On the Taut String Interpretation and Other Properties of the Rudin–Osher–Fatemi Model in One Dimension

We study the one-dimensional version of the Rudin–Osher–Fatemi (ROF) denoising model and some related TV-minimization problems. A new proof of the equivalence between the ROF model and the so-called taut string algorithm is presented, and a fundamental estimate on the denoised signal in terms of the corrupted signal is derived. Based on duality and the projection theorem in Hilbert space, the proof of the taut string interpretation is strictly elementary with the existence and uniqueness of solutions (in the continuous setting) to both models following as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. The taut string interpretation plays an essential role in the proof of the fundamental estimate. This estimate implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the corrupted signal as the regularization parameter vanishes. It can also be used to prove semi-group properties of the denoising model. Finally, it is indicated how the methods developed can be applied to related problems such as the fused lasso model, isotonic regression and signal restoration with higher-order total variation regularization

On the taut string interpretation of the one-dimensional rudin–osher–fatemi model

A new proof of the equivalence of the Taut String Algorithm and the one-dimensional Rudin–Osher–Fatemi model is presented. Based on duality and the projection theorem in Hilbert space, the proof is strictly elementary. Existence and uniqueness of solutions (in the continuous case) to both denoising models follow as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. Moreover, a new and fundamental estimate on the denoised signal is derived. It implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the in-signal as the regularization parameter vanishes

Designing for minimum elongation

We reconsider the variational problem of finding the shape of a vertically hanging rope such that its elongation, due to the rope’s own weight and that of a load attached at its lower end, is minimum. The known solution is recalled and the missing proof of optimality is supplied

Separating rigid motion for continuous shape evolution

A method is proposed for the construction of descent directions for the minimization of energy functionals defined for plane curves. The method is potentially useful in a number of image analysis problems, such as image registration and shape warping, where the standard gradient descent curve evolutions are not always feasible. The descent direction is constructed by taking a weighted average of the three components of the gradient which correspond to translation, rotation, and deformation. Our approach differs from previous work in the field by the use of implicit representation of curves and the notion of normal velocity of a curve evolution. Thus our theory is morphological and well suited for implementation in the level set framework. The results are easily generalized to evolution of surfaces. 1

Separating Rigid Motion for Continuous Shape Evolution

A method is proposed for the construction of descent directions for the minimization of energy functionals defined for plane curves. The method is potentially useful in a number of image analysis problems, such as image registration and shape warping, where the standard gradient descent curve evolutions are not always feasible. The descent direction is constructed by taking a weighted average of the three components of the gradient corresponding to translation, rotation, and deformation. Our approach differs from previous work in the field by the use of implicit representation of curves and the notion of normal velocity of a curve evolution. Thus our theory is morphological and well suited for implementation in the level set framework

Rayleigh segmentation of the endocardium in ultrasound images

In this paper we present the coupled active contours (CAC) model, which is applied to segmentation of the endocardium in ultrasonic images assuming Rayleigh distributed intensities. Comparative experiments, both real and synthetic, with a standard prior model are presented. In the CAC model the prior acts, by affine transformation, on the same image information as the active contour, in addition to the traditional interaction between prior and active contour. By this higher convergence rate and robustness, w.r.t artifacts and poor initialization, is achieved

Pose Invariant Shape Prior Segmentation Using Continuous Cuts and Gradient Descent on Lie Groups

This paper proposes a novel formulation of the Chan-Vese model for pose invariant shape prior segmentation as a continuous cut problem. The model is based on the classic L 2 shape dissimilarity measure and with pose invariance under the full (Lie-) group of similarity transforms in the plane. To overcome the common numerical problems associated with step size control for translation, rotation and scaling in the discretization of the pose model, a new gradient descent procedure for the pose estimation is introduced. This procedure is based on the construction of a Riemannian structure on the group of transformations and a derivation of the corresponding pose energy gradient. Numerically, this amounts to an adaptive step size selection in the discretization of the gradient descent equations. Together with efficient numerics for TV-minimization we get a fast and reliable implementation of the model. Moreover, the theory introduced is generic and reliable enough for application to more general segmentation- and shape-models

Optimizing Parametric Total Variation Models

One of the key factors for the success of recent energy minimization methods is that they seek to compute global solutions. Even for non-convex energy functionals, optimization methods such as graph cuts have proven to produce high-quality solutions by iterative minimization based on large neighborhoods, making them less vulnerable to local minima. Our approach takes this a step further by enlarging the search neighborhood with one dimension. In this paper we consider binary total variation problems that depend on an additional set of parameters. Examples include: (i) the Chan-Vese model that we solve globally (ii) ratio and constrained minimization which can be formulated as parametric problems, and (iii) variants of the Mumford-Shah functional. Our approach is based on a recent theorem of Chambolle which states that solving a one-parameter family of binary problems amounts to solving a single convex variational problem. We prove a generalization of this result and show how it can be applied to parametric optimization. 1