We propose using Partial Differential Equation (PDE) techniques in wavelet based image processing to remove noise and reduce edge artifacts generated by wavelet thresholding. We employ a variational framework, in particular the minimization of total variation (TV), to select and modify the retained wavelet coefficients so that the reconstructed images have fewer oscillations near edges while no...

Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bi-partitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. This paper presents a general framework for multiclass total variation clustering that does not rely...

We prove existence and uniqueness of weak solutions for the minimizing Total Variation ow with initial data in L1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t ! 1. We also prove that local maxima strictly decrease with time...

We introduce a simple new regularizer for auto-encoders whose hidden-unit activation functions contain at least one zero-gradient (saturated) region. This regularizer explicitly encourages activations in the saturated region(s) of the corresponding activation function. We call these Saturating Auto-Encoders (SATAE). We show that the saturation regularizer explicitly limits the SATAE’s ability t...

The novel concept of total generalized variation of a function u is introduced and some of its essential properties are proved. Differently from the bounded variation semi-norm, the new concept involves higher order derivatives of u. Numerical examples illustrate the high quality of this functional as a regularization term for mathematical imaging problems. In particular this functional selecti...

In this work, inspired by (Bühler & Hein, 2009), (Strang, 1983), and (Zhang et al., 2009), we give a continuous relaxation of the Cheeger cut problem on a weighted graph. We show that the relaxation is actually equivalent to the original problem. We then describe an algorithm for finding good cuts suggested by the similarities of the energy of the relaxed problem and various well studied energi...

Analysis sparsity is a common prior in inverse problem or machine learning including special cases such as Total Variation regularization, Edge Lasso and Fused Lasso. We study the geometry of solution set (a polyhedron) analysis l1-regularization (with l2 data fidelity term) when it not reduced to singleton without any assumption dictionary nor degradation operator. In contrast with most theore...

Total-variation denoising (TVD) is a robust algorithm for reconstructing noisy images. Because it imposes an L1 penalty on differences between adjacent pixels, it tends to result in images with piecewise constant regions. The goal of image segmentation is to assign pixels to clusters such that pixels within each cluster are similar. Our goal is to see if TVD can be used to perform segmentation,...

We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any Lipschitz continuous monotone Φ : R→ R, we have that |Φ(u)|TV (R) is nonincreasing in time. We call this property Total Oscillation Diminishing because it is in contradiction with the oscillations observed recently on some numerical computations based on TVD ...

Regression is one of the core problems tackled in supervised learning. Neural networks with rectified linear units generate continuous and piecewise-linear (CPWL) mappings are state-of-the-art approach for solving regression problems. In this paper, we propose an alternative method that leverages expressivity CPWL functions. contrast to deep neural networks, our parameterization guarantees stab...