168 research outputs found
Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions
No full textWe prove well-posedness results for the solution to an initial and boundary-value problem for an Allen-Cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain. The dynamic boundary conditions for the order parameter have been recently proposed by some physicists to account for interactions with the walls. We show our results using suitable regularizations of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments
A scaled, inexact and adaptive Fast Iterative Soft-Thresholding Algorithm for convex image restoration
No full textInternational audienceIn this note, we consider a special instance of the scaled, inexact and adaptive generalised Fast Iterative Soft-Thresholding Algorithm (SAGE-FISTA) recently proposed in (Rebegoldi, Calatroni, '21) for the efficient solution of strongly convex composite optimisation problems. In particular, we address here the sole (non-strongly) convex optimisation scenario, which is frequently encountered in many imaging applications. The proposed inexact S-FISTA algorithm shows analogies to the variable metric and inexact version of FISTA studied in (Bonettini, Rebegoldi, Ruggiero, '19), the main difference being the use of an adaptive (non-monotone) backtracking strategy allowing for the automatic adjustment of the algorithmic step-size along the iterations (see (Scheinberg, Goldfarb, Bai, '14, Calatroni, Chambolle, '19)). A quadratic convergence result in function values depending on the backtracking parameters and the upper and lower bounds on the spectrum of the variable metric operators is given. Experimental results on TV image deblurring problems with Poisson noise are then reported for numerical validation, showing improved computational efficiency and precision
Digital cultural heritage imaging via osmosis filtering
No full textIn Cultural Heritage (CH) imaging, data acquired within different spectral regions are often used to inspect surface and sub-surface features. Due to the experimental setup, these images may suffer from intensity inhomogeneities, which may prevent conservators from distinguishing the physical properties of the object under restoration. Furthermore, in multi-modal imaging, the transfer of information between one modality to another is often used to integrate image contents. In this paper, we apply the image osmosis model proposed in [4, 10, 12] to solve correct these problems arising when diagnostic CH imaging techniques based on reflectance, emission and fluorescence mode in the optical and thermal range are used. For an efficient computation, we use stable operator splitting techniques to solve the discretised model. We test our methods on real artwork datasets: the thermal measurements of the mural painting “Monocromo” by Leonardo Da Vinci, the UV-VIS-IR imaging of an ancient Russian icon and the Archimedes Palimpsest dataset
A continuous, non-convex & sparse super-resolution approach for fluorescence microscopy data with Poisson noise
No full textWe propose a non-convex sparsity-promoting variational
model for the problem of super-resolution in Single
Molecule Localization Microscopy (SMLM). Namely, we study
a continuous non-convex relaxation of a non-continuous and
non-convex variational model where a weighted-L2 data fidelity
modeling signal-dependent Poisson noise is combined with an
L0-regularization to promote signal sparsity. The proposed relaxation
is obtained by adapting the Continuous Exact L0 (CEL0)
relaxation of the analogous `2`0 problem with Gaussian noise
to the Poisson scenario, which is more realistic in fluorescence
microscopy applications. The associated optimization problem is
then solved by an iterative reweighted L1 (IRL1) algorithm. The
weighted-L2 data fidelity leads to a challenging estimation of the
algorithmic parameters for which efficient computation strategies
are detailed. To validate our approach, we report qualitative
and quantitative localization results for a simulated dataset,
showing that the proposed weighted-CEL0 (WCEL0) model is
well suited and capable to deal with Poisson measurements with
high accuracy and precision
A flexible space-variant anisotropic regularisation for image restoration with automated parameter selection
No full textWe propose a new space-variant anisotropic regular ization term for variational image restoration, based on the statistical assumption that the gradients of the target image distribute locally according to a bivariate generalized Gaussian distribution. The highly flexible variational structure of the corresponding regularizer encodes several free parameters which hold the potential for faithfully modeling the local geometry in the image and describing local orientation preferences. For an automatic estimation of such parameters, we design a robust maximum likelihood approach and report results on its reliability on synthetic data and natural images. For the numerical solution of the corresponding image restoration model, we use an iterative algorithm based on the alternating direction method of multipliers. A suitable preliminary variable splitting together with a novel result in multivariate nonconvex proximal calculus yield a very efficient minimization algorithm. Several numerical results showing significant quality improvement of the proposed model with respect to some related state-of-the-art competitors are reported, in particular, in terms of texture and detail preservation
Special issue on optimization and learning methods for inverse problems in microscopy: in memory of Mario Bertero
No full textEditorial in memory of Mario Berter
Scaled, inexact and adaptive generalized FISTA for strongly convex optimization
No full textInternational audienceWe consider a variable metric and inexact version of the FISTA-type algorithm considered in (Chambolle, Pock, 2016, Calatroni, Chambolle, 2019) for the minimization of the sum of two (possibly strongly) convex functions. The proposed algorithm is combined with an adaptive (non-monotone) backtracking strategy, which allows for the adjustment of the algorithmic step-size along the iterations in order to improve the convergence speed. We prove a linear convergence result for the function values, which depends on both the strong convexity moduli of the two functions and the upper and lower bounds on the spectrum of the variable metric operators. We validate the proposed algorithm, named Scaled Adaptive GEneralized FISTA (SAGE-FISTA), on exemplar image denoising and deblurring problems where edge-preserving Total Variation (TV) regularization is combined with Kullback-Leibler-type fidelity terms, as it is common in applications where signal-dependent Poisson noise is assumed in the data
Non-convex super-resolution of oct images via sparse representation
No full textWe propose a non-convex variational model for the superresolution
of Optical Coherence Tomography (OCT) images
of the murine eye, by enforcing sparsity with respect to suitable
dictionaries learnt from high-resolution OCT data. The
statistical characteristics of OCT images motivate the use of
-stable distributions for learning dictionaries, by considering
the non-Gaussian case, = 1. The sparsity-promoting cost
function relies on a non-convex penalty - Cauchy-based or
Minimax Concave Penalty (MCP) - which makes the problem
particularly challenging. We propose an efficient algorithm
for minimizing the function based on the forward-backward
splitting strategy which guarantees at each iteration the existence
and uniqueness of the proximal point. Comparisons
with standard convex `1-based reconstructions show the better
performance of non-convex models, especially in view of
further OCT image analysis
3D Image Super-Resolution by Fluorophore Fluctuations and MA-TIRF Microscopy Reconstruction (3D-COL0RME)
No full textWhiteness-Based Bilevel Learning of Regularization Parameters in Imaging
No full textWe consider an unsupervised bilevel optimization
strategy for learning regularization parameters in the context
of imaging inverse problems in the presence of additive white
Gaussian noise. Compared to supervised and weakly-supervised
metrics relying either on the prior knowledge of reference data
and/or on some (partial) knowledge on the noise statistics, the
proposed approach optimizes the whiteness of the residual between the observed data and the observation model with no need
of ground-truth data. We validate the approach on standard Total
Variation-regularized image deconvolution problems which show
that the proposed quality metric provides estimates close to the
mean-square error oracle and to discrepancy-based principles
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