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Showing papers in "Journal of Mathematical Imaging and Vision in 2011"


Journal ArticleDOI
TL;DR: A first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure can achieve O(1/N2) convergence on problems, where the primal or the dual objective is uniformly convex, and it can show linear convergence, i.e. O(ωN) for some ω∈(0,1), on smooth problems.
Abstract: In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. In particular we show that we can achieve O(1/N 2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(? N ) for some ??(0,1), on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation.

4,487 citations


Journal ArticleDOI
TL;DR: Multi-scale total variation models for image restoration introduce a spatially dependent regularization parameter in order to enhance image regions containing details while still sufficiently smoothing homogeneous features and compares with popular total variation based restoration methods.
Abstract: Multi-scale total variation models for image restoration are introduced. The models utilize a spatially dependent regularization parameter in order to enhance image regions containing details while still sufficiently smoothing homogeneous features. The fully automated adjustment strategy of the regularization parameter is based on local variance estimators. For robustness reasons, the decision on the acceptance or rejection of a local parameter value relies on a confidence interval technique based on the expected maximal local variance estimate. In order to improve the performance of the initial algorithm a generalized hierarchical decomposition of the restored image is used. The corresponding subproblems are solved by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques. The paper ends by a report on numerical tests, a qualitative study of the proposed adjustment scheme and a comparison with popular total variation based restoration methods.

142 citations


Journal ArticleDOI
TL;DR: The theories presented show that it is possible to describe a general set of Gaussian and/or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area.
Abstract: This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms). The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives. The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and/or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area.

120 citations


Journal ArticleDOI
TL;DR: A Riemannian framework for smoothing data that are constrained to live in $\mathcal{P}(n)$, the space of symmetric positive-definite matrices of order n, which is equivalent to the heat flow or isotropic linear diffusion which smooths data everywhere.
Abstract: In this paper we present a Riemannian framework for smoothing data that are constrained to live in $\mathcal{P}(n)$ , the space of symmetric positive-definite matrices of order n. We start by giving the differential geometry of $\mathcal{P}(n)$ , with a special emphasis on $\mathcal{P}(3)$ , considered at a level of detail far greater than heretofore. We then use the harmonic map and minimal immersion theories to construct three flows that drive a noisy field of symmetric positive-definite data into a smooth one. The harmonic map flow is equivalent to the heat flow or isotropic linear diffusion which smooths data everywhere. A modification of the harmonic flow leads to a Perona-Malik like flow which is a selective smoother that preserves edges. The minimal immersion flow gives rise to a nonlinear system of coupled diffusion equations with anisotropic diffusivity. Some preliminary numerical results are presented for synthetic DT-MRI data.

106 citations


Journal ArticleDOI
TL;DR: This paper shows that an image can be decomposed into a sum of a “periodic component” and a ‘smooth component’, which brings a simple and computationally efficient answer to this problem.
Abstract: When the Discrete Fourier Transform of an image is computed, the image is implicitly assumed to be periodic. Since there is no reason for opposite borders to be alike, the "periodic" image generally presents strong discontinuities across the frame border. These edge effects cause several artifacts in the Fourier Transform, in particular a well-known "cross" structure made of high energy coefficients along the axes, which can have strong consequences on image processing or image analysis techniques based on the image spectrum (including interpolation, texture analysis, image quality assessment, etc.). In this paper, we show that an image can be decomposed into a sum of a "periodic component" and a "smooth component", which brings a simple and computationally efficient answer to this problem. We discuss the interest of such a decomposition on several applications.

100 citations


Journal ArticleDOI
TL;DR: It is proved the existence and uniqueness of the minimizer for the variational problem, and it is shown that the solution of the evolution equation converges weakly in BV and strongly in L2 to the minimizers as t→∞.
Abstract: In this paper we study a variational model to deal with the speckle noise in ultrasound images. We prove the existence and uniqueness of the minimizer for the variational problem, and derive the existence and uniqueness of weak solutions for the associated evolution equation. Furthermore, we show that the solution of the evolution equation converges weakly in BV and strongly in L 2 to the minimizer as t??. Finally, some numerical results illustrate the effectiveness of the proposed model for multiplicative noise removal.

96 citations



Journal ArticleDOI
TL;DR: The proposed algorithm successfully detects people occluding each other given severely degraded extracted features, while outperforming state-of-the-art people localization techniques.
Abstract: This paper addresses the problem of localizing people in low and high density crowds with a network of heterogeneous cameras. The problem is recast as a linear inverse problem. It relies on deducing the discretized occupancy vector of people on the ground, from the noisy binary silhouettes observed as foreground pixels in each camera. This inverse problem is regularized by imposing a sparse occupancy vector, i.e., made of few non-zero elements, while a particular dictionary of silhouettes linearly maps these non-empty grid locations to the multiple silhouettes viewed by the cameras network. The proposed framework is (i) generic to any scene of people, i.e., people are located in low and high density crowds, (ii) scalable to any number of cameras and already working with a single camera, (iii) unconstrained by the scene surface to be monitored, and (iv) versatile with respect to the camera's geometry, e.g., planar or omnidirectional. Qualitative and quantitative results are presented on the APIDIS and the PETS 2009 Benchmark datasets. The proposed algorithm successfully detects people occluding each other given severely degraded extracted features, while outperforming state-of-the-art people localization techniques.

79 citations


Journal ArticleDOI
TL;DR: A deblurring algorithm that explicitly takes into account the sparse characteristics of natural images and does not entail solving a numerically ill-conditioned backward-diffusion is proposed.
Abstract: We propose a deblurring algorithm that explicitly takes into account the sparse characteristics of natural images and does not entail solving a numerically ill-conditioned backward-diffusion. The key observation is that the sparse coefficients that encode a given image with respect to an over-complete basis are the same that encode a blurred version of the image with respect to a modified basis. Following an "analysis-by-synthesis" approach, an explicit generative model is used to compute a sparse representation of the blurred image, and its coefficients are used to combine elements of the original basis to yield a restored image.

74 citations


Journal ArticleDOI
TL;DR: This paper shows how to use the proposed framework in practice on the example of constrained connectivity, which allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.
Abstract: We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice on the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.

73 citations


Journal ArticleDOI
TL;DR: The elements necessary to build a specific algebraic solver are given in this paper, allowing for a real-time implementation and the results on real and synthetic data show the efficiency of this method.
Abstract: This paper presents a new method to solve the relative pose between two images, using three pairs of homologous points and the knowledge of the vertical direction. The vertical direction can be determined in two ways: The first requires direct physical measurements such as the ones provided by an IMU (inertial measurement unit). The other uses the automatic extraction of the vanishing point corresponding to the vertical direction in an image. This knowledge of the vertical direction solves two unknowns among the three parameters of the relative rotation, so that only three homologous couples of points are requested to position a couple of images. Rewriting the coplanarity equations thus leads to a much simpler solution. The remaining unknowns resolution is performed by "hiding a variable" approach. The elements necessary to build a specific algebraic solver are given in this paper, allowing for a real-time implementation. The results on real and synthetic data show the efficiency of this method.

Journal ArticleDOI
TL;DR: This paper first provides closed form expressions for several important multicomponent proximity operators and then derive extensions of existing proximal algorithms to the multicomponents setting that are applied to stereoscopic image recovery, multispectral image denoising, and image decomposition into texture and geometry components.
Abstract: In recent years, proximal splitting algorithms have been applied to various monocomponent signal and image recovery problems. In this paper, we address the case of multicomponent problems. We first provide closed form expressions for several important multicomponent proximity operators and then derive extensions of existing proximal algorithms to the multicomponent setting. These results are applied to stereoscopic image recovery, multispectral image denoising, and image decomposition into texture and geometry components.

Journal ArticleDOI
TL;DR: In this paper, the authors study the Monge-Kantorovich problem between discrete distributions on the unit circle S 1, in the case where the ground distance between two points x and y is defined as h(d(x,y)), where d is the geodesic distance on the circle and h a convex and increasing function.
Abstract: This paper is devoted to the study of the Monge-Kantorovich theory of optimal mass transport, in the special case of one-dimensional and circular distributions. More precisely, we study the Monge-Kantorovich problem between discrete distributions on the unit circle S 1, in the case where the ground distance between two points x and y is defined as h(d(x,y)), where d is the geodesic distance on the circle and h a convex and increasing function. This study complements previous results in the literature, holding only for a ground distance equal to the geodesic distance d. We first prove that computing a Monge-Kantorovich distance between two given sets of pairwise different points boils down to cut the circle at a well chosen point and to compute the same distance on the real line. This result is then used to obtain a dissimilarity measure between 1-D and circular discrete histograms. In a last part, a study is conducted to compare the advantages and drawbacks of transportation distances relying on convex or concave cost functions, and of the classical L 1 distance. Simple retrieval experiments based on the hue component of color images are shown to illustrate the interest of circular distances. The framework is eventually applied to the problem of color transfer between images.

Journal ArticleDOI
TL;DR: The problem of finding an interpolating image between two given images in an image sequence is formulated as an optimal control problem governed by a transport equation and the existence of optimal controls is proven and necessary conditions are derived.
Abstract: The problem of finding an interpolating image between two given images in an image sequence is considered. The problem is formulated as an optimal control problem governed by a transport equation, i.e. we aim at finding a flow field which transports the first image as close as possible to the second image. This approach bears similarities with the Horn and Schunck method for optical flow calculation but in fact the model is quite different. The images are modeled in the space of functions of bounded variation and an analysis of solutions of transport equations in this space is included. Moreover, the existence of optimal controls is proven and necessary conditions are derived. Finally, two algorithms are given and numerical results are compared with existing methods. The new method is competitive with state-of-the-art methods and even outperforms several existing methods.

Journal ArticleDOI
TL;DR: This paper presents an innovative three dimensional occlusion detection and restoration strategy for the recognition of three dimensional faces partially occluded by unforeseen, extraneous objects and demonstrates the robustness and feasibility of the approach.
Abstract: This paper presents an innovative three dimensional occlusion detection and restoration strategy for the recognition of three dimensional faces partially occluded by unforeseen, extraneous objects. The detection method considers occlusions as local deformations of the face that correspond to perturbations in a space designed to represent non-occluded faces. Once detected, occlusions represent missing information, or "holes" in the faces. The restoration module exploits the information provided by the non-occluded part of the face to recover the whole face, using an appropriate basis for the space in which non-occluded faces lie. The restoration strategy does not depend on the method used to detect occlusions and can also be applied to restore faces in the presence of noise and missing pixels due to acquisition inaccuracies. The strategy has been experimented on the occluded acquisitions taken from the Bosphorus 3D face database. A method for the generation of real-looking occlusions is also presented. Artificial occlusions, applied to the UND database, allowed for an in-depth analysis of the capabilities of our approach. Experimental results demonstrate the robustness and feasibility of our approach.

Journal ArticleDOI
TL;DR: Two new higher order diffusion models for removing noise from images are presented, modifications of an existing fourth order partial differential equation (PDE) model which was developed by You and Kaveh as a generalization of the well-known second order Perona-Malik equation.
Abstract: This paper presents two new higher order diffusion models for removing noise from images. The models employ fractional derivatives and are modifications of an existing fourth order partial differential equation (PDE) model which was developed by You and Kaveh as a generalization of the well-known second order Perona-Malik equation. The modifications serve to cure the ill-posedness of the You-Kaveh model without sacrificing performance. Also proposed in this paper is a simple smoothing technique which can be used in numerical experiments to improve denoising and reduce processing time. Numerical experiments are shown for comparison.

Journal ArticleDOI
TL;DR: This paper presents a reformulation of Normalized Cut segmentation that in a unified way can handle linear equality constraints for an arbitrary number of classes and provides a principled way to perform multi-class segmentation for tasks like interactive segmentation.
Abstract: Indisputably Normalized Cuts is one of the most popular segmentation algorithms in pattern recognition and computer vision. It has been applied to a wide range of segmentation tasks with great success. A number of extensions to this approach have also been proposed, including ones that can deal with multiple classes or that can incorporate a priori information in the form of grouping constraints. However, what is common for all these methods is that they are noticeably limited in the type of constraints that can be incorporated and can only address segmentation problems on a very specific form. In this paper, we present a reformulation of Normalized Cut segmentation that in a unified way can handle linear equality constraints for an arbitrary number of classes. This is done by restating the problem and showing how linear constraints can be enforced exactly in the optimization scheme through duality. This allows us to add group priors, for example, that certain pixels should belong to a given class. In addition, it provides a principled way to perform multi-class segmentation for tasks like interactive segmentation. The method has been tested on real data showing good performance and improvements compared to standard normalized cuts.

Journal ArticleDOI
TL;DR: It is proved that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates the partial differential equation of self-snakes, a structure-adaptive morphological image filter.
Abstract: This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We establish this correspondence by analysing a space-continuous formulation of iterated amoeba median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates the partial differential equation of self-snakes. This result holds true under very general assumptions on the metric used to construct the amoebas. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.

Journal ArticleDOI
TL;DR: A new implementation of the 3-D fast curvelet transform is presented, which is nearly 2.5 less redundant than the Curvelab (wrapping-based) implementation as originally proposed in Ying et al. (Proceedings of wavelets XI conference, San Diego, 2005), which makes it more suitable to applications including massive data sets.
Abstract: In this paper, we first present a new implementation of the 3-D fast curvelet transform, which is nearly 2.5 less redundant than the Curvelab (wrapping-based) implementation as originally proposed in Ying et al. (Proceedings of wavelets XI conference, San Diego, 2005) and Candes et al. (SIAM Multiscale Model. Simul. 5(3):861---899, 2006), which makes it more suitable to applications including massive data sets. We report an extensive comparison in denoising with the Curvelab implementation as well as other 3-D multi-scale transforms with and without directional selectivity. The proposed implementation proves to be a very good compromise between redundancy, rapidity and performance. Secondly, we exemplify its usefulness on a variety of applications including denoising, inpainting, video de-interlacing and sparse component separation. The obtained results are good with very simple algorithms and virtually no parameter to tune.

Journal ArticleDOI
TL;DR: An algorithm based on left and right side partial derivatives that is computationally efficient as an alternative to conventional algorithms is proposed, and the sensitivity of circle fits for different types of data is evaluated.
Abstract: Geometric fitting is present in different fields of sciences, engineering and astronomy. In particular, circular arc primitives are some of the most commonly employed geometric features in digital image analysis and visual pattern recognition. In this paper, a robust geometric method based on mean absolute error to fit a set of points is proposed. Most geometric and algebraic methods are sensitive to noise and outlier points and so the results are not usually acceptable. It is well known that the least absolute error criterion leads to robust estimations. However, the objective function is non differentiable and thus algorithms based on gradient cannot be applied. We propose an algorithm based on left and right side partial derivatives that is computationally efficient as an alternative to conventional algorithms, and evaluate the sensitivity of circle fits for different types of data.

Journal ArticleDOI
TL;DR: This study puts in evidence that both the uneven distribution of the LBP histogram and the high occurrence of uniform patterns are direct consequences of the mathematical structure of the method rather than an intrinsic property of real textures.
Abstract: It is well-known that local binary pattern (LBP) histograms of real textures exhibit a markedly uneven distribution, which is dominated by the so-called uniform patterns. The widely accepted interpretation of this phenomenon is that uniform patterns correspond to texture microfeatures, such as edges, corners, and spots. In this paper we present a theoretical study about the relative occurrence of LBPs based on the consideration that the LBP operator partitions the set of grayscale patterns into an ensemble of disjoint multidimensional polytopes. We derive exact prior probabilities of LBPs by calculating the volume of such polytopes. Our study puts in evidence that both the uneven distribution of the LBP histogram and the high occurrence of uniform patterns are direct consequences of the mathematical structure of the method rather than an intrinsic property of real textures.

Journal ArticleDOI
TL;DR: This work reformulates the variational model as a constrained optimization problem, presents an augmented Lagrangian method and a projection Lagrangians method to solve the constrained model and proposes two gradient-type algorithms based on the semi-implicit additive operator splitting scheme.
Abstract: Interface evolution problems are often solved elegantly by the level set method, which generally requires the time-consuming reinitialization process. In order to avoid reinitialization, we reformulate the variational model as a constrained optimization problem. Then we present an augmented Lagrangian method and a projection Lagrangian method to solve the constrained model and propose two gradient-type algorithms. For the augmented Lagrangian method, we employ the Uzawa scheme to update the Lagrange multiplier. For the projection Lagrangian method, we use the variable splitting technique and get an explicit expression for the Lagrange multiplier. We apply the two approaches to the Chan-Vese model and obtain two efficient alternating iterative algorithms based on the semi-implicit additive operator splitting scheme. Numerical results on various synthetic and real images are provided to compare our methods with two others, which demonstrate effectiveness and efficiency of our algorithms.

Journal ArticleDOI
TL;DR: The aim of this paper is to define an extension of the analytic signal for a color image to mappings with values in the vectorial part of the Clifford algebra ℝ5,0.
Abstract: The aim of this paper is to define an extension of the analytic signal for a color image. We generalize the construction of the so-called monogenic signal to mappings with values in the vectorial part of the Clifford algebra ?5,0. Solving a Dirac equation in this context leads to a multiscale signal (relatively to the Poisson scale-space) which contains both structure and color information. The color monogenic signal can be used in a wide range of applications. Two examples are detailed: the first one concerns a multiscale geometric segmentation with respect to a given color; the second one is devoted to the extraction of the optical flow from moving objects of a given color.

Journal ArticleDOI
TL;DR: It is shown that RCMs generally give state of the art results when applied to a range of different vision tasks and evaluated on the leading benchmarked datasets.
Abstract: This paper describes and reviews a class of hierarchical probabilistic models of images and objects Visual structures are represented in a hierarchical form where complex structures are composed of more elementary structures following a design principle of recursive composition Probabilities are defined over these structures which exploit properties of the hierarchy--eg long range spatial relationships can be represented by local potentials at the upper levels of the hierarchy The compositional nature of this representation enables efficient learning and inference algorithms In particular, parts can be shared between different object models Overall the architecture of Recursive Compositional Models (RCMs) provides a balance between statistical and computational complexity The goal of this paper is to describe the basic ideas and common themes of RCMs, to illustrate their success on a range of vision tasks, and to gives pointers to the literature In particular, we show that RCMs generally give state of the art results when applied to a range of different vision tasks and evaluated on the leading benchmarked datasets

Journal ArticleDOI
TL;DR: The theoretical and practical version of the signed distance transform algorithm, GBDT, returns the exact value of the distance from the geometrically defined object boundary, and it is proved that this algorithm can be used to find, in linear time, the largest possible distance between any two of its elements.
Abstract: In 2003, Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265---270, 2003) published a paper describing an algorithm that computes the exact distance transform in linear time (with respect to image size) for the rectangular binary images in the k-dimensional space ? k and distance measured with respect to L p -metric for 1?p??, which includes Euclidean distance L 2. In this paper we discuss this algorithm from theoretical and practical points of view. On the practical side, we concentrate on its Euclidean distance version, discuss the possible ways of implementing it as signed distance transform, and experimentally compare implemented algorithms. We also describe the parallelization of these algorithms and discuss the computational time savings associated with them. All these implementations will be made available as a part of the CAVASS software system developed and maintained in our group (Grevera et al. in J. Digit. Imaging 20:101---118, 2007). On the theoretical side, we prove that our version of the signed distance transform algorithm, GBDT, returns the exact value of the distance from the geometrically defined object boundary. We provide a complete proof (which was not given of Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265---270, 2003) that all these algorithms work correctly for L p -metric with 1

Journal ArticleDOI
TL;DR: This paper proposes a new accurate depth dependent lens distortion model and applies it to camera calibration in planar view scenarios (that is 3D scenarios where the objects of interest lie on a plane).
Abstract: In order to calibrate cameras in an accurate manner, lens distortion models have to be included in the calibration procedure. Usually, the lens distortion models used in camera calibration depend on radial functions of image pixel coordinates. Such models are well-known, simple and can be estimated using just image information. However, these models do not take into account an important physical constraint of lens distortion phenomena, namely: the amount of lens distortion induced in an image point depends on the scene point depth with respect to the camera projection plane. In this paper we propose a new accurate depth dependent lens distortion model. To validate this approach, we apply the new lens distortion model to camera calibration in planar view scenarios (that is 3D scenarios where the objects of interest lie on a plane). We present promising experimental results on planar pattern images and on sport event scenarios. Nevertheless, although we emphasize the feasibility of the method for planar view scenarios, the proposed model is valid in general and can be used in any scenario where the point depth can be estimated.

Journal ArticleDOI
TL;DR: A new method for the orienting of shapes is demonstrated to be superior with respect to the standard method in several situations and to be applicable to a range of applications.
Abstract: In this paper we propose a measure which defines the degree to which a shape differs from a square. The new measure is easy to compute and being area based, is robust--e.g., with respect to noise or narrow intrusions. Also, it satisfies the following desirable properties: it ranges over (0,1] and gives the measured squareness equal to 1 if and only if the measured shape is a square; it is invariant with respect to translations, rotations and scaling. In addition, we propose a generalisation of the new measure so that shape squareness can be computed while controlling the impact of the relative position of points inside the shape. Such a generalisation enables a tuning of the behaviour of the squareness measure and makes it applicable to a range of applications. A second generalisation produces a measure, parameterised by ?, that ranges in the interval (0,1] and equals 1 if and only if the measured shape is a rhombus whose diagonals are in the proportion 1:?. The new measures (the initial measure and the generalised ones) are naturally defined and theoretically well founded--consequently, their behaviour can be well understood. As a by-product of the approach we obtain a new method for the orienting of shapes, which is demonstrated to be superior with respect to the standard method in several situations. The usefulness of the methods described in the manuscript is illustrated on three large shape databases: diatoms (ADIAC), MPEG-7 CE-1, and trademarks.

Journal ArticleDOI
TL;DR: A variational model for the determination of the optic-flow in a general setting of non-smooth domains is considered and it is proved that the error indicators provide, as, a by-product, a confidence measure which shows the effects of regularization and serves to compute sparse solutions.
Abstract: We consider a variational model for the determination of the optic-flow in a general setting of non-smooth domains This problem is ill-posed and its solution with PDE techniques includes a regularization procedure The goal of this paper is to study a method to solve the optic flow problem and to control the effects of the regularization by allowing, locally and adaptively the choice of its parameters The regularization in our approach is not controlled by a single parameter but by a function of the space variable This results in a dynamical selection of the variational model which evolves with the variations of this function Such method consists of new adaptive finite element discretization and an a posteriori strategy for the control of the regularization in order to achieve a trade-off between the data and the smoothness terms in the energy functional We perform the convergence analysis and the a posteriori analysis, and we prove that the error indicators provide, as, a by-product, a confidence measure which shows the effects of regularization and serves to compute sparse solutions We perform several numerical experiments, to show the efficiency and the reliability of the method in the computations of optic flow, with high accuracy and of low density

Journal ArticleDOI
TL;DR: An iterative algorithm alternates between depth and ego-motion estimation for fast computation of 3D information from motion in image sequences for 3D reconstruction from synthetic and natural omnidirectional images.
Abstract: We address the problem of depth and ego-motion estimation from omnidirectional images. We propose a correspondence-free structure-from-motion problem for sequences of images mapped on the 2-sphere. A novel graph-based variational framework is first proposed for depth estimation between pairs of images. The estimation is cast as a TV-L1 optimization problem that is solved by a fast graph-based algorithm. The ego-motion is then estimated directly from the depth information without explicit computation of the optical flow. Both problems are finally addressed together in an iterative algorithm that alternates between depth and ego-motion estimation for fast computation of 3D information from motion in image sequences. Experimental results demonstrate the effective performance of the proposed algorithm for 3D reconstruction from synthetic and natural omnidirectional images.

Journal ArticleDOI
TL;DR: A simple and universal adaptive weighting of the SSD resolves the fattening problem at all smooth disparity points, and the optimal disparity function is the result of the convolution of the real disparity with a prefixed kernel.
Abstract: Block matching along epipolar lines is the core of most stereovision algorithms in geographic information systems. The usual distances between blocks are the sum of squared distances in the block (SSD) or the correlation. Minimizing these distances causes the fattening effect, by which the center of the block inherits the disparity of the more contrasted pixels in the block. This fattening error occurs everywhere in the image, and not just on strong depth discontinuities. The fattening effect at strong depth edges is a particular case of fattening, called foreground fattening effect. A theorem proved in the present paper shows that a simple and universal adaptive weighting of the SSD resolves the fattening problem at all smooth disparity points (a Spanish patent has been applied for by Universitat de Illes Balears (Reference P25155ES00, UIB, 2009)). The optimal SSD weights are nothing but the inverses of the squares of the image gradients in the epipolar direction. With these adaptive weights, it is shown that the optimal disparity function is the result of the convolution of the real disparity with a prefixed kernel. Experiments on simulated and real pairs prove that the method does what the theorem predicts, eliminating surface bumps caused by fattening. However, the method does not resolve the foreground fattening.