08:30
MS53: Advanced Numerical Methods in Image Processing (Part 1)
Chair: Damiana Lazzaro
08:30
25 mins
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Learning the Invisible in Inverse Problems
Tatiana Bubba, Gitta Kutyniok, Matti Lassas, Maximilian März, Wojciech Samek, Samuli Siltanen, Vignesh Srinivasan
Abstract: Due to the increasing complexity of problems in imaging, model-based methods are often today not sufficient anymore. At the same time, we witness the tremendous success of data-based methodologies such as deep neural networks for certain problems. However, such methods usually act as a black box without any theoretical understanding.
In this talk, we will demonstrate the success of hybrid approaches while focussing on the problem of limited-angle computed tomography. We will develop a solver for this severely ill-posed inverse problem by combining the model-based method of sparse regularization by shearlets with the data-driven method of deep learning. Our approach is faithful in the sense that we only learn the part which cannot be handled by model-based methods, while applying the theoretically controllable sparse regularization technique to all other parts. We further show that our algorithm significantly outperforms previous methodologies.
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08:55
25 mins
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Biorthogonal Boundary Multiwavelets
Fritz Keinert
Abstract: Wavelets and multiwavelets are normally defined on the entire real
line. One way to implement the discrete wavelet transform on a finite
interval is by using special boundary functions. These boundary
functions can be constructed in two different ways: as linear
combinations of interior scaling functions, or directly from boundary
recursion relations.
In this talk, I will discuss several things related to this topic:
\begin{itemize}
\item The connection between the two approaches
\item How to determine properties of the boundary wavelets from the
boundary recursion coefficients
\item Construction of boundary functions from given interior
functions, as refinable linear combinations
\item Construction of recursion coefficients of boundary functions
from interior recursion coefficients, by a linear algebra
approach. In general, these coefficients do not correspond to
functions
\item A method for imposing minimal continuity and approximation
order on the initial completion, which produces actual boundary
functions
\end{itemize}
All these results will be presented for biorthogonal multiwavelets. This
generalizes previous similar results for orthogonal multiwavelets.
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09:20
25 mins
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A Convex-Nonconvex Variational Method for the Additive Decomposition of Functions on Surfaces
Martin Huska, Alessandro Lanza, Serena Morigi, Ivan Selesnick
Abstract: We present a Convex-NonConvex variational approach for the additive decomposition of noisy scalar fields defined over triangulated surfaces into piece-wise constant and smooth components. The energy functional to be minimized is defined by the weighted sum of three terms, namely an \ell_2 fidelity term for the noise component, a Tikhonov regularization term for the smooth component and a Total Variation (TV)-like non-convex term for the piece-wise constant component.
The last term is parametrized such that the free scalar parameter allows to tune its degree of non-convexity and, hence, to separate the piece-wise constant component more effectively than by using a classical convex TV regularizer without renouncing to convexity of the total energy functional. A method is also presented for selecting the two regularization parameters. The unique solution of the proposed variational model is determined by means of an efficient ADMM-based minimization algorithm. Numerical experiments show a nearly ideal separation of the different components.
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09:45
25 mins
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Automatic parameter selection for weighted-TV image reconstruction problems
Alessandro Lanza, Luca Calatroni, Monica Pragiola, Fiorella Sgallari
Abstract: We present an efficient estimation technique for the automatic selection of locally-adaptive Total Variation (TV) regularisation parameters based on an hybrid strategy which combines a local Maximum-Likelihood (ML) approach estimating space-variant image scales with a global discrepancy principle related to noise statistics. The local closed-form formula obtained by the presented ML approach is extremely handy and, together with a minimisation algorithm based on the Alternating Directions Method of Multipliers (ADMM), makes our proposal very efficient. We verify the effectiveness of the proposed approach solving some exemplar image reconstruction problems and show its outperformance in comparison to two classes of state-of-the-art parameter estimation strategies, the former weighting locally the fit with the data, the latter relying on a bilevel learning paradigm.
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