Tubular and membranous shapes display a wide range of morphologies that are difficult to analyze within a common framework. By generalizing the classical Helfrich energy of biomembranes, we model them as solutions to a curvature optimization problem in which the principal curvatures may play asymmetric roles. We then give a novel phase-field formulation to approximate this geometric problem, and study

Shape analysis of objects in images is a critical area of research, and several approaches, including those that utilize elastic Riemannian metrics, have been proposed. While elastic techniques for shape analysis of curves are pretty advanced, the corresponding results for higher-dimensional objects (surfaces and disks) are less developed. This paper studies shapes of solid planar objects that are

We present an information-theoretic approach to the registration of images with directional information, especially for diffusion-weighted images (DWIs), with explicit optimization over the directional scale. We call it locally orderless registration with directions (LORDs). We focus on normalized mutual information as a robust information-theoretic similarity measure for DWI. The framework is an extension

We present a new supervised image classification method applicable to a broad class of image deformation models. The method makes use of the previously described Radon Cumulative Distribution Transform (R-CDT) for image data, whose mathematical properties are exploited to express the image data in a form that is more suitable for machine learning. While certain operations such as translation, scaling

Recently in this journal, Wang et al. (J Math Imaging Vis 62(5): 1214–1226, 2020) reported an interesting multi-solution phenomenon in P3P (perspective-3-point) problem: A pair of point-sharing solutions appears always in companionship with a pair of side-sharing solutions, and they also gave the necessary and sufficient condition for the existence of such solution pairs. Although the conclusions are

Different from the traditional watermarking schemes, zero-watermarking schemes are lossless embedding methods, which are applicable to be used in medical, military, remote sensing and other fields requiring high-integrity image copyright protection. However, most of the existing zero-watermarking schemes only provide copyright protection for one image at a time, which has certain limitations. This

The surface area of a set which is only observed as a binary pixel image is often estimated by a weighted sum of pixel configurations counts. In this paper we examine these estimators in a design based setting—we assume that the observed set is shifted uniformly randomly. Bounds for the difference between the essential supremum and the essential infimum of such an estimator are derived, which imply

To tackle the problem of modeling and shape designing of complex engineering surfaces, the continuity constraints between generalized hybrid trigonometric Bézier (GHT-Bézier for short) surfaces with three different shape parameters are proposed in this study. Initially, we describe the basic properties of GHT-Bézier surface and influence of shape parameters. Some special triangular surfaces and biangular

The quantification of uncertainties in image segmentation based on the Mumford–Shah model is studied. The aim is to address the error propagation of noise and other error types in the original image to the restoration result and especially the reconstructed edges (sharp image contrasts). Analytically, we rely on the Ambrosio–Tortorelli approximation and discuss the existence of measurable selections

In discrete tomography, ghosts represent indeterminate locations of a reconstruction when there is insufficient projection information to admit a unique solution. Our previous work presented maximal ghosts, which are tilings of \(2^N\) connected points of \(\pm 1\) values with zero line sums over N directions. These directions are given by the recursion \(v_{n+1}=v_n + 2\epsilon _{n+1}v_{n-1}\) with

Geometrical measurements of biological objects form the basis of many quantitative analyses. Hausdorff measures such as the volume and the area of objects are simple and popular descriptors of individual objects; however, for most biological processes, the interaction between objects cannot be ignored, and the shape and function of neighboring objects are mutually influential. In this paper, we present

Deformable image registration aims at estimating a proper displacement field from a fixed image and a moving one. Variational deformable registration models often consist of a data term of the images and a regularization term of the estimated displacement field. In this paper, we propose a variational model for registering uni-modal medical images with intensity biases. Precisely, the proposed model

This work contributes an efficient algorithm to compute the relative pose problem (RPp) between calibrated cameras and certify the optimality of the solution, given a set of pair-wise feature correspondences affected by noise and probably corrupted by wrong matches. We propose a family of certifiers that is shown to increase the ratio of detected optimal solutions. This set of certifiers is incorporated

This paper first proposes a statistic for measuring the correlation between two random variables. Because the data are usually polluted by noise and the feature of the data is usually task-relevant, this paper proposes to perform a transformation on the data before measuring their correlation. In addition, since the kernel method is a commonly used data transformation method in the field of machine

This paper addresses a general problem of computing inversion-free maps between continuous and discrete domains that induce minimal geometric distortions. We will refer to this problem as optimal mapping problem. Finding a good solution to the optimal mapping problem is a key part in many applications in geometry processing and computer vision, including: parameterization of surfaces and volumetric

The distribution of the solutions to the Perspective 3-Point Problem (P3P) has been studied for a few decades, and some understanding of this issue has emerged. However, the present article is the first to comprehensively describe, for a given location p in space, the number of other points that solve the same P3P setup that p solves, and where to find these related points. A dynamic approach is employed

Penalized least squares are widely used in signal and image processing. Yet, it suffers from a major limitation since it requires fine-tuning of the regularization parameters. Under assumptions on the noise probability distribution, Stein-based approaches provide unbiased estimator of the quadratic risk. The Generalized Stein Unbiased Risk Estimator is revisited to handle correlated Gaussian noise

The paper shows that the moment invariants proposed recently in this journal by Hjouji et al. (J Math Imaging Vis 62:606–624, 2020) are incomplete, which leads to a limited discriminability. We prove this by means of circular projection of the image. In a broader context, we demonstrate that completeness of the invariants leads to a better recognition power.

Pairwise rigid registration can be developed by comparing local geometry encoded by intrinsic second-order orientation tensors which allows to model the registration problem using the associated Riemannian geometry or related group structures. Everything starts by representing those tensor fields as multivariate normal models that permit us to manipulate Gaussians in two ways: using the Riemannian

Studying the changes of shape is a common concern in many scientific fields. We address here two problems: (1) quantifying the deformation between two given shapes and (2) transporting this deformation to morph a third shape. These operations can be done with or without point correspondence, depending on the availability of a surface matching algorithm, and on the type of mathematical procedure adopted

In this paper, we address the problem of denoising images obtained under low-light conditions for the Poisson shot noise model. Under such conditions, the variance stabilization transform (VST) is no longer applicable, so that the state-of-the-art algorithms which are proficient for the additive white Gaussian noise cannot be applied. We first introduce an oracular non-local algorithm and prove its

Variational and Bayesian methods are two widely used set of approaches to solve image denoising problems. In a Bayesian setting, these approaches correspond, respectively, to using maximum a posteriori estimators and posterior mean estimators for reconstructing images. In this paper, we propose novel theoretical connections between Hamilton–Jacobi partial differential equations (HJ PDEs) and a broad

Remote photoplethysmography (rPPG) has been at the forefront recently, thanks to its capacity in estimating non-contact physiological parameters such as heart rate and heart rate variability (Wang et al. in FBB 6:33, 2018). rPPG signals are typically extracted from facial videos by performing spatial averaging to obtain temporal RGB traces. Although this spatial averaging simplifies computation, it

Image registration under small displacements is the keystone of several image analysis tasks such as optical flow estimation, stereoscopic imaging, or full-field displacement estimation in photomechanics. A popular approach consists in locally modeling the displacement field between two images by a parametric transformation and performing least-squares estimation afterward. This procedure is known

We propose a bilevel optimization approach for the estimation of parameters in nonlocal image denoising models. The parameters we consider are both the fidelity weight and weights within the kernel of the nonlocal operator. In both cases, we investigate the differentiability of the solution operator in function spaces and derive a first-order optimality system that characterizes local minima. For the

A correction to this paper has been published: https://doi.org/10.1007/s10851-021-01028-0

Complex analyses involving multiple, dependent random quantities often lead to graphical models—a set of nodes denoting variables of interest, and corresponding edges denoting statistical interactions between nodes. To develop statistical analyses for graphical data, especially towards generative modeling, one needs mathematical representations and metrics for matching and comparing graphs, and subsequent

In recent years, digital image zero-watermarking algorithm has made great progress. But there are still some problems. First, most algorithms only focus on robustness and ignore discrimination. Second, most methods have good robustness against conventional signal attacks but poor robustness against geometric attacks. Thirdly, most of the existing zero-watermarking algorithms only focus on gray-scale

The present work proposes a solution to the challenging problem of registering two partial point sets of the same object with very limited overlap. We leverage the fact that most objects found in man-made environments contain a plane of symmetry. By reflecting the points of each set with respect to the plane of symmetry, we can largely increase the overlap between the sets and therefore boost the registration

We propose a mathematical description of human color perception that relies on a hyperbolic structure of the space \({\mathcal {P}}\) of perceived colors. We show that hyperbolicity allows us to reconcile both trichromaticity, from a Riemannian point of view, and color opponency, from a quantum viewpoint. In particular, we will underline how the opponent behavior can be represented by a rebit, a real

This paper develops a new active contour (AC) model capable of multiple complex objects segmentation in the presence of heavy noise. The model segments images in the framework of two types of partial differential equations (PDEs): the Euler–Lagrange and Poisson PDEs. The former is used to build an evolution algorithm, while the Poisson solution gradient vector field (PGVF) directs the evolution toward

Principal component analysis (PCA) is known to be sensitive to outliers, so that various robust PCA variants were proposed in the literature. A recent model, called reaper, aims to find the principal components by solving a convex optimization problem. Usually the number of principal components must be determined in advance and the minimization is performed over symmetric positive semi-definite matrices

In positron emission tomography, movement leads to blurry reconstructions when not accounted for. Whether known a priori or estimated jointly to reconstruction, motion models are increasingly defined in continuum rather that in discrete, for example by means of diffeomorphisms. The present work provides both a statistical and functional analytic framework suitable for handling such models. It is based

While the equality of differential signatures (Calabi et al., Int. J. Comput. Vis. 26: 107–135, 1998) is known to be a necessary condition for congruence, it is not sufficient (Musso and Nicolodi, J. Math Imaging Vis. 35: 68–85, 2009). Hickman (J. Math Imaging Vis. 43: 206–213, 2012, Theorem 2) claimed that for non-degenerate planar curves, equality of Euclidean signatures implies congruence. We prove

Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by-now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing, this strategy leads to a nested optimization problem (known as bilevel optimization)

Locality preserving projections (LPP) is a popular unsupervised dimensionality reduction method based on manifold learning. As a supervised version of the LPP method, discriminant locality preserving projections (DLPP) method has been recently proposed and paid much attention to by researchers. However, the DLPP method has the small-sample-size (SSS) problem. In this paper, in the view of the eigenvalues

The family of PDE-constrained Large Deformation Diffeomorphic Metric Mapping (LDDMM) methods is emerging as a particularly interesting approach for physically meaningful diffeomorphic transformations. The original combination of Gauss–Newton–Krylov optimization and Runge–Kutta integration shows excellent numerical accuracy and fast convergence rate. However, its most significant limitation is the huge

Processes such as growth and atrophy cause changes through time that can be visible in a series of medical images, following the hypothesis that form follows function. As was hypothesised by D’Arcy Thompson more than 100 years ago, models of the changes inherent in these actions can aid understanding of the processes at work. We consider how image registration using finite-dimensional planar Lie groups

Periodic noise degrades the image quality by overlaying similar patterns. This noise appears as peaks in the image spectrum. In this research, a method based on fuzzy transform has been developed to identify and reduce the peaks adaptively. We convert the periodic noise removal task as image compression and a smoothing problem. We first utilize the direct and inverse fuzzy transform of the spectrum

The inverse problem in acousto-electric tomography concerns the reconstruction of the electric conductivity in a body from knowledge of the power density function in the interior of the body. This interior power density results from currents prescribed at boundary electrodes, and it can be obtained through electro-static boundary measurements together with auxiliary acoustic probing. Previous works

We focus on the minimization of the least square loss function under a k-sparse constraint encoded by a \(\ell _0\) pseudo-norm. This is a non-convex, non-continuous and NP-hard problem. Recently, for the penalized form (sum of the least square loss function and a \(\ell _0\) penalty term), a relaxation has been introduced which has strong results in terms of minimizers. This relaxation is continuous

A novel formulation called Virtual Interest Point is presented and used to register point clouds. An implicit quadric surface representation is first used to model the point cloud segments. Macaulay’s resultant then provides the intersection of three such quadrics, which forms a virtual interest point (VIP). A unique feature descriptor for each VIP is computed, and correspondences in descriptor space

In this work, we deal with the region control of \(C^2\) interpolation curves and surfaces using a class of rational interpolation splines in one and two dimensions. Simple sufficient data-dependent constraints are derived on the local control parameters to generate \(C^2\) interpolation curves lying strictly between two given piecewise linear curves and \(C^2\) interpolation surfaces lying strictly

Online optimisation revolves around new data being introduced into a problem while it is still being solved; think of deep learning as more training samples become available. We adapt the idea to dynamic inverse problems such as video processing with optical flow. We introduce a corresponding predictive online primal-dual proximal splitting method. The video frames now exactly correspond to the algorithm

Storage systems often rely on multiple copies of the same compressed data, enabling recovery in case of binary data errors, of course, at the expense of a higher storage cost. In this paper, we show that a wiser method of duplication entails great potential benefits for data types tolerating approximate representations, like images and videos. We propose a method to produce a set of distinct compressed

Discrete tomography reconstructs an image of an object on a grid from its discrete projections along relatively few directions. When the resulting system of linear equations is under-determined, the reconstructed image is not unique. Ghosts are arrays of signed pixels that have zero sum projections along these directions; they define the image pixel locations that have non-unique solutions. In general

Gray balance is one of the key issues of color balance and color appearance adjustment during printing. It takes into account the overlap of two or more colorants to achieve a visually neutral tone. It works in parallel with the adjustment of the main color channels. In electrophotography, the setting of color channels separately is usually carried out using tone reproduction curves. However, this

Image segmentation is an important median level vision topic. Accurate and efficient multiphase segmentation for images with intensity inhomogeneity is still a great challenge. We present a new two-stage multiphase segmentation method trying to tackle this, where the key is to compute an inhomogeneity-free approximate image. For this, we propose to use a new non-Lipschitz variational decomposition

Up to an orientation-preserving symmetry, photographic images are produced by a central projection of a restricted area in the space into the image plane. To obtain reliable information about physical objects and the environment through the process of recording is the basic problem of photogrammetry. We present a reconstruction process based on distances from the center of projection and incidence

Particle estimation is a classical problem arising in many science fields, such as biophysics, fluid mechanics and biomedical imaging. Many interesting applications in these areas involve 3D imaging data: This work presents a technique to estimate the 3D coordinates of the center of spherical particles. This procedure has its core in the processing of the images of the scanned volume: It firstly applies

We consider the Blum medial axis of a region in \(\mathbb R^n\) with piecewise smooth boundary and examine its “rigidity properties,”by which we mean properties preserved under diffeomorphisms of the regions preserving the medial axis. There are several possible versions of rigidity depending on what features of the Blum medial axis we wish to retain. We use a form of the cross ratio from projective

We have solved an old problem posed by Santaló of determining the size distribution of particles derived from the size distribution of their sections. We give an explicit form of particle-size distributions of convex similar bodies for random planes and random lines, which naturally generalize famous Wicksell’s corpuscle problem. The results are achieved by applying the method of model solutions for

Non-local patch-based methods were until recently the state of the art for image denoising but are now outperformed by CNNs. In video denoising, however, they are still competitive with CNNs, as they can effectively exploit the video temporal redundancy, which is a key factor to attain high denoising performance. The problem is that CNN architectures are not compatible with the search for self-similarities

Optimal transport (OT) distances between probability distributions are parameterized by the ground metric they use between observations. Their relevance for real-life applications strongly hinges on whether that ground metric parameter is suitably chosen. The challenge of selecting it adaptively and algorithmically from prior knowledge, the so-called ground metric learning (GML) problem, has therefore

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