Infimal Convolution Research Articles

A note on 𝐶^{1,𝛼}-smooth approximation of Lipschitz functions

We show that on super-reflexive spaces a Moreau-Yosida type of regularisation by infimal convolution together with a known insertion-type theorem (a variant of Ilmanen’s lemma) easily give an approximation of a Lipschitz function by a C 1 , α C^{1,\alpha } -smooth Lipschitz function with the same Lipschitz constant. This is a generalisation of the well-known theorem of J.-M. Lasry and P.-L. Lions from Hilbert spaces. It also gives a new self-contained and probably simpler proof of the Lasry-Lions theorem.

The Investigation of Some Essential Concepts of Extended Fuzzy-Valued Convex Functions and Their Applications

In this paper, we are thus motivated to define and introduce the extended fuzzy-valued convex functions that can take the singleton fuzzy values −∞˜ and +∞˜ at some points. Such functions can be characterized using the notions of effective domain and epigraph. In this way, we study important concepts such as fuzzy indicator function and fuzzy infimal convolution for extended fuzzy-valued functions. Finally, we introduce the concept of directional generalized derivative for extended above functions and its properties. Eventually, we give a practical example that will illustrate well the directional g-derivative for the extended fuzzy-valued convex function.

Risk sharing with deep neural networks

We consider the problem of optimally sharing a financial position among agents with potentially different reference risk measures. The problem is equivalent to computing the infimal convolution of the risk metrics and finding the so-called optimal allocations. We propose a neural network-based framework to solve the problem and we prove the convergence of the approximated inf-convolution, as well as the approximated optimal allocations, to the corresponding theoretical values. We support our findings with several numerical experiments.

A combined first and fractional order regularization method for mixed Poisson-white spike noisy image restoration

In this paper we study the problem of restoring images affected by mixed Poisson-Square Cauchy noise. A multi-convex variational model is derived by integrating infimal convolution likelihood into a process of joint maximum a posteriori estimation, where a modified four-directional fractional-order total variation is united with the first order total variation to characterize the image prior. To solve the proposed model numerically, a block coefficient descent based algorithm is derived, in which variable splitting and alternating direction minimization of multipliers are utilized along with anisotropic diffusion and additive operator splitting to gain efficiency and quality. The obtained numerical results are compared with results obtained from two other total variation regularized models with different fidelities. The numerical discuss confirms the flexibility and validity of the proposed model.

Infimal convolution and AM-GM majorized total variation-based integrated approach for biosignal denoising

Infimal convolution and AM-GM majorized total variation-based integrated approach for biosignal denoising

On the framework of Lp summations for functions

We develop the framework of Lp operations for functions: one based on an Lp analogue of the supremal convolution and another based on an Lp analogue of the infimal convolution. On one hand, we establish the general forms of the Lp-Borell-Brascamp-Lieb inequality for various functionals on the space of s-concave functions. On the other hand, we propose the j-th quermassintegral for convex functions and prove its associated variational formula in the rotationally invariant case.

First-order primal–dual algorithm for image restoration corrupted by mixed Poisson–Gaussian noise

The total variation infimal convolution (TV-IC) model combining Kullback–Leibler and ℓ2-norm data fidelity term works well for the inverse problem of mixed Poisson–Gaussian noise. Most existing algorithms for solving the TV-IC model rely on the Newton method to solve a nonlinear optimization subproblem, which inevitably increases the computation burden. In this study, we apply the first-order primal–dual Chambolle–Pock algorithm to solve the TV-IC model. In particular, we present an effective algorithm to solve the subproblem of the joint proximal operator with the Kullback–Leibler divergence and ℓ2-norm, which is based on the bilinear constraint alternating direction multiplier method. Numerical experiment results demonstrate that the proposed algorithm outperforms the state-of-the-art methods for mixed Poisson–Gaussian denoising and deblurring problems.

Boundedness and Unboundedness in Total Variation Regularization

We consider whether minimizers for total variation regularization of linear inverse problems belong to L^infty even if the measured data does not. We present a simple proof of boundedness of the minimizer for fixed regularization parameter, and derive the existence of uniform bounds for sufficiently small noise under a source condition and adequate a priori parameter choices. To show that such a result cannot be expected for every fidelity term and dimension we compute an explicit radial unbounded minimizer, which is accomplished by proving the equivalence of weighted one-dimensional denoising with a generalized taut string problem. Finally, we discuss the possibility of extending such results to related higher-order regularization functionals, obtaining a positive answer for the infimal convolution of first and second order total variation.

A generalization of the Moreau–Yosida regularization

In many applications, one deals with nonsmooth functions, e.g., in nonsmooth dynamical systems, nonsmooth mechanics, or nonsmooth optimization. In order to establish theoretical results, it is often beneficial to regularize the nonsmooth functions in an intermediate step. In this work, we investigate the properties of a generalization of the Moreau–Yosida regularization on a normed space where we replace the quadratic kernel in the infimal convolution with a more general function. More precisely, for a function f:X→(−∞,+∞] defined on a normed space (X,‖⋅‖) and given parameters p>1 and ε>0, we investigate the properties of the generalized Moreau–Yosida regularization given byfε(u)=infv∈X⁡{1pε‖u−v‖p+f(v)},u∈X. We show that the generalized Moreau–Yosida regularization satisfies the same properties as in the classical case for p=2, provided that X is not a Hilbert space. We further establish a convergence result in the sense of Mosco-convergence as the regularization parameter ε tends to zero.

Characteristic-constrained accelerating MR T1rho mapping with blockwise infimal convolution of matrix elastic-net regularization.

Magnetic resonance parameter mapping (MRPM) plays an important role in clinical applications and biomedical researches. However, the acceleration of MRPM remains a major challenge for achieving furtherimprovements. In this work, a new undersampled k-space based joint multi-contrast image reconstruction approach named CC-IC-LMEN is proposed for accelerating MR T1rhomapping. The reconstruction formulation of the proposed CC-IC-LMEN method imposes a blockwise low-rank assumption on the characteristic-image series (c-p space) and utilizes infimal convolution (IC) to exploit and balance the generalized low-rank properties in low-and high-order c-p spaces, thereby improving the accuracy. In addition, matrix elastic-net (MEN) regularization based on the nuclear and Frobenius norms is incorporated to obtain stable and exact solutions in cases with large accelerations and noisy observations. This formulation results in a minimization problem, that can be effectively solved using a numerical algorithm based on the alternating direction method of multipliers (ADMM). Finally, T1rho maps are then generated according to the reconstructed images using nonlinear least-squares (NLSQ) curve fitting with an established relaxometrymodel. The relative l2 -norm error (RLNE) and structural similarity (SSIM) in the regions of interest (ROI) show that the CC-IC-LMEN approach is more accurate than other competing methods even in situations with heavy undersampling or noisyobservation. Our proposed CC-IC-LMEN method provides accurate and robust solutions for accelerated MR T1rhomapping.

Optimization of first-order Nicoletti boundary value problem with discrete and differential inclusions and duality

This paper is devoted to optimal control of the first-order Nicoletti boundary value problem (BVP) with discrete and differential inclusions (DFIs) and duality. First, we define Nicoletti-type problem with discrete inclusions, formulate optimality conditions for it and, based on the concept of infimal convolution, dual problems. Then, using the auxiliary discrete-approximate problem, we construct dual problems for Nicoletti DFIs and prove the duality theorems. Here, for the transition to the continuous problem, some results on the equivalence of locally adjoint mappings and support functions to the graph of set-valued mapping are proved. It turns out that the Euler–Lagrange type inclusions are ‘duality relations’ for both primal and dual problems, which means that a pair consisting of solutions to the primal and dual problems satisfies this extremal relation and vice versa. Finally, as an appication of the results obtained, we consider the first-order Nicoletti BVP with polyhedral DFIs.

Regularization graphs—a unified framework for variational regularization of inverse problems

We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: regularization graphs. Regularization graphs allow to construct functionals using as building blocks linear operators and convex functionals, assembled by means of operators that can be seen as generalizations of classical infimal convolution operators. This class of functionals exhaustively covers existing regularization approaches and it is flexible enough to craft new ones in a simple and constructive way. We provide well-posedness and convergence results with the proposed class of functionals in a general setting. Further, we consider a bilevel optimization approach to learn optimal weights for such regularization graphs from training data. We demonstrate that this approach is capable of optimizing the structure and the complexity of a regularization graph, allowing, for example, to automatically select a combination of regularizers that is optimal for given training data.

A General Non-Lipschitz Infimal Convolution Regularized Model: Lower Bound Theory and Algorithm

The convex infimal convolution model proposed in Chambolle and Lions [Numer. Math., 76 (1997), pp. 167--188] is a fundamental model to extract two useful components from a single input image and has various low level vision applications. In many of them, one target component has an (approximately) piecewise constant structure and the other is a smoothly varying function or repeated texture pattern. In this paper, we propose and study a general non-Lipschitz infimal convolution (GnLIC) regularization model, which covers most existing applications in this type. Therein the non-Lipschitz regularization enforces the piecewise constant property of the first component. For this GnLIC model, we prove a lower bound theory for its local minimizers and a local version for its stationary points. Motivated by these, we naturally extend previous works to design an inexact iterative support shrinking algorithm with proximal linearization for our GnLIC model (InISSAPL-GnLIC). Moreover, we establish the sequence convergence property and a sequence lower bound theory for InISSAPL-GnLIC, provided that an inexact subgradient condition generated by a subsolver holds. The subsolver is constructed by efficient ADMM and a specially designed feasibilization operation. We finally give numerical experiments in two low level vision applications: Retinex and cartoon-texture decomposition. These tests demonstrate that our non-Lipschitz regularization based method can indeed extract the piecewise constant component better than existing approaches, which is consistent with the established lower bound theory.

Image Reconstruction in Light-Sheet Microscopy: Spatially Varying Deconvolution and Mixed Noise

We study the problem of deconvolution for light-sheet microscopy, where the data is corrupted by spatially varying blur and a combination of Poisson and Gaussian noise. The spatial variation of the point spread function of a light-sheet microscope is determined by the interaction between the excitation sheet and the detection objective PSF. We introduce a model of the image formation process that incorporates this interaction and we formulate a variational model that accounts for the combination of Poisson and Gaussian noise through a data fidelity term consisting of the infimal convolution of the single noise fidelities, first introduced in L. Calatroni et al. (SIAM J Imaging Sci 10(3):1196–1233, 2017). We establish convergence rates and a discrepancy principle for the infimal convolution fidelity and the inverse problem is solved by applying the primal–dual hybrid gradient (PDHG) algorithm in a novel way. Numerical experiments performed on simulated and real data show superior reconstruction results in comparison with other methods.

Dualization and Automatic Distributed Parameter Selection of Total Generalized Variation via Bilevel Optimization

Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, a Newton type method for the solution of the lower level problem, i.e. the image reconstruction problem, and a bilevel TGV algorithm are introduced. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.

Dual-Mode Volumetric Optoacoustic and Contrast Enhanced Ultrasound Imaging With Spherical Matrix Arrays.

Spherical matrix arrays represent an advantageous tomographic detection geometry for non-invasive deep tissue mapping of vascular networks and oxygenation with volumetric optoacoustic tomography (VOT). Hybridization of VOT with ultrasound (US) imaging remains difficult with this configuration due to the relatively large inter-element pitch of spherical arrays. We suggest a new approach for combining VOT and US contrast-enhanced 3D imaging employing injection of clinically-approved microbubbles. Power Doppler (PD) and US localization imaging were enabled with a sparse US acquisition sequence and model-based inversion based on infimal convolution of total variation (ICTV) regularization. In vitro experiments in tissue-mimicking phantoms and in living mouse brain demonstrate the powerful capabilities of the new dual-mode imaging approach attaining 80 μm spatial resolution and a more than 10 dB signal to noise improvement with respect to a classical delay and sum beamformer. Microbubble localization and tracking allowed for flow velocity mapping up to 40 mm/s.

Multi-directional rain streak removal based on infimal convolution of oscillation TGV

Image deraining is essential in both image processing and computer vision. However it is still challenging to remove rain streaks, due to the fact that they are complex and usually have multiple directions. In this paper, we propose a novel infimal convolution type functional based model to remove multi-directional rain streaks. Specifically, we employ total generalized variation (TGV) to represent the rain-free background and m-fold infimal convolution of oscillation TGV (ICTGVmosci) to capture the rain streaks with m different directions. The existence of solutions to the proposed model is proved. We also design an algorithm using the classical first-order Primal–Dual method for solving the proposed model. Numerical experiments on both synthetic and real images demonstrate that the proposed model outperforms the state-of-the-art methods quantitatively and qualitatively.

A stochastic variance reduction algorithm with Bregman distances for structured composite problems

We develop a novel stochastic primal–dual splitting method with Bregman distances for solving structured composite problems involving infimal convolutions in non-Euclidean spaces. The sublinear convergence in expectation of the primal–dual gap is proved under mild conditions on stepsize for the general case. The linear convergence rate is obtained under additional condition like the strong convexity relative to Bregman functions.

A variational method for Abel inversion tomography with mixed Poisson-Laplace-Gaussian noise

<p style='text-indent:20px;'>Abel inversion tomography plays an important role in dynamic experiments, while most known studies are started with a single Gaussian assumption. This paper proposes a mixed Poisson-Laplace-Gaussian distribution to characterize the noise in charge-coupled-device (CCD) sensed radiographic data, and develops a multi-convex optimization model to address the reconstruction problem. The proposed model is derived by incorporating varying amplitude Gaussian approximation and expectation maximization algorithm into an infimal convolution process. To solve it numerically, variable splitting and augmented Lagrangian method are integrated into a block coordinate descent framework, in which anisotropic diffusion and additive operator splitting are employed to gain edge preserving and computation efficiency. Supplementarily, a space of functions of adaptive bounded Hessian is introduced to prove the existence and uniqueness of solution to a higher-order regularized, quadratic subproblem. Moreover, a simplified algorithm with higher order regularizer is derived for Poisson noise removal. To illustrate the performance of the proposed algorithms, numerical tests on synthesized and real digital data are performed.</p>

A variational image denoising model under mixed Cauchy and Gaussian noise

<abstract><p>In this article, we propose a novel variational model for restoring images in the presence of the mixture of Cauchy and Gaussian noise. The model involves a novel data-fidelity term that features the mixed noise as an infimal convolution of two noise distributions and total variation regularization. This data-fidelity term contributes to suitable separation of Cauchy noise and Gaussian noise components, facilitating simultaneous removal of the mixed noise. Besides, the total variation regularization enables adequate denoising in homogeneous regions while conserving edges. Despite the nonconvexity of the model, the existence of a solution is proven. By employing an alternating minimization approach and the alternating direction method of multipliers, we present an iterative algorithm for solving the proposed model. Experimental results validate the effectiveness of the proposed model compared to other existing models according to both visual quality and some image quality measurements.</p></abstract>