Section:
New Results
Exact biconvex reformulation of the minimization problem
Participants :
Gilles Aubert, Arne Henrik Bechensteen, Laure Blanc-Féraud.
We focus on the problem of minimizing the least-squares loss function
under the constraint that the reconstructed signal is at maximum
k-sparse. This is called the - constrained
problem. The pseudo-norm counts the number of non-zero
elements in a vector. The minimization problem is of interest in
signal processing, with a wide range of applications such as compressed
sensing, source separation, and super-resolution imaging.
Based on the results of [20], we reformulate
the pseudo-norm exactly as a convex minimization problem by
introducing an auxiliary variable. We then propose an exact biconvex
reformulation of the constrained and penalized
problems. We give correspondence results between minimizer of the
initial function and the reformulated ones. The reformulation is
biconvex. This property is used to derive a minimization algorithm.
We apply the algorithm to the problem of Single-Molecule Localization
Microscopy and compare the results with the well-known IHT algorithm
[13]. Both visually and numerically the
biconvex reformulations perform better. This work has been presented at the iTWIST 2018 workshop [5].
Furthermore, the algorithm has been compared to the IRL1-CEL0
[14] and Deep-STORM
[15] (see figure 1). The IRL1-CEL0 minimizes an exact relaxation
[19] of the penalized form
and Deep-STORM is an algorithm that uses deep-learning and
convolutional network to localize the molecules. This work has been
accepted to the ISBI 2019 conference.
Figure
1. Reconstruction by the different algorithms. Data set from ISBI 2013 challenge [18].
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