or casting. Both design and manufacturing engineers have benefited from these efforts because of the close-to-optimal and friendly-to-manufacture design solutions. Recently, additive manufacturing (AM) has received significant attention from both academia and industry. AM is characterized by producing geometrically complex components layer-by-layer, and greatly reduces the geometric complexity restrictions imposed on topology optimization by conventional manufacturing. In other words, AM can make near-full use of the freeform structural evolution of topology optimization. Even so, new rules and restrictions emerge due to the diverse and intricate AM processes, which should be carefully addressed when developing the AM-specific topology optimization algorithms. Therefore, the motivation of this perspective paper is to summarize the state-of-art topology optimization methods for a variety of AM topics. At the same time, this paper also expresses the authors' perspectives on the challenges and opportunities in these topics. The hope is to inspire both researchers and engineers to meet these challenges with innovative solutions.

/pdf/current-and-future-trends-in-topology-optimization-for-7k1902v47j.pdf

Current and future trends in topology optimization for additive manufacturing

In this paper, we propose a PDE-based level set method. Traditionally, interfaces are represented by the zero level set of continuous level set functions. Instead, we let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can at convergence only take two values, i.e., it can only be 1 or -1; thus, our method is related to phase-field methods. Some of the properties of standard level set methods are preserved in the proposed method, while others are not. Using this new method for interface problems, we need to minimize a smooth convex functional under a quadratic constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth. We show numerical results using the method for segmentation of digital images.

/pdf/a-binary-level-set-model-and-some-applications-to-mumford-2trhq3hfmf.pdf

A binary level set model and some applications to Mumford-Shah image segmentation

In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of n level set functions are utilized to identify up to 2" phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If 2" phases should be identified, the level set function must approach 2" predetermined constants. We just need one level set function to represent 2" unique phases, and this gains in storage capacity. Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the Delta and Heaviside functions. Numerical examples are given, and we also compare our method with related approaches.

/pdf/a-variant-of-the-level-set-method-and-applications-to-image-3h4o2ekeni.pdf

A variant of the level set method and applications to image segmentation

We generalize the total variation restoration model, introduced by Rudin, Osher, and Fatemi in 1992, to matrix-valued data, in particular, to diffusion tensor images (DTIs). Our model is a natural extension of the color total variation model proposed by Blomgren and Chan in 1998. We treat the diffusion matrix D implicitly as the product D = LLT, and work with the elements of L as variables, instead of working directly on the elements of D. This ensures positive definiteness of the tensor during the regularization flow, which is essential when regularizing DTI. We perform numerical experiments on both synthetical data and 3D human brain DTI, and measure the quantitative behavior of the proposed model.

/pdf/total-variation-regularization-of-matrix-valued-images-3xh5ggsx28.pdf

Total Variation Regularization of Matrix-Valued Images

In this work we discuss variants of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead use piecewise constant level set functions, and let interfaces be represented by discontinuities. Some of the properties of the standard level set function are preserved in the proposed method. Using the methods for interface problems, we minimize a smooth locally convex functional under a constraint. We show numerical results using the methods for image segmentation.

Piecewise constant level set methods and image segmentation

Error minimization of global functionals provides a natural setting for analyzing image processing and regularization. This approach leads to scale spaces, which in the continuous formulation are the solution of nonlinear partial differential equations. In this work we derive properties for a class of inverse scale space methods. The main contribution of this paper is the development of a proof that the methods considered are convergent for convex regularization operators. The proof is based on energy methods and the Bregman distance. Further, estimates for convergence toward a clean image with noisy forcing data is provided in terms of both the L 2 norm and Bregman distances. This leads to natural estimates of optimal stopping scale for the inverse scale space method. These analytical results are discussed in the context of a numerical example.

Johan Lie

Papers

A binary level set model and some applications to Mumford-Shah image segmentation

A variant of the level set method and applications to image segmentation

Total Variation Regularization of Matrix-Valued Images

Piecewise constant level set methods and image segmentation

Inverse Scale Spaces for Nonlinear Regularization