Abstract: In this work, we study the topic of high-resolution adaptive sampling of a given deterministic and differentiable signal and establish a connection with classic approaches to high-rate quantization. Specifically, we formulate solutions for the task of optimal high-resolution sampling, counterparts of well-known results for high-rate quantization. Our results reveal that the optimal high-resolution sampling structure is determined by the density of the signal-gradient energy, just as the probability density function defines the optimal high-rate quantization form. This paper has three main contributions: The first is establishing a fundamental paradigm bridging the topics of sampling and quantization. The second is a theoretical analysis of nonuniform sampling, for arbitrary signal dimension, relevant to the emerging field of high-resolution signal processing. The third is a new practical approach to nonuniform sampling of one-dimensional signals that enables reconstruction based only on the sampling time points and the signal extrema locations and values. Experiments for signal sampling and coding showed that our method outperforms an optimized tree-structured sampling technique. PubDate: 2019-03-11 DOI: 10.1007/s10851-019-00880-5

Abstract: We propose a model of plane ovals based on two foci defined in the 3D space. Geometrically, some of these ovals are the isocontours of a planar specular highlight for a point light source according to Phong’s reflectance model. We, hence, call them the highlight ovals. More precisely, we define a geometric highlight oval as the locus of points p whose product of distances to the foci is proportional to the dot product of the vectors from p to the foci. When the proportionality is taken up to sign, this gives a quartic, whose roots define the algebraic highlight oval. Similarly to the Cartesian oval, the algebraic formulation hence includes a pair of geometric highlight ovals. Similarly to the Cassinian oval, the geometric definition includes non-convex closed curves. The highlight ovals have four intrinsic parameters, namely the height of the two foci relative to the oval’s plane, the distance between the two foci’s projection on the oval’s plane and a scale factor fixing the proportionality and related to the isocontour’s intensity. We thoroughly study the topology and convexity of the family of algebraic highlight ovals for any combination of its four intrinsic parameters and show how this characterizes the geometric highlight ovals, and thus the isocontours of specular highlights. We report an extensive experimental evaluation, showing that the highlight ovals form a reliable model of specular highlights. PubDate: 2019-03-04 DOI: 10.1007/s10851-019-00878-z

Abstract: Mathematical morphology is a framework composed by a set of well-known image processing techniques, widely used for binary and grayscale images, but less commonly used to process color or multivariate images. In this paper, we generalize fuzzy mathematical morphology to process multivariate images in such a way that overcomes the problem of defining an appropriate order among colors. We introduce the soft color erosion and the soft color dilation, which are the foundations of the rest of operators. Besides studying their theoretical properties, we analyze their behavior and compare them with the corresponding morphological operators from other frameworks that deal with color images. The soft color morphology outstands when handling images in the CIEL \({}^*a{}^*b{}^*\) color space, where it guarantees that no colors with different chromatic values to the original ones are created. The soft color morphological operators prove to be easily customizable but also highly interpretable. Besides, they are fast operators and provide smooth outputs, more visually appealing than the crisp color transitions provided by other approaches. PubDate: 2019-03-01 DOI: 10.1007/s10851-018-0849-2

Abstract: In this paper, a dual online subspace-based learning method called dual-generalized discriminative common vectors (Dual-GDCV) is presented. The method extends incremental GDCV by exploiting simultaneously both the concepts of incremental and decremental learning for supervised feature extraction and classification. Our methodology is able to update the feature representation space without recalculating the full projection or accessing the previously processed training data. It allows both adding information and removing unnecessary data from a knowledge base in an efficient way, while retaining the previously acquired knowledge. The proposed method has been theoretically proved and empirically validated in six standard face recognition and classification datasets, under two scenarios: (1) removing and adding samples of existent classes, and (2) removing and adding new classes to a classification problem. Results show a considerable computational gain without compromising the accuracy of the model in comparison with both batch methodologies and other state-of-art adaptive methods. PubDate: 2019-03-01 DOI: 10.1007/s10851-018-0837-6

Abstract: Planar ornaments, a.k.a. wallpapers, are regular repetitive patterns which exhibit translational symmetry in two independent directions. There are exactly 17 distinct planar symmetry groups. We present a fully automatic method for complete analysis of planar ornaments in 13 of these groups, specifically, the groups called p6, p6m, p4g, p4m, p4, p31m, p3m, p3, cmm, pgg, pg, p2 and p1. Given the image of an ornament fragment, we present a method to simultaneously classify the input into one of the 13 groups and extract the so-called fundamental domain, the minimum region that is sufficient to reconstruct the entire ornament. A nice feature of our method is that even when the given ornament image is a small portion such that it does not contain multiple translational units, the symmetry group as well as the fundamental domain can still be defined. This is because, in contrast to common approach, we do not attempt to first identify a global translational repetition lattice. Though the presented constructions work for quite a wide range of ornament patterns, a key assumption we make is that the perceivable motifs (shapes that repeat) alone do not provide clues for the underlying symmetries of the ornament. In this sense, our main target is the planar arrangements of asymmetric interlocking shapes, as in the symmetry art of Escher. PubDate: 2019-03-01 DOI: 10.1007/s10851-018-0835-8

Abstract: We prove that the 8-point algorithm always fails to reconstruct a unique fundamental matrix F independent on the camera positions, when its inputs are image point configurations that are perspective projections of the intersection of three quadrics in \(\mathbb {P}^3\) . This generalizes a multiple results of the degeneracies of the 8-point algorithm. We give an algorithm that improves the 7- and 8-point algorithm in such a pathological situation. Additionally, we analyze the regions of focal point positions where a reconstruction of F is possible at all, when the world points are the vertices of a combinatorial cube in \(\mathbb {R}^3\) . PubDate: 2019-03-01 DOI: 10.1007/s10851-018-0841-x

Abstract: In this work, doubly stochastic Poisson (Cox) processes and convolutional neural net (CNN) classifiers are used to estimate the number of instances of an object in an image. Poisson processes are well suited to model events that occur randomly in space, such as the location of objects in an image or the enumeration of objects in a scene. The proposed algorithm selects a subset of bounding boxes in the image domain, then queries them for the presence of the object of interest by running a pre-trained CNN classifier. The resulting observations are then aggregated, and a posterior distribution over the intensity of a Cox process is computed. This intensity function is summed up, providing an estimator of the number of instances of the object over the entire image. Despite the flexibility and versatility of Cox processes, their application to large datasets is limited as their computational complexity and storage requirements do not easily scale with image size, typically requiring \(O(n^3)\) computation time and \(O(n^2)\) storage, where n is the number of observations. To mitigate this problem, we employ the Kronecker algebra, which takes advantage of direct product structures. As the likelihood is non-Gaussian, the Laplace approximation is used for inference, employing the conjugate gradient and Newton’s method. Our approach has then close to linear performance, requiring only \(O(n^{3/2})\) computation time and O(n) memory. Results are presented on simulated data and on images from the publicly available MS COCO dataset. We compare our counting results with the state-of-the-art detection method, Faster RCNN, and demonstrate superior performance. PubDate: 2019-03-01 DOI: 10.1007/s10851-018-0838-5

Abstract: Among the various existing and mathematically equivalent definitions of the skeleton, we consider the set of critical points of the Euclidean distance transform of the shape. The problem of detecting these points and using them to generate a skeleton that is stable, thin and homotopic to the shape has been the focus of numerous papers. Skeleton branches correspond to ridges of the distance map, i.e., continuous lines of points that are local maxima of the distance in at least one direction. Extracting these ridges is a non-trivial task on a discrete grid. In this context, the average outward flux, used in the Hamilton–Jacobi skeleton (Siddiqi et al. in Int J Comput Vis 48(3):215–231, 2002), and the ridgeness measure (Leborgne et al. in J Vis Commun Image Represent 31:165–176, 2015) have been proposed as ridge detectors. We establish the mathematical relation between these detectors and, extending the work in Dimitrov et al. (Computer vision and pattern recognition, pp 835–841, 2003), we study various local shape configurations, on which closed-form expressions or approximations of the average outward flux and ridgeness can be derived. In addition, we conduct experiments to assess the accuracy of skeletons generated using these measures and study the influence of their respective parameters. PubDate: 2019-03-01 DOI: 10.1007/s10851-018-0836-7

Abstract: This article presents a novel discretization of the Laplace–Beltrami operator on digital surfaces. We adapt an existing convolution technique proposed by Belkin et al. (in: Teillaud (ed) Proceedings of the 24th ACM symposium on computational geometry, College Park, MD, USA, pp 278–287, 2008, https://doi.org/10.1145/1377676.1377725) for triangular meshes to topological border of subsets of \(\mathbb {Z}^n\) . The core of the method relies on first-order estimation of measures associated with our discrete elements (such as length, area etc.). We show strong consistency (i.e., pointwise convergence) of the operator and compare it against various other discretizations. PubDate: 2019-03-01 DOI: 10.1007/s10851-018-0839-4

Abstract: A long-time, hand-crafted features governed by a directional image analysis have been the base for the best performing pedestrian detection algorithms. In the past few years, approaches using convolutional neural networks have taken over the leadership concerning detection quality. We investigate in which way shearlets can be used for an improved hand-crafted feature computation in order to reduce the gap to CNNs. Shearlets are a relatively new mathematical framework for multiscale signal analysis, which can be seen as an extension of the wavelet framework. Shearlets are designed to capture directional information and can therefore be used for detecting the orientation of edges in images. We use this characteristic to compute image features with high informative content for pedestrian detection. Furthermore, we provide experimental results using these features and show that they outperform the results obtained by the currently best performing hand-crafted features for pedestrian detection. PubDate: 2019-03-01 DOI: 10.1007/s10851-018-0834-9

Abstract: Polynomial functions are a usual choice to model the nonlinearity of lenses. Typically, these models are obtained through physical analysis of the lens system or on purely empirical grounds. The aim of this work is to facilitate an alternative approach to the selection or design of these models based on establishing a priori the desired geometrical properties of the distortion functions. With this purpose we obtain all the possible isotropic linear models and also those that are formed by functions with symmetry with respect to some axis. In this way, the classical models (decentering, thin prism distortion) are found to be particular instances of the family of models found by geometric considerations. These results allow to find generalizations of the most usually employed models while preserving the desired geometrical properties. Our results also provide a better understanding of the geometric properties of the models employed in the most usual computer vision software libraries. PubDate: 2019-03-01 DOI: 10.1007/s10851-018-0833-x

Abstract: Single-image nonparametric blind super-resolution is a fundamental image restoration problem yet largely ignored in the past decades among the computational photography and computer vision communities. An interesting phenomenon is observed that learning-based single-image super-resolution (SR) has been experiencing a rapid development since the boom of the sparse representation in 2005s and especially the representation learning in 2010s, wherein the high-res image is generally blurred by a supposed bicubic or Gaussian blur kernel. However, the parametric assumption on the form of blur kernels does not hold in most practical applications because in real low-res imaging a high-res image can undergo complex blur processes, e.g., Gaussian-shaped kernels of varying sizes, ellipse-shaped kernels of varying orientations, curvilinear kernels of varying trajectories. The paper is mainly motivated by one of our previous works: Shao and Elad (in: Zhang (ed) ICIG 2015, Part III, Lecture notes in computer science, Springer, Cham, 2015). Specifically, we take one step further in this paper and present a type of adaptive heavy-tailed image priors, which result in a new regularized formulation for nonparametric blind super-resolution. The new image priors can be expressed and understood as a generalized integration of the normalized sparsity measure and relative total variation. Although it seems that the proposed priors are simple, the core merit of the priors is their practical capability for the challenging task of nonparametric blur kernel estimation for both super-resolution and deblurring. Harnessing the priors, a higher-quality intermediate high-res image becomes possible and therefore more accurate blur kernel estimation can be accomplished. A great many experiments are performed on both synthetic and real-world blurred low-res images, demonstrating the comparative or even superior performance of the proposed algorithm convincingly. Meanwhile, the proposed priors are demonstrated quite applicable to blind image deblurring which is a degenerated problem of nonparametric blind SR. PubDate: 2019-02-27 DOI: 10.1007/s10851-019-00876-1

Abstract: In this paper, we propose a novel accelerated alternating optimization scheme to solve block biconvex nonsmooth problems whose objectives can be split into smooth (separable) regularizers and simple coupling terms. The proposed method performs a Bregman distance-based generalization of the well-known forward–backward splitting for each block, along with an inertial strategy which aims at getting empirical acceleration. We discuss the theoretical convergence of the proposed scheme and provide numerical experiments on image colorization. PubDate: 2019-02-23 DOI: 10.1007/s10851-019-00877-0

Abstract: We derived analytical forward and inverse solution of thermoacoustic wave equation for nonhomogeneous medium. We modelled the nonhomogeneous medium as a multi-layer planar medium and defined initial conditions, continuity conditions on the layer boundaries and radiation conditions at infinity assuming the source distribution existing in all layers. These solutions of thermoacoustic wave equation are based on the method of Green’s functions for layered planar media. For qualitative testing and comparison of the point-spread functions associated with the homogeneous and layered solutions, we performed numerical simulations. Our simulation results showed that the conventional inverse solution based on homogeneous medium assumption, as expected, produced incorrect locations of point sources, whereas our inverse solution involving the multi-layer planar medium produced point sources at the correct source locations. Also, we examined whether the performance of our layered inverse solution is sensitive to medium parameters used as priority information in the measured data. Our inverse solutions based on multi-layer planar media are applicable for cross-sectional two-dimensional imaging of abdominal structure and the organs such as breast and skin. PubDate: 2019-02-20 DOI: 10.1007/s10851-019-00875-2

Abstract: In 2013, Najman and Géraud proved that by working on a well-composed discrete representation of a gray-level image, we can compute what is called its tree of shapes, a hierarchical representation of the shapes in this image. This way, we can proceed to morphological filtering and to image segmentation. However, the authors did not provide such a representation for the non-cubical case. We propose in this paper a way to compute a well-composed representation of any gray-level image defined on a discrete surface, which is a more general framework than the usual cubical grid. Furthermore, the proposed representation is self-dual in the sense that it treats bright and dark components in the image the same way. This paper can be seen as an extension to gray-level images of the works of Daragon et al. on discrete surfaces. PubDate: 2019-02-12 DOI: 10.1007/s10851-019-00873-4

Abstract: We present an approach for variational regularization of inverse and imaging problems for recovering functions with values in a set of vectors. We introduce regularization functionals, which are derivative-free double integrals of such functions. These regularization functionals are motivated from double integrals, which approximate Sobolev semi-norms of intensity functions. These were introduced in Bourgain et al. (Another look at Sobolev spaces. In: Menaldi, Rofman, Sulem (eds) Optimal control and partial differential equations-innovations and applications: in honor of professor Alain Bensoussan’s 60th anniversary, IOS Press, Amsterdam, pp 439–455, 2001). For the proposed regularization functionals, we prove existence of minimizers as well as a stability and convergence result for functions with values in a set of vectors. PubDate: 2019-02-07 DOI: 10.1007/s10851-018-00869-6

Abstract: Component-trees are classical tree structures for grey-level image modelling. Component-graphs are defined as a generalization of component-trees to images taking their values in any (totally or partially) ordered sets. Similar to component-trees, component-graphs are a lossless image model; then, they can allow for the development of various image processing approaches. However, component-graphs are not trees, but directed acyclic graphs. This makes their construction non-trivial, leading to nonlinear time cost and resulting in nonlinear space data structures. In this theoretical article, we discuss the notion(s) of component-graph, and we propose a strategy for their efficient building and representation, which are necessary conditions for further involving them in image processing approaches. PubDate: 2019-02-07 DOI: 10.1007/s10851-019-00872-5

Abstract: Measuring attenuation coefficients is a fundamental problem that can be solved with diverse techniques such as X-ray or optical tomography and lidar. We propose a novel approach based on the observation of a sample from a few different angles. This principle can be used in existing devices such as lidar or various types of fluorescence microscopes. It is based on the resolution of a nonlinear inverse problem. We propose a specific computational approach to solve it and show the well-foundedness of the approach on simulated data. Some of the tools developed are of independent interest. In particular, we propose an efficient method to correct attenuation defects, new robust solvers for the lidar equation as well as new efficient algorithms to compute the proximal operator of the logsumexp function in dimension 2. PubDate: 2019-02-06 DOI: 10.1007/s10851-019-00871-6