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Author: Pavan K. Turaga
Publisher: Springer
ISBN: 3319229575
Category : Technology & Engineering
Languages : en
Pages : 382
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Book Description
This book presents a comprehensive treatise on Riemannian geometric computations and related statistical inferences in several computer vision problems. This edited volume includes chapter contributions from leading figures in the field of computer vision who are applying Riemannian geometric approaches in problems such as face recognition, activity recognition, object detection, biomedical image analysis, and structure-from-motion. Some of the mathematical entities that necessitate a geometric analysis include rotation matrices (e.g. in modeling camera motion), stick figures (e.g. for activity recognition), subspace comparisons (e.g. in face recognition), symmetric positive-definite matrices (e.g. in diffusion tensor imaging), and function-spaces (e.g. in studying shapes of closed contours).
Author: Pavan K. Turaga
Publisher: Springer
ISBN: 3319229575
Category : Technology & Engineering
Languages : en
Pages : 382
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Book Description
This book presents a comprehensive treatise on Riemannian geometric computations and related statistical inferences in several computer vision problems. This edited volume includes chapter contributions from leading figures in the field of computer vision who are applying Riemannian geometric approaches in problems such as face recognition, activity recognition, object detection, biomedical image analysis, and structure-from-motion. Some of the mathematical entities that necessitate a geometric analysis include rotation matrices (e.g. in modeling camera motion), stick figures (e.g. for activity recognition), subspace comparisons (e.g. in face recognition), symmetric positive-definite matrices (e.g. in diffusion tensor imaging), and function-spaces (e.g. in studying shapes of closed contours).
Author: Hà Quang Minh
Publisher: Springer
ISBN: 3319450263
Category : Computers
Languages : en
Pages : 216
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Book Description
This book presents a selection of the most recent algorithmic advances in Riemannian geometry in the context of machine learning, statistics, optimization, computer vision, and related fields. The unifying theme of the different chapters in the book is the exploitation of the geometry of data using the mathematical machinery of Riemannian geometry. As demonstrated by all the chapters in the book, when the data is intrinsically non-Euclidean, the utilization of this geometrical information can lead to better algorithms that can capture more accurately the structures inherent in the data, leading ultimately to better empirical performance. This book is not intended to be an encyclopedic compilation of the applications of Riemannian geometry. Instead, it focuses on several important research directions that are currently actively pursued by researchers in the field. These include statistical modeling and analysis on manifolds,optimization on manifolds, Riemannian manifolds and kernel methods, and dictionary learning and sparse coding on manifolds. Examples of applications include novel algorithms for Monte Carlo sampling and Gaussian Mixture Model fitting, 3D brain image analysis,image classification, action recognition, and motion tracking.
Author: Masoud Faraki
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
A diverse number of tasks in computer vision and machine learning enjoy from representations of data that are compact yet discriminative, informative and robust to critical measurements. Two notable representations are offered by Region Covariance Descriptors (RCovD) and linear subspaces which are naturally analyzed through the manifold of Symmetric Positive Definite (SPD) matrices and the Grassmann manifold, respectively, two widely used types of Riemannian manifolds in computer vision. As our first objective, we examine image and video-based recognition applications where the local descriptors have the aforementioned Riemannian structures, namely the SPD or linear subspace structure. Initially, we provide a solution to compute Riemannian version of the conventional Vector of Locally aggregated Descriptors (VLAD), using geodesic distance of the underlying manifold as the nearness measure. Next, by having a closer look at the resulting codes, we formulate a new concept which we name Local Difference Vectors (LDV). LDVs enable us to elegantly expand our Riemannian coding techniques to any arbitrary metric as well as provide intrinsic solutions to Riemannian sparse coding and its variants when local structured descriptors are considered. We then turn our attention to two special types of covariance descriptors namely infinite-dimensional RCovDs and rank-deficient covariance matrices for which the underlying Riemannian structure, i.e. the manifold of SPD matrices is out of reach to great extent. Generally speaking, infinite-dimensional RCovDs offer better discriminatory power over their low-dimensional counterparts. To overcome this difficulty, we propose to approximate the infinite-dimensional RCovDs by making use of two feature mappings, namely random Fourier features and the Nystrom method. As for the rank-deficient covariance matrices, unlike most existing approaches that employ inference tools by predefined regularizers, we derive positive definite kernels that can be decomposed into the kernels on the cone of SPD matrices and kernels on the Grassmann manifolds and show their effectiveness for image set classification task. Furthermore, inspired by attractive properties of Riemannian optimization techniques, we extend the recently introduced Keep It Simple and Straightforward MEtric learning (KISSME) method to the scenarios where input data is non-linearly distributed. To this end, we make use of the infinite dimensional covariance matrices and propose techniques towards projecting on the positive cone in a Reproducing Kernel Hilbert Space (RKHS). We also address the sensitivity issue of the KISSME to the input dimensionality. The KISSME algorithm is greatly dependent on Principal Component Analysis (PCA) as a preprocessing step which can lead to difficulties, especially when the dimensionality is not meticulously set. To address this issue, based on the KISSME algorithm, we develop a Riemannian framework to jointly learn a mapping performing dimensionality reduction and a metric in the induced space. Lastly, in line with the recent trend in metric learning, we devise end-to-end learning of a generic deep network for metric learning using our derivation.
Author: Gayan Sadeep Jayasuman Hirimbura Matara Kankanamge
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
Several branches of modern computer vision research make heavy use of machine learning techniques. Machine learning for computer vision generally deals with Euclidean data. However, with the advances in the field, mathematical objects lying in non-Euclidean spaces that can be naturally modeled as Riemannian manifolds are now commonly encountered in computer vision. Therefore, machine learning methods on Riemannian manifolds has become an interesting area of computer vision research. Many Euclidean machine learning methods cannot be directly utilized on data lying in a Riemannian manifold. Generalizing Euclidean methods to Riemannian manifolds is not straightforward either due to differences in geometries. This thesis targets at solving this problem of learning on manifold-valued data, by performing kernel methods on Riemannian manifolds. More specifically, we aim to introduce a superior class of learning algorithms on manifold-valued data by proposing positive definite kernels on Riemannian manifolds and by designing improved kernel learning algorithms on them. We work on a number of Riemannian manifolds encountered in computer vision research, namely, the unit n-sphere, the Riemannian manifold of symmetric positive definite matrices, the Grassmann manifold and the shape manifold. A key component in kernel methods is the positive definite kernel employed. In the earlier chapters of the thesis we introduce positive definite kernels on these manifolds while giving rigorous proofs for their positive definiteness. Being able to define positive definite kernels on these manifolds enables us to use powerful Euclidean algorithms such as support vector machines and principle component analysis on manifold-valued data. This approach significantly reduces the complexities associated with learning on manifold-valued data while simultaneously yielding much better results. In the later chapters, we tackle a more advanced problem: automatically learning the optimal kernel on a manifold for a given computer vision task. The ability to learn the optimal kernel automatically eliminates the need to manually select kernels and the risk of using a sub-optimal kernel, which can significantly degrade the performance of kernel methods. We demonstrate applications of our algorithms on a variety of computer vision tasks such as pedestrian detection, object recognition, image-set recognition, segmentation, clustering, shape recognition and shape retrieval. Experimental evaluations towards the end of each chapter provide evidence that using kernel methods on manifolds achieves superior performance compared to state-of-the-art learning methods on manifold-valued data.
Author: Xavier Pennec
Publisher: Academic Press
ISBN: 0128147261
Category : Computers
Languages : en
Pages : 636
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Book Description
Over the past 15 years, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data. Riemannian Geometric Statistics in Medical Image Analysis is a complete reference on statistics on Riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. It provides an introduction to the core methodology followed by a presentation of state-of-the-art methods. Beyond medical image computing, the methods described in this book may also apply to other domains such as signal processing, computer vision, geometric deep learning, and other domains where statistics on geometric features appear. As such, the presented core methodology takes its place in the field of geometric statistics, the statistical analysis of data being elements of nonlinear geometric spaces. The foundational material and the advanced techniques presented in the later parts of the book can be useful in domains outside medical imaging and present important applications of geometric statistics methodology Content includes: The foundations of Riemannian geometric methods for statistics on manifolds with emphasis on concepts rather than on proofs Applications of statistics on manifolds and shape spaces in medical image computing Diffeomorphic deformations and their applications As the methods described apply to domains such as signal processing (radar signal processing and brain computer interaction), computer vision (object and face recognition), and other domains where statistics of geometric features appear, this book is suitable for researchers and graduate students in medical imaging, engineering and computer science. A complete reference covering both the foundations and state-of-the-art methods Edited and authored by leading researchers in the field Contains theory, examples, applications, and algorithms Gives an overview of current research challenges and future applications
Author:
Publisher:
ISBN:
Category : Computer vision
Languages : en
Pages : 146
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Book Description
The nonlinear nature of many compute vision tasks involves analysis over curved nonlinear spaces embedded in higher dimensional Euclidean spaces. Such spaces are known as manifolds and can be studied using the theory of differential geometry. In this thesis we develop two algorithms which can be applied over manifolds. The nonlinear mean shift algorithm is a generalization of the original mean shift, a popular feature space analysis method for vector spaces. Nonlinear mean shift can be applied to any Riemannian manifold and is provably convergent to the local maxima of an appropriate kernel density. This algorithm is used for motion segmentation with different motion models and for the filtering of complex image data. The projection based M-estimator is a robust regression algorithm which does not require a user supplied estimate of the scale, the level of noise corrupting the inliers. We build on the connections between kernel density estimation and robust M-estimators and develop data driven rules for scale estimation. The method can be generalized to handle heteroscedastic data and subspace estimation. The results of using pbM for affine motion estimation, fundamental matrix estimation and multibody factorization are presented. A new sensor fusion method which can handle heteroscedastic data and incomplete estimates of parameters is also discussed. The method is used to combine image based pose estimates with inertial sensors.
Author: Cuneyt Oncel Tuzel
Publisher:
ISBN:
Category : Computer vision
Languages : en
Pages : 156
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Book Description
Classical machine learning techniques provide effective methods for analyzing data when the parameters of the underlying process lie in a Euclidean space. However, various parameter spaces commonly occurring in computer vision problems violate this assumption. We derive novel learning methods for parameter spaces having Riemannian manifold structure and present several practical applications for scene analysis. The mean shift algorithm on Lie groups is a generalization of the mean shift procedure which is also an unsupervised learning technique for vector spaces. The derived procedure can be used to cluster data points which form a matrix Lie group. We present an application of the new algorithm for multiple 3D rigid motion estimation problem from noisy point correspondences in the presence of outliers. The approach performs simultaneous estimation of all the motions and does not require prior specification of the number of motion groups. We present a novel geometric framework to learn a supervised classification model for data points lying on a connected Riemannian manifold. The structure of the classifier is an additive model, where the weak learners are trained on the tangent spaces of the manifold. The derived algorithm is applied to pedestrian detection problem which is known to be among the hardest examples of the detection tasks. We describe a regression model where the response parameters form a Lie group. The model is utilized for affine tracking problem where the motion is estimated as a parameter of the image observations. We present generalization of the learning model to build an invariant object detector from an existing pose dependent detector. The proposed model can accurately detect objects in various poses, where the size of the search space is only a fraction compared to the existing detection methods. The other contributions of the thesis include a novel region descriptor and an online learning algorithm for estimating background statistics of a scene which are utilized for several challenging problems such as matching, tracking, texture classification and low frame rate tracking.
Author: C. Udriste
Publisher: Springer Science & Business Media
ISBN: 9401583900
Category : Mathematics
Languages : en
Pages : 365
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Book Description
The object of this book is to present the basic facts of convex functions, standard dynamical systems, descent numerical algorithms and some computer programs on Riemannian manifolds in a form suitable for applied mathematicians, scientists and engineers. It contains mathematical information on these subjects and applications distributed in seven chapters whose topics are close to my own areas of research: Metric properties of Riemannian manifolds, First and second variations of the p-energy of a curve; Convex functions on Riemannian manifolds; Geometric examples of convex functions; Flows, convexity and energies; Semidefinite Hessians and applications; Minimization of functions on Riemannian manifolds. All the numerical algorithms, computer programs and the appendices (Riemannian convexity of functions f:R ~ R, Descent methods on the Poincare plane, Descent methods on the sphere, Completeness and convexity on Finsler manifolds) constitute an attempt to make accesible to all users of this book some basic computational techniques and implementation of geometric structures. To further aid the readers,this book also contains a part of the folklore about Riemannian geometry, convex functions and dynamical systems because it is unfortunately "nowhere" to be found in the same context; existing textbooks on convex functions on Euclidean spaces or on dynamical systems do not mention what happens in Riemannian geometry, while the papers dealing with Riemannian manifolds usually avoid discussing elementary facts. Usually a convex function on a Riemannian manifold is a real valued function whose restriction to every geodesic arc is convex.
Author: Gabriel Peyré
Publisher: Now Publishers Inc
ISBN: 1601983964
Category : Computers
Languages : en
Pages : 213
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Book Description
Reviews the emerging field of geodesic methods and features the following: explanations of the mathematical foundations underlying these methods; discussion on the state of the art algorithms to compute shortest paths; review of several fields of application, including medical imaging segmentation, 3-D surface sampling and shape retrieval
Author: David Fleet
Publisher: Springer
ISBN: 9783319106045
Category : Computers
Languages : en
Pages : 854
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Book Description
The seven-volume set comprising LNCS volumes 8689-8695 constitutes the refereed proceedings of the 13th European Conference on Computer Vision, ECCV 2014, held in Zurich, Switzerland, in September 2014. The 363 revised papers presented were carefully reviewed and selected from 1444 submissions. The papers are organized in topical sections on tracking and activity recognition; recognition; learning and inference; structure from motion and feature matching; computational photography and low-level vision; vision; segmentation and saliency; context and 3D scenes; motion and 3D scene analysis; and poster sessions.
