arXiv subjects

Tuomo Valkonen

Publications and source records attributed to Tuomo Valkonen.

48 records · Page 3Linked to original sources

arXiv2016

Acceleration of the PDHGM on strongly convex subspaces

We propose several variants of the primal-dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, $O(1/N^2)$ with respect to initialisation and $O(1/N)$ with respect to the dual sequence, and the residual part of the primal sequence. We demonstrate the efficacy of the proposed methods on image processing problems lacking strong convexity, such as total generalised variation denoising and total variation deblurring.

math.OC↗

arXiv2016

Explorations on anisotropic regularisation of dynamic inverse problems by bilevel optimisation

We explore anisotropic regularisation methods in the spirit of [Holler & Kunisch, 14]. Based on ground truth data, we propose a bilevel optimisation strategy to compute the optimal regularisation parameters of such a model for the application of video denoising. The optimisation poses a challenge in itself, as the dependency on one of the regularisation parameters is non-linear such that the standard existence and convergence theory does not apply. Moreover, we analyse numerical results of the proposed parameter learning strategy based on three exemplary video sequences and discuss the impact of these results on the actual modelling of dynamic inverse problems.

math.NA↗

arXiv2016

Diffusion tensor imaging with deterministic error bounds

Errors in the data and the forward operator of an inverse problem can be handily modelled using partial order in Banach lattices. We present some existing results of the theory of regularisation in this novel framework, where errors are represented as bounds by means of the appropriate partial order. We apply the theory to Diffusion Tensor Imaging, where correct noise modelling is challenging: it involves the Rician distribution and the nonlinear Stejskal-Tanner equation. Linearisation of the latter in the statistical framework would complicate the noise model even further. We avoid this using the error bounds approach, which preserves simple error structure under monotone transformations.

cs.CV↗

arXiv2015

Preconditioned ADMM with nonlinear operator constraint

We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI).

math.NA↗

arXiv2015

Bilevel approaches for learning of variational imaging models

We review some recent learning approaches in variational imaging, based on bilevel optimisation, and emphasize the importance of their treatment in function space. The paper covers both analytical and numerical techniques. Analytically, we include results on the existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are studied for the solution of the problems at hand, combining them with sampling techniques in case of large databases. The computational verification of the developed techniques is extensively documented, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.

math.OC↗

arXiv2015

The structure of optimal parameters for image restoration problems

We study the qualitative properties of optimal regularisation parameters in variational models for image restoration. The parameters are solutions of bilevel optimisation problems with the image restoration problem as constraint. A general type of regulariser is considered, which encompasses total variation (TV), total generalized variation (TGV) and infimal-convolution total variation (ICTV). We prove that under certain conditions on the given data optimal parameters derived by bilevel optimisation problems exist. A crucial point in the existence proof turns out to be the boundedness of the optimal parameters away from $0$ which we prove in this paper. The analysis is done on the original -- in image restoration typically non-smooth variational problem -- as well as on a smoothed approximation set in Hilbert space which is the one considered in numerical computations. For the smoothed bilevel problem we also prove that it $Γ$ converges to the original problem as the smoothing vanishes. All analysis is done in function spaces rather than on the discretised learning problem.

math.OC↗

arXiv2015

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

Let $u \in \mbox{BV}(Ω)$ solve the total variation denoising problem with $L^2$-squared fidelity and data $f$. Caselles et al. [Multiscale Model. Simul. 6 (2008), 879--894] have shown the containment $\mathcal{H}^{m-1}(J_u \setminus J_f)=0$ of the jump set $J_u$ 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, curvature-based, and other advanced geometric regularisers, such as total generalised variation (TGV) and Euler's elastica. These have received increased attention in recent times due to their better practical regularisation properties compared to conventional total variation or wavelets. We prove analogous jump set containment properties for a general class of regularisers. We do this with novel Lipschitz transformation techniques, and do not require the co-area formula. In the present Part 1 we demonstrate the general technique on first-order regularisers, while in Part 2 we will extend it to higher-order regularisers. In particular, we concentrate in this part on TV and, as a novelty, Huber-regularised TV. We also demonstrate that the technique would apply to non-convex TV models as well as the Perona-Malik anisotropic diffusion, if these approaches were well-posed to begin with.

math.FA↗

arXiv2015

Asymptotic behaviour of total generalised variation

The recently introduced second order total generalised variation functional $\mathrm{TGV}_{β,α}^{2}$ has been a successful regulariser for image processing purposes. Its definition involves two positive parameters $α$ and $β$ whose values determine the amount and the quality of the regularisation. In this paper we report on the behaviour of $\mathrm{TGV}_{β,α}^{2}$ in the cases where the parameters $α, β$ as well as their ratio $β/α$ becomes very large or very small. Among others, we prove that for sufficiently symmetric two dimensional data and large ratio $β/α$, $\mathrm{TGV}_{β,α}^{2}$ regularisation coincides with total variation ($\mathrm{TV}$) regularisation.

math.OC↗

arXiv2014

Limiting aspects of non-convex ${TV}^ϕ$ models

Recently, non-convex regularisation models have been introduced in order to provide a better prior for gradient distributions in real images. They are based on using concave energies $ϕ$ in the total variation type functional ${TV}^ϕ(u) := \int ϕ(|\nabla u(x)|) d x$. In this paper, it is demonstrated that for typical choices of $ϕ$, functionals of this type pose several difficulties when extended to the entire space of functions of bounded variation, ${BV}(Ω)$. In particular, if $ϕ(t)=t^q$ for $q \in (0, 1)$ and ${TV}^ϕ$ is defined directly for piecewise constant functions and extended via weak* lower semicontinuous envelopes to ${BV}(Ω)$, then still ${TV}^ϕ(u)=\infty$ for $u$ not piecewise constant. If, on the other hand, ${TV}^ϕ$ is defined analogously via continuously differentiable functions, then ${TV}^ϕ\equiv 0$, (!). We study a way to remedy the models through additional multiscale regularisation and area strict convergence, provided that the energy $ϕ(t)=t^q$ is linearised for high values. The fact, that this kind of energies actually better matches reality and improves reconstructions, is demonstrated by statistics and numerical experiments.

math.FA↗

arXiv2014

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 $\mathcal{H}^{m-1}(J_u \setminus J_f)=0$ 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 variation (TGV) based on the symmetrised gradient for the second-order term. We show that the property holds under three conditions: First, the solution $u$ is locally bounded. Second, the second-order variable is of locally bounded variation, $w \in \mbox{BV}_\mbox{loc}(Ω; \mathbb{R}^m)$, instead of just bounded deformation, $w \in \mbox{BD}(Ω)$. Third, $w$ does not jump on $J_u$ parallel to it. The second condition can be achieved for non-symmetric TGV. Both the second and third condition can be achieved if we change the Radon (or $L^1$) norm of the symmetrised gradient $Ew$ into an $L^p$ norm, $p>1$, in which case Korn's inequality holds. We also consider the application of the technique to infimal convolution TV, and study the limiting behaviour of the singular part of $D u$, as the second parameter of $\mbox{TGV}^2$ goes to zero. Unsurprisingly, it vanishes, but in numerical discretisations the situation looks quite different. Finally, our work additionally includes a result on TGV-strict approximation in $\mbox{BV}(Ω)$.

math.FA↗

arXiv2014

A primal-dual hybrid gradient method for non-linear operators with applications to MRI

We study the solution of minimax problems $\min_x \max_y G(x) + \langle K(x),y\rangle - F^*(y)$ in finite-dimensional Hilbert spaces. The functionals $G$ and $F^*$ we assume to be convex, but the operator $K$ we allow to be non-linear. We formulate a natural extension of the modified primal-dual hybrid gradient method (PDHGM), originally for linear $K$, due to Chambolle and Pock. We prove the local convergence of the method, provided various technical conditions are satisfied. These include in particular the Aubin property of the inverse a monotone operator at the solution. Of particular interest to us is the case arising from reformulation of regularisation problems $\min_x \|f-T(x)\|^2/2 + αR(x)$ with the operator $T$ non-linear. For such problems, we show that our general local convergence result holds when the noise level of the data $f$ is low, and the regularisation parameter $α$ is correspondingly small. We verify the numerical performance of the method by applying it to problems from magnetic resonance imaging (MRI) in chemical engineering and medicine. The specific applications are in diffusion tensor imaging (DTI) and MR velocity imaging. These numerical studies show very promising performance.

math.OC↗

arXiv2014

Imaging with Kantorovich-Rubinstein discrepancy

We propose the use of the Kantorovich-Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularisation model endowed with a Kantorovich-Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition. We point out connections of this approach to several other recently proposed methods such as total generalized variation and norms capturing oscillating patterns. We also show that the respective optimization problem can be turned into a convex-concave saddle point problem with simple constraints and hence, can be solved by standard tools. Numerical examples exhibit interesting features and favourable performance for denoising and cartoon-texture decomposition.

cs.CV↗