Abstract. Deep equilibrium models (DEQs) are implicit neural networks with fixed points that have recently gained attention for learning image regularization functionals, particularly in settings involving Gaussian fidelities, where assumptions on the forward operator ensure contractiveness of standard (proximal) gradient descent operators. In this work, we extend the application of DEQs to Poisson inverse problems, where the data fidelity term is more appropriately modeled by the Kullback–Leibler divergence. To this end, we introduce a novel DEQ formulation based on mirror descent defined in terms of a tailored non-Euclidean geometry that naturally adapts with the structure of the data term. This enables the learning of neural regularizers within a principled training framework. We derive sufficient conditions and establish refined convergence results based on the Kurdyka–Łojasiewicz framework for functions with nonclosed domains to guarantee the convergence of the learned reconstruction scheme and propose computational strategies that enable both efficient training and parameter-free inference. Numerical experiments show that our method outperforms traditional model-based approaches, and it is comparable to the performance of Bregman plug-and-play methods, while mitigating their typical drawbacks, such as time-consuming tuning of hyperparameters. The code is publicly available at https://github.com/christiandaniele/DEQ-MD .

Abstract We consider the minimization of $$\ell _0$$ ℓ 0 -regularized criteria involving non-quadratic data terms such as the Kullback-Leibler divergence and the logistic regression, possibly combined with an $$\ell _2$$ ℓ 2 regularization. We first prove the existence of global minimizers for such problems and characterize their local minimizers. Then, we propose a new class of continuous relaxations of the $$\ell _0$$ ℓ 0 pseudo-norm, termed as $$\ell _0$$ ℓ 0 Bregman Relaxations (B-rex). They are defined in terms of suitable Bregman distances and lead to exact continuous relaxations of the original $$\ell _0$$ ℓ 0 -regularized problem in the sense that they do not alter its set of global minimizers and reduce its non-convexity by eliminating certain local minimizers. Both features make such relaxed problems more amenable to be solved by standard non-convex optimization algorithms. In this spirit, we consider the proximal gradient algorithm and provide explicit computation of proximal points for the B-rex penalty in several cases. Finally, we report a set of numerical results illustrating the geometrical behavior of the proposed B-rex penalty for different choices of the underlying Bregman distance, its relation with convex envelopes, as well as its exact relaxation properties in 1D/2D and higher dimensions.

Growing evidence indicates that only a sparse subset from a pool of sensory neurons is active for the encoding of visual stimuli at any instant in time. Traditionally, to replicate such biological sparsity, generative models have been using the ℓ1 norm as a penalty due to its convexity, which makes it amenable to fast and simple algorithmic solvers. In this work, we use biological vision as a test-bed and show that the soft thresholding operation associated to the use of the ℓ1 norm is highly suboptimal compared to other functions suited to approximating ℓp with 0 ≤ p < 1 (including recently proposed continuous exact relaxations), in terms of performance. We show that ℓ1 sparsity employs a pool with more neurons, i.e. has a higher degree of overcompleteness, in order to maintain the same reconstruction error as the other methods considered. More specifically, at the same sparsity level, the thresholding algorithm using the ℓ1 norm as a penalty requires a dictionary of ten times more units compared to the proposed approach, where a non-convex continuous relaxation of the ℓ0 pseudo-norm is used, to reconstruct the external stimulus equally well. At a fixed sparsity level, both ℓ0- and ℓ1-based regularization develop units with receptive field (RF) shapes similar to biological neurons in V1 (and a subset of neurons in V2), but ℓ0-based regularization shows approximately five times better reconstruction of the stimulus. Our results in conjunction with recent metabolic findings indicate that for V1 to operate efficiently it should follow a coding regime which uses a regularization that is closer to the ℓ0 pseudo-norm rather than the ℓ1 one, and suggests a similar mode of operation for the sensory cortex in general.

We propose a smooth, non-convex and content-adaptive regularisation model for single-image super-resolution of murine Optical Coherence Tomography (OCT) data. We follow a sparse-representation approach where sparsity is modelled with respect to a suitable dictionary generated from high-resolution OCT data. To do so, we employ a pre-learned dictionary tailored to model α-stable statistics in the non-Gaussian case, i.e. α<2. The image reconstruction problem renders here particularly challenging due to the high level of noise degradation and to the heterogeneity of the data at hand. As a regulariser, we employ a separable Cauchy-type penalty. To favour adaptivity to image contents, we propose a space-variant modelling by which the local degree of non-convexity given by the local Cauchy shape parameter is estimated via maximum likelihood. For the solution of the reconstruction problem, we consider an extension of the cautious Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm where the descent direction is suitably updated depending on the local convexity of the functional. Our numerical results show that the combination of a space-variant modelling with a tailored optimisation strategy improves reconstruction results and allows for an effective segmentation with standard approaches.

We propose a new variational model for non-linear image fusion. Our approach is based on the use of an osmosis energy term related to the one studied in Vogel et al. [44] and Weickert et al. [45]. The minimization of the proposed non-convex energy realizes visually plausible image data fusion, invariant to multiplicative brightness changes. On the practical side, it requires minimal supervision and parameter tuning and can encode prior information on the structure of the images to be fused. For the numerical solution of the proposed model, we develop a primal-dual algorithm and we apply the resulting minimization scheme to solve multi-modal face fusion, color transfer and cultural heritage conservation problems. Visual and quantitative comparisons to state-of-the-art approaches prove the out-performance and the flexibility of our method.

We propose a new space-variant anisotropic regularization 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.

In 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.

We consider the problem of image denoising in the presence of noise whose statistical properties are a combination of two different distributions. We focus on noise distributions frequently considered in applications, such as salt & pepper and Gaussian, and Gaussian and Poisson noise mixtures. We derive a variational image denoising model that features a total variation regularization term and a data discrepancy encoding the mixed noise as an infimal convolution of discrepancy terms of the single-noise distributions. We give a statistical derivation of this model by joint maximum a posteriori (MAP) estimation. Classical single-noise models are recovered asymptotically as the weighting parameters go to infinity. The numerical solution of the model is computed using second order Newton-type methods. Numerical results show the decomposition of the noise into its constituting components. The paper is furnished with several numerical experiments, and comparisons with other methods dealing with the mixed noise case are shown.