The jump set under geometric regularisation. Part 2: Higher-order approaches

The jump set under geometric regularisation. Part 2: Higher-order approaches

In Part 1, we developed a new technique based on Lipschitz pushforwards for proving the jump set containment property H(Ju \ Jf ) = 0 of solutions u to total variation denoising. We demonstrated that the technique also applies to Huber-regularised TV. Now, in this Part 2, we extend the technique to higher-order regularisers. We are not quite able to prove the property for total generalised vari...

The jump set under geometric regularisation. Part 1: Basic technique and first-order denoising

Let u ∈ BV(Ω) solve the total variation denoising problem with L-squared fidelity and data f . Caselles et al. [Multiscale Model. Simul. 6 (2008), 879–894] have shown the containment H(Ju \Jf ) = 0 of the jump set Ju of u in that of f . Their proof unfortunately depends heavily on the co-area formula, as do many results in this area, and as such is not directly extensible to higher-order, curva...

The Jump Set under Geometric Regularization. Part 1: Basic Technique and First-Order Denoising

Abstract. Let u ∈ BV(Ω) solve the total variation denoising problem with L2-squared fidelity and data f . Caselles et al. [Multiscale Model. Simul. 6 (2008), 879–894] have shown the containment Hm−1(Ju \ Jf ) = 0 of the jump set Ju of u in that of f . Their proof unfortunately depends heavily on the co-area formula, as do many results in this area, and as such is not directly extensible to high...

On Noncommutative Geometric Regularisation

Studies in string theory and in quantum gravity suggest the existence of a finite lower bound to the possible resolution of lengths which, quantum theoretically, takes the form of a minimal uncertainty in positions ∆x0. A finite minimal uncertainty in momenta ∆p0 has been motivated from the absence of plane waves on generic curved spaces. Both effects can be described as small noncommutative ge...

Geometric and Higher Order Logicin