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We provide a novel sharp-interface analysis via Gamma-convergence for a non-local and non-homogeneous diffuse-interface model for phase transitions, featuring an interplay between a non-local interaction kernel and a spatially dependent double-well potential. This interaction requires the development of new strategies both for the Gamma-liminf inequality and for the construction of recovery sequences. A key element of our approach is an asymptotic calibration, used to establish the Gamma-liminf lower bound. The study of the optimality of the lower bound hinges upon a novel analysis of the regularity dependence of one-dimensional optimal profiles on a family of parameters. In particular, we show how such regularity is influenced by the singularity of the interaction kernel at the origin, providing a precise and previously unexplored link between the two. Our results rely solely on the assumption of H\"older continuity for the moving wells, and also account for the compactness of sequences with equibounded energies.

We propose a novel approach for reconstructing animatable 3D Gaussian avatars from monocular videos captured by commodity devices like smartphones. Photorealistic 3D head avatar reconstruction from such recordings is challenging due to limited observations, which leaves unobserved regions under-constrained and can lead to artifacts in novel views. To address this problem, we introduce a multi-view head diffusion model, leveraging its priors to fill in missing regions and ensure view consistency in Gaussian splatting renderings. To enable precise viewpoint control, we use normal maps rendered from FLAME-based head reconstruction, which provides pixel-aligned inductive biases. We also condition the diffusion model on VAE features extracted from the input image to preserve details of facial identity and appearance. For Gaussian avatar reconstruction, we distill multi-view diffusion priors by using iteratively denoised images as pseudo-ground truths, effectively mitigating over-saturation issues. To further improve photorealism, we apply latent upsampling to refine the denoised latent before decoding it into an image. We evaluate our method on the NeRSemble dataset, showing that GAF outperforms the previous state-of-the-art methods in novel view synthesis by a 5.34\% higher SSIM score. Furthermore, we demonstrate higher-fidelity avatar reconstructions from monocular videos captured on commodity devices., Comment: Paper Video: https://youtu.be/QuIYTljvhyg Project Page: https://tangjiapeng.github.io/projects/GAF

The efficacy of address space layout randomization has been formally demonstrated in a shared-memory model by Abadi et al., contingent on specific assumptions about victim programs. However, modern operating systems, implementing layout randomization in the kernel, diverge from these assumptions and operate on a separate memory model with communication through system calls. In this work, we relax Abadi et al.'s language assumptions while demonstrating that layout randomization offers a comparable safety guarantee in a system with memory separation. However, in practice, speculative execution and side-channels are recognized threats to layout randomization. We show that kernel safety cannot be restored for attackers capable of using side-channels and speculative execution, and introduce enforcement mechanisms that can guarantee speculative kernel safety for safe system calls in the Spectre era. We implement two suitable mechanisms and we use them to compile the Linux kernel in order to evaluate their performance overhead., Comment: arXiv admin note: substantial text overlap with arXiv:2406.07278

In this paper we prove a strong two-scale approximation result for sphere-valued maps in $L^2(\Omega;W^{1,2}_0(Q_0;\mathbb{S}^2))$, where $\Omega\subset \mathbb{R}^3$ is an open domain and $Q_0\subset Q$ an open subset of the unit cube $Q=(0,1)^3$. The proof relies on a generalization of the seminal argument by F. Bethuel and X.M. Zheng to the two-scale setting. We then present an application to a variational problem in high-contrast micromagnetics., Comment: 24 pages

We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class of continuous finite-difference approximants, we show the compactness of deformations with equibounded energies, as well as their Gamma-convergence. The compactness analysis relies on a generalization of a Fr\'echet-Kolmogorov approach previously introduced by two of the authors. An essential difficulty is the identification of the limiting space of admissible deformations. We show that if the approximants involve superlinear contributions, a limiting GSBD representation can be ensured, whereas a further integral geometric term appears in the limiting functional in the linear case. We eventually discuss the connection between this latter setting and an open problem in the theory of integral geometric measures.

Despite the recent advances in computer vision research, estimating the 3D human pose from single RGB images remains a challenging task, as multiple 3D poses can correspond to the same 2D projection on the image. In this context, depth data could help to disambiguate the 2D information by providing additional constraints about the distance between objects in the scene and the camera. Unfortunately, the acquisition of accurate depth data is limited to indoor spaces and usually is tied to specific depth technologies and devices, thus limiting generalization capabilities. In this paper, we propose a method able to leverage the benefits of depth information without compromising its broader applicability and adaptability in a predominantly RGB-camera-centric landscape. Our approach consists of a heatmap-based 3D pose estimator that, leveraging the paradigm of Privileged Information, is able to hallucinate depth information from the RGB frames given at inference time. More precisely, depth information is used exclusively during training by enforcing our RGB-based hallucination network to learn similar features to a backbone pre-trained only on depth data. This approach proves to be effective even when dealing with limited and small datasets. Experimental results reveal that the paradigm of Privileged Information significantly enhances the model's performance, enabling efficient extraction of depth information by using only RGB images., Comment: ECCV 2024 Workshop T-CAP: TOWARDS A COMPLETE ANALYSIS OF PEOPLE: FINE-GRAINED UNDERSTANDING FOR REAL-WORLD APPLICATIONS

We study a nonlocal Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects and degenerate mobility. The nonlocality is described by means of a symmetric singular kernel. We define a notion of weak solution adapted to possible degeneracies and prove, as our first main result, its global-in-time existence. The proof relies on an application of the formal gradient flow structure of the system (to overcome the lack of a-priori estimates), combined with an extension of the boundedness-by-entropy method, in turn involving a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired nonlocal weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment. Finally, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts.

We investigate the primordial power spectra (PPS) of scalar and tensor perturbations, derived through the slow-roll approximation. By solving the Mukhanov-Sasaki equation and the tensor perturbation equation with Green's function techniques, we extend the PPS calculations to third-order corrections, providing a comprehensive perturbative expansion in terms of slow-roll parameters. We investigate the accuracy of the analytic predictions with the numerical solutions of the perturbation equations for a selection of single-field slow-roll inflationary models. We derive the constraints on the Hubble flow functions $\epsilon_i$ from Planck, ACT, SPT, and BICEP/Keck data. We find an upper bound $\epsilon_1 \lesssim 0.002$ at 95\% CL dominated by BICEP/Keck data and robust to all the different combination of datasets. We derive the constraint $\epsilon_2 \simeq 0.031 \pm 0.004$ at 68\% confidence level (CL) from the combination of Planck data and late-time probes such as baryon acoustic oscillations, redshift space distortions, and supernovae data at first order in the slow-roll expansion. The uncertainty on $\epsilon_2$ gets larger including second- and third-order corrections, allowing for a non-vanishing running and running of the running respectively, leading to $\epsilon_2 \simeq 0.034 \pm 0.007$ at 68\% CL. We find $\epsilon_3 \simeq 0.1 \pm 0.4$ at 95\% CL both at second and at third order in the slow-roll expansion of the spectra. $\epsilon_4$ remains always unconstrained. The combination of Planck and SPT data leads to slightly tighter constraints on $\epsilon_2$ and $\epsilon_3$. On the contrary, the combination of Planck data with ACT measurements, which point to higher values of the scalar spectral index compared to Planck findings, leads to shifts in the means and maximum likelihood values for $\epsilon_2$ and $\epsilon_3$., Comment: 50 pages, 11 figures, 5 tables

We introduce eRHL, a program logic for reasoning about relational expectation properties of pairs of probabilistic programs. eRHL is quantitative, i.e., its pre- and post-conditions take values in the extended non-negative reals. Thanks to its quantitative assertions, eRHL overcomes randomness alignment restrictions from prior logics, including PRHL, a popular relational program logic used to reason about security of cryptographic constructions, and apRHL, a variant of PRHL for differential privacy. As a result, eRHL is the first relational probabilistic program logic to be supported by non-trivial soundness and completeness results for all almost surely terminating programs. We show that eRHL is sound and complete with respect to program equivalence, statistical distance, and differential privacy. We also show that every PRHL judgment is valid iff it is provable in eRHL. We showcase the practical benefits of eRHL with examples that are beyond reach of PRHL and apRHL.

The efficacy of address space layout randomization has been formally demonstrated in a shared-memory model by Abadi et al., contingent on specific assumptions about victim programs. However, modern operating systems, implementing layout randomization in the kernel, diverge from these assumptions and operate on a separate memory model with communication through system calls. In this work, we relax Abadi et al.'s language assumptions while demonstrating that layout randomization offers a comparable safety guarantee in a system with memory separation. However, in practice, speculative execution and side-channels are recognized threats to layout randomization. We show that kernel safety cannot be restored for attackers capable of using side-channels and speculative execution and introduce a new condition, that allows us to formally prove kernel safety in the Spectre era. Our research demonstrates that under this condition, the system remains safe without relying on layout randomization. We also demonstrate that our condition can be sensibly weakened, leading to enforcement mechanisms that can guarantee kernel safety for safe system calls in the Spectre era.

Separation logic is a substructural logic which has proved to have numerous and fruitful applications to the verification of programs working on dynamic data structures. Recently, Barthe, Hsu and Liao have proposed a new way of giving semantics to separation logic formulas in which separating conjunction is interpreted in terms of probabilistic independence. The latter is taken in its exact form, i.e., two events are independent if and only if the joint probability is the product of the probabilities of the two events. There is indeed a literature on weaker notions of independence which are computational in nature, i.e. independence holds only against efficient adversaries and modulo a negligible probability of success. The aim of this work is to explore the nature of computational independence in a cryptographic scenario, in view of the aforementioned advances in separation logic. We show on the one hand that the semantics of separation logic can be adapted so as to account for complexity bounded adversaries, and on the other hand that the obtained logical system is useful for writing simple and compact proofs of standard cryptographic results in which the adversary remains hidden. Remarkably, this allows for a fruitful interplay between independence and pseudorandomness, itself a crucial notion in cryptography., Comment: to be published in CSF'24

We investigate a model for the accretive growth of an elastic solid. The reference configuration of the body is accreted in its normal direction, with space- and deformation-dependent accretion rate. The time-dependent reference configuration is identified via the level sets of the unique viscosity solution of a suitable generalized eikonal equation. After proving the global-in-time well-posedness of the quasistatic equilibrium under prescribed growth, we prove the existence of a local-in-time solution for the coupled equilibrium-growth problem, where both mechanical displacement and time-evolving set are unknown. A distinctive challenge is the limited regularity of the growing body, which calls for proving a new uniform Korn inequality.

About a hundred years ago the Russian biologist A. Gurwitsch, based on his experiments with onion plants by measuring their growth rate, made the hypothesis that plants emitted a weak electromagnetic field which somehow influenced cell growth. This interesting observation remained fundamentally ignored by the scientific community and only in the 1950s the electromagnetic emission from some plants was measured using a photomultiplier used in single counting mode. Later, in the 80s several groups in the world started some extensive work to understand the origin and role of this ultra-weak emission, hereby called biophotons, coming from living organisms. Biophotons are an endogenous very small production of photons in the visible energy range in and from cells and organism, and this emission is characteristic of alive organisms. Today there is no doubt that biophotons really exist, this emission has in fact been measured by many groups and on many different living organisms, from humans to bacteria. On the contrary, the origin of biophotons and whether organisms use them in some way to exchange information is not yet well known; no model proposed since now is really capable of reproducing and interpreting the great variety of experimental data coming from the many different living systems measured so far. In this brief review we present our experimental work on biophotons coming from germinating seeds, the main experimental results and some methods we are using to analyze the data in order to open the door for interpretative models of this phenomenon and clarifying its function in the regulation and communication between cells and living organisms. We also discuss some ideas on how to increase the signal-to-noise ratio of the measured signal to have new experimental possibilities that allow the measurement and the characterization of currently unmeasurable quantities., Comment: 19 pages, 7 figures

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as \textit{minimizing-movements scheme}, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, we prove the existence of a solution for a general class of integrodifferential operators, not necessarily linear or symmetric, which include fractional versions of the $q$-Laplacian. In the second part of the paper, we adapt the argument in order to prove the existence of solutions in the case of regional fractional operators. As a byproduct, this yields an existence result in the interesting cases of homogeneous fractional Neumann boundary conditions or periodic boundary conditions., Comment: 80 pages

Let $u$ be the unique viscosity solution of $\alpha(x)|\nabla u|=1$ in the external domain ${\mathbb R}^{ n} \setminus K$ with $u=0$ on $K$. In case $\alpha$ is continuous, bounded, and uniformly positive and $K$ is a bounded John domain, we prove that all superlevels of $u$ are John domains, too. Moreover, we give counterexamples showing that John regularity is sharp in this setting., Comment: 13 pages, 6 figures

We introduce GaussianAvatars, a new method to create photorealistic head avatars that are fully controllable in terms of expression, pose, and viewpoint. The core idea is a dynamic 3D representation based on 3D Gaussian splats that are rigged to a parametric morphable face model. This combination facilitates photorealistic rendering while allowing for precise animation control via the underlying parametric model, e.g., through expression transfer from a driving sequence or by manually changing the morphable model parameters. We parameterize each splat by a local coordinate frame of a triangle and optimize for explicit displacement offset to obtain a more accurate geometric representation. During avatar reconstruction, we jointly optimize for the morphable model parameters and Gaussian splat parameters in an end-to-end fashion. We demonstrate the animation capabilities of our photorealistic avatar in several challenging scenarios. For instance, we show reenactments from a driving video, where our method outperforms existing works by a significant margin., Comment: Project page: https://shenhanqian.github.io/gaussian-avatars

In this paper we study nonlocal-to-local asymptotics for a tumor-growth model coupling a viscous Cahn-Hilliard equation describing the tumor proportion with a reaction-diffusion equation for the nutrient phase parameter. First, we prove that solutions to the nonlocal Cahn-Hilliard system converge, as the nonlocality parameter tends to zero, to solutions to its local counterpart. Second, we provide first-order optimality conditions for an optimal control problem on the local model, accounting also for chemotaxis, and both for regular or singular potentials, without any additional regularity assumptions on the solution operator. The proof is based on an approximation of the local control problem by means of suitable nonlocal ones, and on proving nonlocal-to-local convergence both for the corresponding dual systems and for the associated first-order optimality conditions.

This paper presents a framework addressing the challenge of global localization in autonomous mobile robotics by integrating LiDAR-based descriptors and Wi-Fi fingerprinting in a pre-mapped environment. This is motivated by the increasing demand for reliable localization in complex scenarios, such as urban areas or underground mines, requiring robust systems able to overcome limitations faced by traditional Global Navigation Satellite System (GNSS)-based localization methods. By leveraging the complementary strengths of LiDAR and Wi-Fi sensors used to generate predictions and evaluate the confidence of each prediction as an indicator of potential degradation, we propose a redundancy-based approach that enhances the system's overall robustness and accuracy. The proposed framework allows independent operation of the LiDAR and Wi-Fi sensors, ensuring system redundancy. By combining the predictions while considering their confidence levels, we achieve enhanced and consistent performance in localization tasks., Comment: 7 pages, 5 figures. Accepted for publication in the 21st International Conference on Advanced Robotics (ICAR 2023)

Relevance of research on telecommunications networks is predicated upon the implementations which it explicitly claims or implicitly subsumes. This paper supports researchers through a survey of Communications Service Providers current implementations within the metro area, and trends that are expected to shape the next-generation metro area network. The survey is composed of a quantitative component, complemented by a qualitative component carried out among field experts. Among the several findings, it has been found that service providers with large subscriber base sizes, are less agile in their response to technological change than those with smaller subscriber base sizes: thus, copper media are still an important component in the set of access network technologies. On the other hand, service providers with large subscriber base sizes are strongly committed to deploying distributed access architectures, notably using remote access nodes like remote OLT and remote MAC-PHY. This study also shows that the extent of remote node deployment for multi-access edge computing is about the same as remote node deployment for distributed access architectures, indicating that these two aspects of metro area networks are likely to be co-deployed., Comment: 70 pages

We perform a dimension reduction analysis for a coupled rate-dependent/rate-independent adhesive-contact model in the setting of visco-elastodynamic plates. We work with a weak solvability notion inspired by the theory of (purely) rate-independent processes, and accordingly term the related solutions `Semistable Energetic'. For Semistable Energetic solutions, the momentum balance holds in a variational sense, whereas the flow rule for the adhesion parameter is replaced by a semi-stability condition coupled with an energy-dissipation inequality. Prior to addressing the dimension reduction analysis, we show that Semistable Energetic solutions to the three-dimensional damped adhesive contact model converge, as the viscosity term tends to zero, to three-dimensional Semistable Energetic solutions for the undamped corresponding system. We then perform a dimension reduction analysis, both in the case of a vanishing viscosity tensor (leading, in the limit, to an undamped model), and in the complementary setting in which the damping is assumed to go to infinity as the thickness of the plate tends to zero. In both regimes, the presence of adhesive contact yields a nontrivial coupling of the in-plane and out-of-plane contributions. In the undamped scenario we obtain in the limit an energy-dissipation inequality and a semistability condition. In the damped case, instead, we achieve convergence to an enhanced notion of solution, fulfilling an energy-dissipation balance.

We perform a stochastic-homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere, and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii's predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure.

Biophotons are an ultra-weak emission of photons in the visible energy range from living matter. In this work, we study the emission from germinating seeds using an experimental technique designed to detect light of extremely small intensity. The emission from lentil seeds and single bean was analyzed during the whole germination process in terms of both the different spectral components through low pass filters and the different count distributions in the various stages of the germination process. Although the shape of the emission spectrum appears to be very similar in the two samples used in our experiment, our analysis is able to highlight the differences present in the two cases. In this way, it was possible to correlate the various types of emissions to the degree of development of the seed during germination., Comment: 18 pages including references, 8 figures, this work has been presented to the DCP22 held in 2022 in Pisa

Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best-known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image pairs. In this work, we consider such methods with space-dependent parameters which are piecewise constant on dyadic grids, with the grid itself being part of the minimization. We prove existence of minimizers for fixed discontinuous parameters under mild assumptions on the data, which lead to existence of finite optimal partitions. We further establish that these assumptions are equivalent to the commonly used box constraints on the parameters. On the numerical side, we consider a simple subdivision scheme for optimal partitions built on top of any other bilevel optimization method for scalar parameters, and demonstrate its improved performance on some representative test images when compared with constant optimized parameters., Comment: 45 pages, 6 figures, 1 table

In the context of Sobolev spaces with variable exponents, Poincar\'e--Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form \begin{equation*} \int_\Omega \left|f(x)-\langle f\rangle_{\Omega}\right|^{p(x)} \ {\mathrm{d} x} \leqslant C \int_\Omega|\nabla f(x)|^{p(x)}{\mathrm{d} x}, \end{equation*} are known to be \emph{false}. As a result, all available modular versions of the Poincar\'e- Wirtinger inequality in the variable-exponent setting always contain extra terms that do not disappear in the constant exponent case, preventing such inequalities from reducing to the classical ones in the constant exponent setting. Our contribution is threefold. First, we establish that a modular Poincar\'e--Wirtinger inequality particularizing to the classical one in the constant exponent case is indeed conceivable. We show that if $\Omega\subset \mathbb{R}^n$ is a bounded Lipschitz domain, and if $p\in L^\infty(\Omega)$, $p \geq 1$, then for every $f\in C^\infty(\bar\Omega)$ the following generalized Poincar\'e--Wirtinger inequality holds \begin{equation*} \int_\Omega \left|f(x)-\langle f\rangle_{\Omega}\right|^{p(x)} \ {\mathrm{d} x} \leq C \int_\Omega\int_\Omega \frac{|\nabla f(z)|^{p(x)}}{|z-x|^{n-1}}\ {\mathrm{d} z}{\mathrm{d} x}, \end{equation*} where $\langle f\rangle_{\Omega}$ denotes the mean of $f$ over $\Omega$, and $C>0$ is a positive constant depending only on $\Omega$ and $\|p\|_{L^\infty(\Omega)}$. Second, our argument is concise and constructive and does not rely on compactness results. Third, we additionally provide geometric information on the best Poincar\'e--Wirtinger constant on Lipschitz domains.

We identify effective models for thin, linearly elastic and perfectly plastic plates exhibiting a microstructure resulting from the periodic alternation of two elastoplastic phases. We study here both the case in which the thickness of the plate converges to zero on a much faster scale than the periodicity parameter and the opposite scenario in which homogenization occurs on a much finer scale than dimension reduction. After performing a static analysis of the problem, we show convergence of the corresponding quasistatic evolutions. The methodology relies on two-scale convergence and periodic unfolding, combined with suitable measure-disintegration results and evolutionary $\Gamma$-convergence., Comment: 48 pages

Following the seminal paper by Bourgain, Brezis and Mironescu, we focus on the asymptotic behavior of some nonlocal functionals that, for each $u\in L^2(\mathbb{R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights $\{\rho_\epsilon\}$, $\epsilon\in (0,1)$, we devise sufficient and necessary conditions on $\{\rho_\epsilon\}$ for the associated nonlocal functionals to converge as $\epsilon\to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.

We introduce a new bounded theory RS^1_2 and show that the functions which are Sigma^b_1-representable in it are precisely random functions which can be computed in polynomial time. Concretely, we pass through a class of oracle functions over string, called POR, together with the theory of arithmetic RS^1_2. Then, we show that functions computed by poly-time PTMs are arithmetically characterized by a class of probabilistic bounded formulas., Comment: 37 pages, pre-print

This work is devoted to the analysis of the interplay between internal variables and high-contrast microstructure in inelastic solids. As a concrete case-study, by means of variational techniques, we derive a macroscopic description for an elastoplastic medium. Specifically, we consider a composite obtained by filling the voids of a periodically perforated stiff matrix by soft inclusions. We study the $\Gamma$-convergence of the related energy functionals as the periodicity tends to zero. The main challenge is posed by the lack of coercivity brought about by the degeneracy of the material properties in the soft part. We prove that the $\Gamma$-limit, which we compute with respect to a suitable notion of convergence, is the sum of the contributions resulting from each of the two components separately. Eventually, convergence of the energy minimizing configurations is obtained., Comment: 39 pages. The content of the paper was originally part of version 1 of arXiv:2204.09084, which has now been split into two parts

An effective model is identified for thin perfectly plastic plates whose microstructure consists of the periodic assembling of two elastoplastic phases, as the periodicity parameter converges to zero. Assuming that the thickness of the plates and the periodicity of the microstructure are comparably small, a limiting description is obtained in the quasistatic regime via simultaneous homogenization and dimension reduction by means of evolutionary $\Gamma$-convergence, two-scale convergence, and periodic unfolding., Comment: 46 pages

We introduce a unified framework based on bi-level optimization schemes to deal with parameter learning in the context of image processing. The goal is to identify the optimal regularizer within a family depending on a parameter in a general topological space. Our focus lies on the situation with non-compact parameter domains, which is, for example, relevant when the commonly used box constraints are disposed of. To overcome this lack of compactness, we propose a natural extension of the upper-level functional to the closure of the parameter domain via Gamma-convergence, which captures possible structural changes in the reconstruction model at the edge of the domain. Under two main assumptions, namely, Mosco-convergence of the regularizers and uniqueness of minimizers of the lower-level problem, we prove that the extension coincides with the relaxation, thus admitting minimizers that relate to the parameter optimization problem of interest. We apply our abstract framework to investigate a quartet of practically relevant models in image denoising, all featuring nonlocality. The associated families of regularizers exhibit qualitatively different parameter dependence, describing a weight factor, an amount of nonlocality, an integrability exponent, and a fractional order, respectively. After the asymptotic analysis that determines the relaxation in each of the four settings, we finally establish theoretical conditions on the data that guarantee structural stability of the models and give examples of when stability is lost.