國立高雄大學圖資館 |

Language: English

Back

Computational Analysis of Deformable...

Rensselaer Polytechnic Institute.

Computational Analysis of Deformable Manifolds: From Geometric Modeling to Deep Learning.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Computational Analysis of Deformable Manifolds: From Geometric Modeling to Deep Learning.
Author:	Schonsheck, Stefan C.
Published:	Ann Arbor : ProQuest Dissertations & Theses, 2020
Description:	137 p.
Notes:	Source: Dissertations Abstracts International, Volume: 82-05, Section: B.
Notes:	Advisor: Lai, Rongjie.
Contained By:	Dissertations Abstracts International82-05B.
Subject:	Applied mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28001226
ISBN:	9798684681127

Computational Analysis of Deformable Manifolds: From Geometric Modeling to Deep Learning.
Schonsheck, Stefan C.

Computational Analysis of Deformable Manifolds: From Geometric Modeling to Deep Learning. - Ann Arbor : ProQuest Dissertations & Theses, 2020 - 137 p.

Source: Dissertations Abstracts International, Volume: 82-05, Section: B.

Thesis (Ph.D.)--Rensselaer Polytechnic Institute, 2020.

This item must not be sold to any third party vendors.

Leo Tolstoy opened his monumental novel Anna Karenina with the now famous words: Happy families are all alike; every unhappy family is unhappy in its own way.A similar notion also applies to mathematical spaces: Every flat space is alike; every unflat space is unflat in its own way. However, rather than being a source of unhappiness, we will show that the diversity of non-flat spaces provides a rich area of study. The genesis of the so-called ’big data era’ and the proliferation of social and scientific databases of increasing size has led to a need for algorithms that can efficiently process, analyze and, even generate high dimensional data. However, the curse of dimensionality leads to the fact that many classical approaches do not scale well with respect to the size of these problems. One technique to avoid some of these ill-effects is to exploit the geometric structure of coherent data. In this thesis, we will explore geometric methods for shape processing and data analysis. More specifically, we will study techniques for representing manifolds and signals supported on them through a variety of mathematical tools including, but not limited to, computational differential geometry, variational PDE modeling and deep learning. First, we will explore non-isometric shape matching through variational modeling. Next, we will use ideas from parallel transport on manifolds to generalize convolution and convolutional neural networks to deformable manifolds. Finally, we conclude by proposing a novel auto-regressive model for capturing the intrinsic geometry and topology of data. Throughout this work, we will use the idea of computing correspondences as a though-line to both motivate our work and analyze our results. One of the advantages of working in this manner is that questions which arise from very specific problems will have far reaching consequences. There are many deep connections between concise models, harmonic analysis, geometry and learning that have only started to emerge in the past few years, and the consequences will continue to shape these fields for many years to come. Our goal in this work is to explore these connections and develop some useful tools for shape analysis, signal processing and representation learning. 

ISBN: 9798684681127Subjects--Topical Terms:

377601
Applied mathematics.
Subjects--Index Terms:

Computational geometry

Computational Analysis of Deformable Manifolds: From Geometric Modeling to Deep Learning.
LDR:03396nmm a2200337 4500 001 594584
005 20210521101701.5
008 210917s2020 ||||||||||||||||| ||eng d
020 $a 9798684681127
035 $a (MiAaPQ)AAI28001226
035 $a AAI28001226
040 $a MiAaPQ $c MiAaPQ
100 1 $a Schonsheck, Stefan C. $3 886612
245 1 0 $a Computational Analysis of Deformable Manifolds: From Geometric Modeling to Deep Learning.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2020
300 $a 137 p.
500 $a Source: Dissertations Abstracts International, Volume: 82-05, Section: B.
500 $a Advisor: Lai, Rongjie.
502 $a Thesis (Ph.D.)--Rensselaer Polytechnic Institute, 2020.
506 $a This item must not be sold to any third party vendors.
520 $a Leo Tolstoy opened his monumental novel Anna Karenina with the now famous words: Happy families are all alike; every unhappy family is unhappy in its own way.A similar notion also applies to mathematical spaces: Every flat space is alike; every unflat space is unflat in its own way. However, rather than being a source of unhappiness, we will show that the diversity of non-flat spaces provides a rich area of study. The genesis of the so-called ’big data era’ and the proliferation of social and scientific databases of increasing size has led to a need for algorithms that can efficiently process, analyze and, even generate high dimensional data. However, the curse of dimensionality leads to the fact that many classical approaches do not scale well with respect to the size of these problems. One technique to avoid some of these ill-effects is to exploit the geometric structure of coherent data. In this thesis, we will explore geometric methods for shape processing and data analysis. More specifically, we will study techniques for representing manifolds and signals supported on them through a variety of mathematical tools including, but not limited to, computational differential geometry, variational PDE modeling and deep learning. First, we will explore non-isometric shape matching through variational modeling. Next, we will use ideas from parallel transport on manifolds to generalize convolution and convolutional neural networks to deformable manifolds. Finally, we conclude by proposing a novel auto-regressive model for capturing the intrinsic geometry and topology of data. Throughout this work, we will use the idea of computing correspondences as a though-line to both motivate our work and analyze our results. One of the advantages of working in this manner is that questions which arise from very specific problems will have far reaching consequences. There are many deep connections between concise models, harmonic analysis, geometry and learning that have only started to emerge in the past few years, and the consequences will continue to shape these fields for many years to come. Our goal in this work is to explore these connections and develop some useful tools for shape analysis, signal processing and representation learning. 
590 $a School code: 0185.
650 4 $a Applied mathematics. $3 377601
653 $a Computational geometry
653 $a Deep learning
653 $a Differential equations
653 $a Variational modeling
690 $a 0364
710 2 $a Rensselaer Polytechnic Institute. $b Mathematics. $3 766181
773 0 $t Dissertations Abstracts International $g 82-05B.
790 $a 0185
791 $a Ph.D.
792 $a 2020
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28001226