The Shape of Things

The Shape of Things
Author: Shawn W. Walker
Publisher: SIAM
Total Pages: 156
Release: 2015-12-17
Genre: Mathematics
ISBN: 1611973953

Many things around us have properties that depend on their shape?for example, the drag characteristics of a rigid body in a flow. This self-contained overview of differential geometry explains how to differentiate a function (in the calculus sense) with respect to a ?shape variable.? This approach, which is useful for understanding mathematical models containing geometric partial differential equations (PDEs), allows readers to obtain formulas for geometric quantities (such as curvature) that are clearer than those usually offered in differential geometry texts. Readers will learn how to compute sensitivities with respect to geometry by developing basic calculus tools on surfaces and combining them with the calculus of variations. Several applications that utilize shape derivatives and many illustrations that help build intuition are included.

Variational Analysis in Sobolev and BV Spaces

Variational Analysis in Sobolev and BV Spaces
Author: Hedy Attouch
Publisher: SIAM
Total Pages: 794
Release: 2014-10-02
Genre: Mathematics
ISBN: 1611973481

This volume is an excellent guide for anyone interested in variational analysis, optimization, and PDEs. It offers a detailed presentation of the most important tools in variational analysis as well as applications to problems in geometry, mechanics, elasticity, and computer vision.

Variational Methods for Shape and Image Registrations

Variational Methods for Shape and Image Registrations
Author: Rachid Fahmi
Publisher:
Total Pages: 342
Release: 2008
Genre: Image processing
ISBN:

One of most important image analysis tools that greatly benefits from the process of registration, and which will be addressed in this dissertation, is the image segmentation. This dissertation addresses the registration problem from a variational point of view, with more focus on shape registration. First, a novel variational framework for global-to-local shape registration is proposed. The input shapes are implicitly represented through their signed distance maps. A new Sum-of-Squared-Differences ( SSD ) criterion which measures the disparity between the implicit representations of the input shapes, is introduced to recover the global alignment parameters. This new criteria has the advantages over some existing ones in accurately handling scale variations. In addition, the proposed alignment model is less expensive computationally. Complementary to the global registration field, the local deformation field is explicitly established between the two globally aligned shapes, by minimizing a new energy functional. This functional incrementally and simultaneously updates the displacement field while keeping the corresponding implicit representation of the globally warped source shape as close to a signed distance function as possible. This is done under some regularization constraints that enforce the smoothness of the recovered deformations. The overall process leads to a set of coupled set of equations that are simultaneously solved through a gradient descent scheme. Several applications, where the developed tools play a major role, are addressed throughout this dissertation. For instance, some insight is given as to how one can solve the challenging problem of three dimensional face recognition in the presence of facial expressions. Statistical modelling of shapes will be presented as a way of benefiting from the proposed shape registration framework. Second, this dissertation will visit the shape-based segmentation problem. The piece-wise constant Chan and Vese segmentation models [1] are chosen as the underlying segmentation models and it will be shown how the proposed global shape registration technique can serve in enhancing the segmentation results of an input image when some prior knowledge of shapes is integrated in the underlying segmentation framework. The resulting paradigm allows the recovery of a segmentation map that is in accordance with the shape prior model as well as an affine transformation between this map and the model. Furthermore, it can deal with noisy, occluded and missing or corrupted data. The classical way of solving the shape-based segmentation problems within a level set framework is by directly solving the underlying Euler-Lagrange equations using a gradient descent scheme. This is very computationally expensive given the non-linear parabolic nature of the corresponding PDE 's. To overcome these difficulties, a fast algorithm is designed and implemented to solve both the two-phase and the multi-phase shape-based segmentation problem. This algorithm exploits the fact that only the sign of the level set function, not its value, is needed to evolve the segmenting interface. The integration of multiple selective shape priors and the segmentation into multiple regions has never been addressed before. Third, a new image/volume non-rigid registration approach based on scale space and level set theories, will be introduced. This contribution is the fruit of a collaborative effort with two other members of the CVIP Lab. New feature descriptors are built as voxel signatures using scale space theory. These descriptors are used to capture the global motion of the imaged object. Local deformations are modelled through an evolution process of equi-spaced closed curves/surfaces (iso-contours/surfaces) which are generated using fast marching level sets and are matched based on a cross correlation measure between neighboring voxels. A novel Finite Element ( FE )-based approach is developed to validate the performance of the proposed image registration techniques. Both two and three dimensional tissue deformations are simulated using the FE method. The registration accuracy with respect to the FE simulations is assessed by co-registering the deformed images with the original ones and comparing the recovered displacement field with the bio-mechanically simulated ones.

Gradient Flows

Gradient Flows
Author: Luigi Ambrosio
Publisher: Springer Science & Business Media
Total Pages: 333
Release: 2008-10-29
Genre: Mathematics
ISBN: 376438722X

The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.

Shape Reconstruction from Apparent Contours

Shape Reconstruction from Apparent Contours
Author: Giovanni Bellettini
Publisher: Springer
Total Pages: 385
Release: 2015-02-25
Genre: Mathematics
ISBN: 3662451913

Motivated by a variational model concerning the depth of the objects in a picture and the problem of hidden and illusory contours, this book investigates one of the central problems of computer vision: the topological and algorithmic reconstruction of a smooth three dimensional scene starting from the visible part of an apparent contour. The authors focus their attention on the manipulation of apparent contours using a finite set of elementary moves, which correspond to diffeomorphic deformations of three dimensional scenes. A large part of the book is devoted to the algorithmic part, with implementations, experiments, and computed examples. The book is intended also as a user's guide to the software code appcontour, written for the manipulation of apparent contours and their invariants. This book is addressed to theoretical and applied scientists working in the field of mathematical models of image segmentation.

Variational Methods in Image Segmentation

Variational Methods in Image Segmentation
Author: Jean-Michel Morel
Publisher: Springer Science & Business Media
Total Pages: 257
Release: 2012-12-06
Genre: Mathematics
ISBN: 1468405675

This book contains both a synthesis and mathematical analysis of a wide set of algorithms and theories whose aim is the automatic segmen tation of digital images as well as the understanding of visual perception. A common formalism for these theories and algorithms is obtained in a variational form. Thank to this formalization, mathematical questions about the soundness of algorithms can be raised and answered. Perception theory has to deal with the complex interaction between regions and "edges" (or boundaries) in an image: in the variational seg mentation energies, "edge" terms compete with "region" terms in a way which is supposed to impose regularity on both regions and boundaries. This fact was an experimental guess in perception phenomenology and computer vision until it was proposed as a mathematical conjecture by Mumford and Shah. The third part of the book presents a unified presentation of the evi dences in favour of the conjecture. It is proved that the competition of one-dimensional and two-dimensional energy terms in a variational for mulation cannot create fractal-like behaviour for the edges. The proof of regularity for the edges of a segmentation constantly involves con cepts from geometric measure theory, which proves to be central in im age processing theory. The second part of the book provides a fast and self-contained presentation of the classical theory of rectifiable sets (the "edges") and unrectifiable sets ("fractals").

Discrete Geometry for Computer Imagery

Discrete Geometry for Computer Imagery
Author: Michel Couprie
Publisher: Springer
Total Pages: 496
Release: 2019-03-19
Genre: Computers
ISBN: 3030140857

This book constitutes the thoroughly refereed proceedings of the 21st IAPR International Conference on Discrete Geometry for Computer Imagery, DGCI 2019, held in Marne-la-Vallée, France, in March 2019. The 38 full papers were carefully selected from 50 submissions. The papers are organized in topical sections on discrete geometric models and transforms; discrete topology; graph-based models, analysis and segmentation; mathematical morphology; shape representation, recognition and analysis; and geometric computation.

The Proceedings of the International Conference on Sensing and Imaging

The Proceedings of the International Conference on Sensing and Imaging
Author: Ming Jiang
Publisher: Springer
Total Pages: 421
Release: 2018-09-18
Genre: Technology & Engineering
ISBN: 3319916599

This book collects ​a number of papers presented at the International Conference on Sensing and Imaging, which was held at Chengdu University of Information Technology on June 5-7, 2017. Sensing and imaging is an interdisciplinary field covering a variety of sciences and techniques such as optics, electricity, magnetism, heat, sound, mathematics, and computing technology. The field has diverse applications of interest such as sensing techniques, imaging, and image processing techniques. This book will appeal to professionals and researchers within the field.