Estimation of vector fields in unconstrained and inequality constrained variational problems for segmentation and registration

Vector fields arise in many problems of computer vision, particularly in non-rigid registration. In this paper, we develop coupled partial differential equations (PDEs) to estimate vector fields that define the deformation between objects, and the contour or surface that defines the segmentation of the objects as well.We also explore the utility of inequality constraints applied to variational problems in vision such as estimation of deformation fields in non-rigid registration and tracking. To solve inequality constrained vector field estimation problems, we apply tools from the Kuhn-Tucker theorem in optimization theory. Our technique differs from recently popular joint segmentation and registration algorithms, particularly in its coupled set of PDEs derived from the same set of energy terms for registration and segmentation. We present both the theory and results that demonstrate our approach

Active Mean Fields for Probabilistic Image Segmentation: Connections with Chan-Vese and Rudin-Osher-Fatemi Models

Segmentation is a fundamental task for extracting semantically meaningful regions from an image. The goal of segmentation algorithms is to accurately assign object labels to each image location. However, image-noise, shortcomings of algorithms, and image ambiguities cause uncertainty in label assignment. Estimating the uncertainty in label assignment is important in multiple application domains, such as segmenting tumors from medical images for radiation treatment planning. One way to estimate these uncertainties is through the computation of posteriors of Bayesian models, which is computationally prohibitive for many practical applications. On the other hand, most computationally efficient methods fail to estimate label uncertainty. We therefore propose in this paper the Active Mean Fields (AMF) approach, a technique based on Bayesian modeling that uses a mean-field approximation to efficiently compute a segmentation and its corresponding uncertainty. Based on a variational formulation, the resulting convex model combines any label-likelihood measure with a prior on the length of the segmentation boundary. A specific implementation of that model is the Chan-Vese segmentation model (CV), in which the binary segmentation task is defined by a Gaussian likelihood and a prior regularizing the length of the segmentation boundary. Furthermore, the Euler-Lagrange equations derived from the AMF model are equivalent to those of the popular Rudin-Osher-Fatemi (ROF) model for image denoising. Solutions to the AMF model can thus be implemented by directly utilizing highly-efficient ROF solvers on log-likelihood ratio fields. We qualitatively assess the approach on synthetic data as well as on real natural and medical images. For a quantitative evaluation, we apply our approach to the icgbench dataset

A stochastic-variational model for soft Mumford-Shah segmentation

In contemporary image and vision analysis, stochastic approaches demonstrate great flexibility in representing and modeling complex phenomena, while variational-PDE methods gain enormous computational advantages over Monte-Carlo or other stochastic algorithms. In combination, the two can lead to much more powerful novel models and efficient algorithms. In the current work, we propose a stochastic-variational model for soft (or fuzzy) Mumford-Shah segmentation of mixture image patterns. Unlike the classical hard Mumford-Shah segmentation, the new model allows each pixel to belong to each image pattern with some probability. We show that soft segmentation leads to hard segmentation, and hence is more general. The modeling procedure, mathematical analysis, and computational implementation of the new model are explored in detail, and numerical examples of synthetic and natural images are presented.Comment: 22 page

A Novel Euler's Elastica based Segmentation Approach for Noisy Images via using the Progressive Hedging Algorithm

Euler's Elastica based unsupervised segmentation models have strong capability of completing the missing boundaries for existing objects in a clean image, but they are not working well for noisy images. This paper aims to establish a Euler's Elastica based approach that properly deals with random noises to improve the segmentation performance for noisy images. We solve the corresponding optimization problem via using the progressive hedging algorithm (PHA) with a step length suggested by the alternating direction method of multipliers (ADMM). Technically, all the simplified convex versions of the subproblems derived from the major framework of PHA can be obtained by using the curvature weighted approach and the convex relaxation method. Then an alternating optimization strategy is applied with the merits of using some powerful accelerating techniques including the fast Fourier transform (FFT) and generalized soft threshold formulas. Extensive experiments have been conducted on both synthetic and real images, which validated some significant gains of the proposed segmentation models and demonstrated the advantages of the developed algorithm

A Posteriori Error Control for the Binary Mumford-Shah Model

The binary Mumford-Shah model is a widespread tool for image segmentation and can be considered as a basic model in shape optimization with a broad range of applications in computer vision, ranging from basic segmentation and labeling to object reconstruction. This paper presents robust a posteriori error estimates for a natural error quantity, namely the area of the non properly segmented region. To this end, a suitable strictly convex and non-constrained relaxation of the originally non-convex functional is investigated and Repin's functional approach for a posteriori error estimation is used to control the numerical error for the relaxed problem in the $L^2$-norm. In combination with a suitable cut out argument, a fully practical estimate for the area mismatch is derived. This estimate is incorporated in an adaptive meshing strategy. Two different adaptive primal-dual finite element schemes, and the most frequently used finite difference discretization are investigated and compared. Numerical experiments show qualitative and quantitative properties of the estimates and demonstrate their usefulness in practical applications.Comment: 18 pages, 7 figures, 1 tabl

Geometrical-based algorithm for variational segmentation and smoothing of vector-valued images

An optimisation method based on a nonlinear functional is considered for segmentation and smoothing of vector-valued images. An edge-based approach is proposed to initially segment the image using geometrical properties such as metric tensor of the linearly smoothed image. The nonlinear functional is then minimised for each segmented region to yield the smoothed image. The functional is characterised with a unique solution in contrast with the Mumford–Shah functional for vector-valued images. An operator for edge detection is introduced as a result of this unique solution. This operator is analytically calculated and its detection performance and localisation are then compared with those of the DroGoperator. The implementations are applied on colour images as examples of vector-valued images, and the results demonstrate robust performance in noisy environments