In many linear regression problems, including ill-posed inverse problems in image restoration, the data exhibit some sparse structures that can be used to regularize the inversion. To this end, a classical path is to use \(\ell _{12}\) block-based regularization. While efficient at retrieving the inherent sparsity patterns of the data—the support—the estimated solutions are known to suffer from a systematical

The curvature regularities are well-known for providing strong priors in the continuity of edges, which have been applied to a wide range of applications in image processing and computer vision. However, these models are usually non-convex, non-smooth, and highly nonlinear, the first-order optimality condition of which are high-order partial differential equations. Thus, numerical computation is extremely

We propose an approach to the problem of global reconstruction of an orientation field. The method is based on a geometric model called bisector line fields, which maps a pair of vector fields to an orientation field, effectively generalizing the notion of doubling phase vector fields. Endowed with a well-chosen energy minimization problem, we provide a polynomial interpolation of a target orientation

Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is used to study microvascular structure and tissue perfusion. In DCE-MRI, a bolus of gadolinium-based contrast agent is injected into the blood stream and spatiotemporal changes induced by the contrast agent flow are estimated from a time series of MRI data. Sufficient time resolution can often only be obtained by using an imaging protocol

Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their

Geometric priors have been shown to be useful in image segmentation to regularize the results. For example, the classical Mumford–Shah functional uses region perimeter as prior. This has inspired much research in the last few decades, with classical approaches like the Rudin–Osher–Fatemi and most graph-cut formulations, which all use a weighted or binary perimeter prior. It has been observed that this

Among the different flavors of well-composednesses on cubical grids, two of them, called, respectively, digital well-composedness (DWCness) and well-composedness in the sense of Alexandrov (AWCness), are known to be equivalent in 2D and in 3D. The former means that a cubical set does not contain critical configurations, while the latter means that the boundary of a cubical set is made of a disjoint

In discrete topology, we like digitally well-composed (shortly DWC) interpolations because they remove pinches in cubical images. Usual well-composed interpolations are local and sometimes self-dual (they treat in a same way dark and bright components in the image). In our case, we are particularly interested in n-D self-dual DWC interpolations to obtain a purely self-dual tree of shapes. However,

In a wide range of applications such as astronomy, biology, and medical imaging, acquired data are usually corrupted by Poisson noise and blurring artifacts. Poisson noise often occurs when photon counting is involved in such imaging modalities as X-ray, positron emission tomography, and fluorescence microscopy. Meanwhile, blurring is also inevitable due to the physical mechanism of an imaging system

Multiplicative noise models are often used instead of additive noise models in cases in which the noise variance depends on the state. Furthermore, when Poisson distributions with relatively small counts are approximated with normal distributions, multiplicative noise approximations are straightforward to implement. There are a number of limitations in the existing approaches to deal with multiplicative

Local M-smoothers are interesting and important signal and image processing techniques with many connections to other methods. In our paper, we derive a family of partial differential equations (PDEs) that result in one, two, and three dimensions as limiting processes from M-smoothers which are based on local order-p means within a ball the radius of which tends to zero. The order p may take any nonzero

We study the inverse problem of model parameter learning for pixelwise image labeling, using the linear assignment flow and training data with ground truth. This is accomplished by a Riemannian gradient flow on the manifold of parameters that determines the regularization properties of the assignment flow. Using the symplectic partitioned Runge–Kutta method for numerical integration, it is shown that

This paper combines image metamorphosis with deep features. To this end, images are considered as maps into a high-dimensional feature space and a structure-sensitive, anisotropic flow regularization is incorporated in the metamorphosis model proposed by Miller and Younes (Int J Comput Vis 41(1):61–84, 2001) and Trouvé and Younes (Found Comput Math 5(2):173–198, 2005). For this model, a variational

It is well known that the P3P problem could have 1, 2, 3 and at most 4 positive solutions under different configurations among its three control points and the position of the optical center. Since in any real applications, the knowledge on the exact number of possible solutions is a prerequisite for selecting the right one among all the possible solutions, and the study on the phenomenon of multiple

Lossy image compression methods based on partial differential equations have received much attention in recent years. They may yield high-quality results but rely on the computationally expensive task of finding an optimal selection of data. For the possible extension to video compression, this data selection is a crucial issue. In this context, one could either analyse the video sequence as a whole

Many mathematical imaging problems are posed as non-convex optimization problems. When numerically tractable global optimization procedures are not available, one is often interested in testing ex post facto whether or not a locally convergent algorithm has found the globally optimal solution. When the problem is formulated in terms of maximizing the likelihood function under a statistical model for

Spectral clustering is one of the most important image processing tools, especially for image segmentation. This specializes at taking local information such as edge weights and globalizing them. Due to its unsupervised nature, it is widely applicable. However, traditional spectral clustering is \({\mathcal {O}}(n^{3/2})\). This poses a challenge, especially given the recent trend of large datasets

Reproducing the perception of a real-world scene on a display device is a very challenging task which requires the understanding of the camera processing pipeline, the display process, and the way the human visual system processes the light it captures. Mathematical models based on psychophysical and physiological laws on color vision, named Retinex, provide efficient tools to handle degradations produced

The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a backward diffusion model which implements a smart stabilisation approach that can be used in combination with an easy-to-handle numerical scheme. So far, existing stabilisation

We propose a new model and a corresponding iterative algorithm for Computed Tomography (CT) when the view angles are uncertain. The uncertainty is described by an additive model discrepancy term which is included in the data fidelity term of a total variation regularized variational model. We approximate the model discrepancy with a Gaussian distribution. Our iterative algorithm alternates between

This paper investigates a new stochastic algorithm to approximate semi-discrete optimal transport for large-scale problem, i.e., in high dimension and for a large number of points. The proposed technique relies on a hierarchical decomposition of the target discrete distribution and the transport map itself. A stochastic optimization algorithm is derived to estimate the parameters of the corresponding

This paper is devoted to the challenging problem of fine structure detection with applications to bituminous surfacing crack recovery. Drogoul (SIAM J Imag Sci 7(4):2700–2731, 2014) shows that such structures can be suitably modeled by a sequence of smooth functions whose Hessian matrices blow up in the perpendicular direction to the crack, while their gradient is null. This observation serves as the

Distance transforms (DTs) are standard tools in image analysis, with applications in image registration and segmentation. The DT is based on extremal (minimal) distance values and is therefore highly sensitive to noise. We present a stochastic distance transform (SDT) based on discrete random sets, in which a model of element-wise probability is utilized and the SDT is computed as the first moment

Many problems in applied computer science can be expressed in a graph setting and solved by finding an appropriate vertex labeling of the associated graph. It is also common to identify the term “appropriate labeling” with a labeling that optimizes some application-motivated objective function. The goal of this work is to present two algorithms that, for the objective functions in a general format

We consider the evolution model proposed in Bertalmío (Front Comput Neurosci 8:71, 2014), Bertalmío et al. (IEEE Trans Image Process 16(4):1058–1072, 2007) to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used Wilson–Cowan equations (Wilson and Cowan in BioPhys J 12(1):1–24, 1972), mainly in terms

A plane-probing algorithm computes the normal vector of a digital plane from a starting point and a predicate “Is a point \({x}\) in the digital plane?”. This predicate is used to probe the digital plane as locally as possible and decide on the fly the next points to consider. However, several existing plane-probing algorithms return the correct normal vector only for some specific starting points

In this paper, we define a new shape-based measure which evaluates how much a given shape is hexagonal. Such an introduced measure ranges through the interval (0, 1] and reaches the maximal possible value 1 if and only if the shape considered is a hexagon. The new measure is also invariant with respect to rotation, translation and scaling transformations. A number of experiments, performed on both

This paper addresses the problem of finding the closest generalized essential matrix from a given \(6\times 6\) matrix, with respect to the Frobenius norm. To the best of our knowledge, this nonlinear constrained optimization problem has not been addressed in the literature yet. Although it can be solved directly, it involves a large number of constraints, and any optimization method to solve it would

An important task when processing sensor data is to distinguish relevant from irrelevant data. This paper describes a method for an iterative singular value decomposition that maintains a model of the background via singular vectors spanning a subspace of the image space, thus providing a way to determine the amount of new information contained in an incoming frame. We update the singular vectors spanning

In this paper, we study three basic novel measures of convexity for shape analysis. The convexity considered here is the so-called Q-convexity, that is, convexity by quadrants. The measures are based on the geometrical properties of Q-convex shapes and have the following features: (1) their values range from 0 to 1; (2) their values equal 1 if and only if the binary image is Q-convex; and (3) they

Persistent homology is a popular data analysis technique that is used to capture the changing homology of an indexed sequence of simplicial complexes. These changes are summarized in persistence diagrams. A natural problem is to contract edges in complexes in the initial sequence to obtain a sequence of simplified complexes while controlling the perturbation between the original and simplified persistence

In this article, we introduce a family of elastic metrics on the space of parametrized surfaces in 3D space using a corresponding family of metrics on the space of vector-valued one-forms. We provide a numerical framework for the computation of geodesics with respect to these metrics. The family of metrics is invariant under rigid motions and reparametrizations; hence, it induces a metric on the “shape

A set \(S \subset \mathbb {Z}^2\) of integer points is digital convex if \({{\,\mathrm{conv}\,}}(S) \cap \mathbb {Z}^2 = S\), where \({{\,\mathrm{conv}\,}}(S)\) denotes the convex hull of S. In this paper, we consider the following two problems. The first one is to test whether a given set S of n lattice points is digital convex. If the answer to the first problem is positive, then the second problem

We propose two minimal solvers to the problem of relative pose estimation for a camera with known relative rotation angle. In practice, such angle can be derived from the readings of a 3D gyroscope. Different from other relative pose formulations fusing a camera and a gyroscope, the use of relative rotation angle does not require extrinsic calibration between the two sensors. The first proposed solver

This article focuses on the classical problem of the control of information loss during the digitization step. The properties proposed in the literature rely on smoothness hypotheses that are not satisfied by the curves including angular points. The notion of turn introduced by Milnor in the article On the Total Curvature of Knots generalizes the notion of integral curvature to continuous curves. Thanks

We characterize proximity operators, that is to say functions that map a vector to a solution of a penalized least-squares optimization problem. Proximity operators of convex penalties have been widely studied and fully characterized by Moreau. They are also widely used in practice with nonconvex penalties such as the \(\ell ^0\) pseudo-norm, yet the extension of Moreau’s characterization to this setting

The goal of this work is to describe an efficient algorithm for finding a binary segmentation of an image such that the indicated object satisfies a novel high-level prior, called local band, LB, constraint; the returned segmentation is optimal, with respect to an appropriate graph-cut measure, among all segmentations satisfying the given LB constraint. The new algorithm has two stages: expanding the

The aim of this paper is to clear up the question of the connection between the geometric moment invariants and the invariant theory, considering a problem of describing the 2D geometric moment invariants as a problem of the classical invariant theory. We give a precise statement of the problem of computation of the 2D geometric invariant moments, introducing the notions of the algebras of simultaneous

We investigate a well-known phenomenon of variational approaches in image processing, where typically the best image quality is achieved when the gradient flow process is stopped before converging to a stationary point. This paradox originates from a tradeoff between optimization and modeling errors of the underlying variational model and holds true even if deep learning methods are used to learn highly

Watershed is a well-established clustering and segmentation method. In this article, we aim to achieve a better theoretical understanding of the hierarchical version of the watershed operator. More precisely, we propose a characterization of hierarchical watersheds in the framework of edge-weighted graphs. The proposed characterization leads to an efficient algorithm to recognize hierarchical watersheds

We propose a novel two-stage method for the classification of hyperspectral images. Pixel-wise classifiers, such as the classical support vector machine (SVM), consider spectral information only. As spatial information is not utilized, the classification results are not optimal and the classified image may appear noisy. Many existing methods, such as morphological profiles, superpixel segmentation

In single image blind deblurring, the blur kernel and latent image are estimated from a single observed blurry image. The associated mathematical problem is ill-posed, and an acceptable solution is difficult to obtain without additional priors or heuristics. Inspired by the nonlocal self-similarity in image denoising problem, we introduce elastic-net regularization as a rank prior to improve the estimation

We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall’s shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical

It is known that the rotation, scaling and translation invariant property of image moments has a high significance in image recognition. For this reason, the seven invariant moments presented by Hu are widely used in the field of image analysis. These moments are of finite order; therefore, they do not comprise a complete set of image descriptors. For this reason, we introduce in this paper another

The use of a single uncalibrated camera is desirable for eye tracking to reduce the overall complexity and cost of the system. Quite often, at least one external light source is used to enhance image quality and generate a corneal reflection used as a reference point to estimate the point-of-gaze (PoG). Though the use of more than one light source has shown to enhance accuracy and robustness to head

High-order data are modeled using matrices whose entries are numerical arrays of a fixed size. These arrays, called t-scalars, form a commutative ring under the convolution product. Matrices with elements in the ring of t-scalars are referred to as t-matrices. The t-matrices can be scaled, added and multiplied in the usual way. There are t-matrix generalizations of positive matrices, orthogonal matrices

In this paper we propose optimisation methods for variational regularisation problems based on discretising the inverse scale space flow with discrete gradient methods. Inverse scale space flow generalises gradient flows by incorporating a generalised Bregman distance as the underlying metric. Its discrete-time counterparts, Bregman iterations and linearised Bregman iterations are popular regularisation

We propose an adaptive implementation of a Crouzeix–Raviart-based discretization of the total variation, which has the property of approximating from below the total variation, with metrication errors only depending on the local curvature, rather than on the orientation as is usual for other approaches.

In view of the exceptional ability of curvature in connecting missing edges and structures, we propose novel sparse reconstruction models via the Euler’s elastica energy. In particular, we firstly extend the Euler’s elastica regularity into the nonlocal formulation to fully take the advantages of the pattern redundancy and structural similarity in image data. Due to its non-convexity, non-smoothness

This work investigates the role of the filters implemented on analog-to-digital converters for the reconstruction of magnetic resonance images. We analyze the effects of these filters both from a theoretical and an experimental point of view and demonstrate how it may lead to severe degradation of the reconstructed images when the distance between consecutive samples is larger than Shannon’s limit

Total variation (TV) signal denoising is a popular nonlinear filtering method to estimate piecewise constant signals corrupted by additive white Gaussian noise. Following a ‘convex non-convex’ strategy, recent papers have introduced non-convex regularizers for signal denoising that preserve the convexity of the cost function to be minimized. In this paper, we propose a non-convex TV regularizer, defined

The problem of jointly segmenting objects, according to a set of labels (of cardinality L), from a set of images (of cardinality K) to produce K individual segmentations plus one joint segmentation, can be cast as a Markov random field model. Coupling terms in the considered energy function enforce the consistency between the individual segmentations and the joint segmentation. However, neither optimality

A structure-adaptive vector filter for removal of impulse noise from color images is presented. The proposed method is based on local orientation estimation. A color image is represented in quaternion form, and then, quaternion Fourier transform is used to compute the orientation of the pattern in a local neighborhood. Since the computation in quaternion frequency domain is extremely time-consuming

This work addresses two main contributions for shape measurement: First, a new circularity measure for planar shapes is introduced based on their geometrical properties in the projection space of Radon transform. Second, a general-purpose evaluation criterion, power of discrimination, for assessing the efficiency of a shape measure is proposed. The new measure ranges over the interval [0, 1] and produces

Structure-preserving image filtering is an image smoothing technique that aims to preserve prominent structures while removing unwanted details in natural images. However, relevant studies mainly focus on small variances/fluctuations suppression and are vulnerable to separate pixels connected by some low-contrast edges or cluster pixels which exhibit strong differences between neighbors in highly textured

Magnetorelaxometry imaging is a highly sensitive technique enabling noninvasive, quantitative detection of magnetic nanoparticles. Electromagnetic coils are sequentially energized, aligning the nanoparticles’ magnetic moments. Relaxation signals are recorded after turning off the coils. The forward model describing this measurement process is reformulated into a severely ill-posed inverse problem that

This paper introduces the unsupervised assignment flow that couples the assignment flow for supervised image labeling (Åström et al. in J Math Imaging Vis 58(2):211–238, 2017) with Riemannian gradient flows for label evolution on feature manifolds. The latter component of the approach encompasses extensions of state-of-the-art clustering approaches to manifold-valued data. Coupling label evolution

The main aim of this paper is to study tensor factorization for low-rank tensor completion in imaging data. Due to the underlying redundancy of real-world imaging data, the low-tubal-rank tensor factorization (the tensor–tensor product of two factor tensors) can be used to approximate such tensor very well. Motivated by the spatial/temporal smoothness of factor tensors in real-world imaging data, we

In this paper, we focus on learning optimized partial differential equation (PDE) models for image filtering. In our approach, the gray-scale images are represented by a vector field of two real-valued functions and the image restoration problem is modeled by an evolutionary process such that the restored image at any time satisfies an initial-boundary value problem of cross-diffusion with reaction

The article was originally published by the journal with an incorrect title.