One-dimensional signal decomposition is a well-established and widely used technique across various scientific fields. It serves as a highly valuable pre-processing step for data analysis. While traditional decomposition techniques often rely on mathematical models, recent research suggests that applying the latest deep learning models to this very ill-posed inverse problem represents an exciting, unexplored area with promising potential. This work presents a novel method for the additive decomposition of one-dimensional signals. We leverage the Transformer architecture to decompose signals into their constituent components: piecewise constant, smooth (trend), highly-oscillatory, and noise components. Our model, trained on synthetic data, achieves excellent accuracy in modeling and decomposing input signals from the same distribution, as demonstrated by the experimental results. • We study additive decomposition of 1D signals with Transformers. • We define a neural architecture for the problem, based on the Transformer encoder. • Our method is more effective and orders of magnitude faster than variational ones. • The proposed method automatically detects the absence of a component.

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 nonconvex variational decomposition model which separates a given image into piecewise-constant, smooth, and oscillatory components. This decomposition is motivated not only by image denoising and structure separation, but also by shadow and spot light removal. The proposed model clearly separates the piecewise-constant structure and smoothly varying harmonic part, thanks to having a separated oscillatory component. The piecewise-constant part is captured by TV-like nonconvex regularization, harmonic term via second-order regularization, and oscillatory (noise and texture) term via a $H^{-1}$-norm penalty. There are interesting interactions between these three regularization terms. We explore the effects of each regularization and the choice of parameters carefully. We propose an efficient alternating direction method of multipliers based minimization for fast numerical computation of the optimization problem. Various experiments are presented to show the robustness against a high level of noise, applications to soft spotlight and shadow removal, and the comparisons with other methods.

A popular strategy for determining solutions to linear least-squares problems relies on using sparsity-promoting regularizers and is widely exploited in image processing applications such as, e.g., image denoising, deblurring, and inpainting. It is well known that, in general, nonconvex regularizers hold the potential for promoting sparsity more effectively than convex regularizers such as, e.g., those involving the $\ell_1$ norm. To avoid the intrinsic difficulties related to non-convex optimization, the convex nonconvex (CNC) strategy has been proposed, which allows the use of nonconvex regularization while maintaining convexity of the total objective function. In this paper, a new CNC variational model is proposed, based on a more general parametric nonconvex nonseparable regularizer. The proposed model is applicable to a greater variety of image processing problems than prior CNC methods. We derive the convexity conditions and related theoretical properties of the presented CNC model, and we analyze existence and uniqueness of its solutions. A primal-dual forward-backward splitting algorithm is proposed for solving the related saddle-point problem. The convergence of the algorithm is demonstrated theoretically and validated empirically. Several numerical experiments are presented which prove the effectiveness of the proposed approach.

In this paper we propose a Group-Sparse Representation based method with applications to Face Recognition (GSR-FR). The novel sparse representation variational model includes a non-convex sparsity-inducing penalty and a robust non-convex loss function. The penalty encourages group sparsity by using approximation of the ℓ0-quasinorm, and the loss function is chosen to make the algorithm robust to noise, occlusions and disguises. The solution of the non-trivial non-convex optimization problem is efficiently obtained by a majorization-minimization strategy combined with forward-backward splitting, which in particular reduces the solution to a sequence of easier convex optimization sub-problems. Extensive experiments on widely used face databases show the potentiality of the proposed model and demonstrate that the GSR-FR algorithm is competitive with state-of-the-art methods based on sparse representation, especially for very low dimensional feature spaces.

Tight-frame, a generalization of orthogonal wavelets, has been used successfully in various problems in image processing, including inpainting, impulse noise removal, and superresolution image restoration. Segmentation is the process of identifying object outlines within images. There are quite a few efficient algorithms for segmentation such as model-based approaches, pattern recognition techniques, tracking-based approaches, and artificial intelligence--based approaches. In this paper, we propose applying the tight-frame approach to automatically identify tube-like structures in medical imaging, with the primary application of segmenting blood vessels in magnetic resonance angiography images. Our method iteratively refines a region that encloses the potential boundary of the vessels. At each iteration, we apply the tight-frame algorithm to denoise and smooth the potential boundary and sharpen the region. The cost per iteration is proportional to the number of pixels in the image. We prove that the iteration converges in a finite number of steps to a binary image whereby the segmentation of the vessels can be done straightforwardly. Numerical experiments on synthetic and real two-dimensional (2D) and three-dimensional (3D) images demonstrate that our method is more accurate when compared with some representative segmentation methods, and it usually converges within a few iterations.

Variational and PDE-based methods have been widely used over the past two decades for edge-preserving denoising of images. However, in general, these methods fail to preserve textural and other fine scale features but typically remove them in a similar manner as noise. We propose a strategy which fully exploits the prior information available when the noise is known to be white in order to better preserve fine scale features during the denoising process. In particular, based on the assumption that noise is additive and white, we propose a novel fidelity functional to be used in a variational framework in order to enforce whiteness of the residue image. Unlike the classical total variation $L_2$ functional, whose variational analysis yields local differential terms in the resulting Euler--Lagrange equation, our whiteness term exhibits a global first variation, thus transforming the classical Euler--Lagrange PDE into an integro-differential equation which is most conveniently solved using gradient descent techniques. Numerical results show the effectiveness of the proposed strategy.

Subdivision Surfaces provide a compact way to describe a smooth surface using a mesh model. They are widely used in 3D animation and nearly all modern modeling programs support them. In this work we describe a complete parallel pipeline for real-time interactive editing, processing and rendering of smooth surface primitives on the Cell BE. Our approach makes it possible to edit and render these high-order graphics primitives passing them directly to a parallel pipeline which tessellates them just before rendering. We describe a combination of algorithmic, architectural and back-end optimizations that enable us to render smooth subdivision surfaces in real time and to dynamically deform 3D models represented by subdivision surfaces.

This paper investigates the use of cascadic multiresolution methods for image deblurring. Iterations with a conjugate gradient-type method are carried out on each level, and terminated by a stopping rule based on the discrepancy principle. Prolongation is carried out by nonlinear edge-preserving operators, which are defined via PDEs associated with Perona–Malik or total variation-type models. Computed examples demonstrate the effectiveness of the methods proposed.

Abstract This paper describes an animation interactive system that produces image sequences used to study the numerical results of scientific simulations. The fundamental goal of the system is to provide an easy method for observing and exploring scientific computation during its space‐time evolution. To do that it is necessary to combine simulation processes, algorithmic and key frame animation with geometric transformations, light effects and camera movements. ANIMA allows the researcher to use animation without special programming, with the aid of a graphic user interface. Modelling, rendering, motion control and synchronization of graphic objects are interactively specified by the animator in order to define an animation script. The system development methodology follows an object‐oriented technique using modular programming, abstract data type definition, and graphic metaphor with user interface menus.