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...

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...

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 ...

We prove that isentropic gas flow does not admit non-degenerate TVD fields on any invariant set M(r0, s0) = {r0 < r < s < s0}, where r, s are Riemann coordinates. A TVD field refers to a scalar field whose spatial variation VarX(φ(τ(t,X), u(t,X))) is nonincreasing in time along entropic solutions. The result is established under the assumption that the Riemann problem defined by an overtaking s...

The one–dimensional differential equations for the conservation mixture mass and mixture momentum coupled by a relative velocity conservation equation have been solved for an isentropic mixture of gas and liquid two–phase flow. Under certain restrictions the resulting system of partial differential equations is hyperbolic and allow discontinuous solutions. These equations have been solved by th...

This paper studies the total variation regularization with an L1 fidelity term (TV-L1) model for decomposing an image into features of different scales. We first show that the images produced by this model can be formed from the minimizers of a sequence of decoupled geometry subproblems. Using this result we show that the TV-L1 model is able to separate image features according to their scales,...