## Linear Algebra for Pattern Processing Author: Kenichi Kanatani
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Size: 20.34 MB
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Pages : 155
Linear algebra is one of the most basic foundations of a wide range of scientific domains, and most textbooks of linear algebra are written by mathematicians. However, this book is specifically intended to students and researchers of pattern information processing, analyzing signals such as images and exploring computer vision and computer graphics applications. The author himself is a researcher of this domain. Such pattern information processing deals with a large amount of data, which are represented by high-dimensional vectors and matrices. There, the role of linear algebra is not merely numerical computation of large-scale vectors and matrices. In fact, data processing is usually accompanied with "geometric interpretation." For example, we can think of one data set being "orthogonal" to another and define a "distance" between them or invoke geometric relationships such as "projecting" some data onto some space. Such geometric concepts not only help us mentally visualize abstract high-dimensional spaces in intuitive terms but also lead us to find what kind of processing is appropriate for what kind of goals. First, we take up the concept of "projection" of linear spaces and describe "spectral decomposition," "singular value decomposition," and "pseudoinverse" in terms of projection. As their applications, we discuss least-squares solutions of simultaneous linear equations and covariance matrices of probability distributions of vector random variables that are not necessarily positive definite. We also discuss fitting subspaces to point data and factorizing matrices in high dimensions in relation to motion image analysis. Finally, we introduce a computer vision application of reconstructing the 3D location of a point from three camera views to illustrate the role of linear algebra in dealing with data with noise. This book is expected to help students and researchers of pattern information processing deepen the geometric understanding of linear algebra.
RELATED BOOKS Language: en
Pages: 155
Authors: Kenichi Kanatani
Categories:
Type: BOOK - Published: 2021-04-30 - Publisher:

Linear algebra is one of the most basic foundations of a wide range of scientific domains, and most textbooks of linear algebra are written by mathematicians. However, this book is specifically intended to students and researchers of pattern information processing, analyzing signals such as images and exploring computer vision and computer graphics applications. The author himself is a researcher of this domain. Such pattern information processing deals with a large amount of data, which are represented by high-dimensional vectors and matrices. There, the role of linear algebra is not merely numerical computation of large-scale vectors and matrices. In fact, data processing is usually accompanied with "geometric interpretation." For example, we can think of one data set being "orthogonal" to another and define a "distance" between them or invoke geometric relationships such as "projecting" some data onto some space. Such geometric concepts not only help us mentally visualize abstract high-dimensional spaces in intuitive terms but also lead us to find what kind of processing is appropriate for what kind of goals. First, we take up the concept of "projection" of linear spaces and describe "spectral decomposition," "singular value decomposition," and "pseudoinverse" in terms of projection. As their applications, we discuss least-squares solutions of simultaneous linear equations and covariance matrices of probability distributions of vector random variables that are not necessarily positive definite. We also discuss fitting subspaces to point data and factorizing matrices in high dimensions in relation to motion image analysis. Finally, we introduce a computer vision application of reconstructing the 3D location of a point from three camera views to illustrate the role of linear algebra in dealing with data with noise. This book is expected to help students and researchers of pattern information processing deepen the geometric understanding of linear algebra. Language: en
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Authors: Sohail A. Dianat, Eli Saber
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Type: BOOK - Published: 2020-05-23 - Publisher: Springer Nature

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Type: BOOK - Published: 1983 - Publisher:

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Pages: 199
Authors: Yen-Wei Chen, Lakhmi C. Jain
Categories: Technology & Engineering
Type: BOOK - Published: 2014-04-07 - Publisher: Springer

This research book provides a comprehensive overview of the state-of-the-art subspace learning methods for pattern recognition in intelligent environment. With the fast development of internet and computer technologies, the amount of available data is rapidly increasing in our daily life. How to extract core information or useful features is an important issue. Subspace methods are widely used for dimension reduction and feature extraction in pattern recognition. They transform a high-dimensional data to a lower-dimensional space (subspace), where most information is retained. The book covers a broad spectrum of subspace methods including linear, nonlinear and multilinear subspace learning methods and applications. The applications include face alignment, face recognition, medical image analysis, remote sensing image classification, traffic sign recognition, image clustering, super resolution, edge detection, multi-view facial image synthesis. Language: en
Pages: 607
Authors: El?bieta P?kalska, Robert P. W. Duin
Categories: Computers
Type: BOOK - Published: 2005 - Publisher: World Scientific Publishing Company Incorporated

1. Introduction. 1.1. Recognizing the pattern. 1.2. Dissimilarities for representation. 1.3. Learning from examples. 1.4. Motivation of the use of dissimilarity representations. 1.5. Relation to kernels. 1.6. Outline of the book. 1.7. In summary -- 2. Spaces. 2.1. Preliminaries. 2.2. A brief look at spaces. 2.3. Generalized topological spaces. 2.4. Generalized metric spaces. 2.5. Vector spaces. 2.6. Normed and inner product spaces. 2.7. Indefinite inner product spaces. 2.8. Discussion -- 3. Characterization of dissimilarities. 3.1. Embeddings, tree models and transformations. 3.2. Tree models for dissimilarities. 3.3. Useful transformations. 3.4. Properties of dissimilarity matrices. 3.5. Linear embeddings of dissimilarities. 3.6. Spatial representation of dissimilarities. 3.7. Summary -- 4. Learning approaches. 4.1. Traditional learning. 4.2. The role of dissimilarity representations. 4.3. Classification in generalized topological spaces. 4.4. Classification in dissimilarity spaces. 4.5. Classification in pseudo-Euclidean spaces. 4.6. On generalized kernels and dissimilarity spaces. 4.7. Discussion -- 5. Dissimilarity measures. 5.1. Measures depending on feature types. 5.2. Measures between populations. 5.3. Dissimilarity measures between sequences. 5.4. Information-theoretic measures. 5.5. Dissimilarity measures between sets. 5.6. Dissimilarity measures in applications. 5.7. Discussion and conclusions -- 6. Visualization. 6.1. Multidimensional scaling. 6.2. Other mappings. 6.3. Examples : getting insight into the data. 6.4. Tree models. 6.5. Summary -- 7. Flirther data exploration. 7.1. Clustering. 7.2. Intrinsic dimension. 7.3. Sampling density. 7.4. Summary -- 8. One-class classifiers. 8.1. General issues. 8.2. Domain descriptors for dissimilarity representations. 8.3. Experiments. 8.4. Conclusions -- 9. Classification. 9.1. Proof of principle. 9.2. Selection of the representation set : the dissimilarity space approach. 9.3. Selection of the representation set : the embedding approach. 9.4. On corrections of dissimilarity measures. 9.5. A few remarks on a simulated missing value problem. 9.6. Existence of zero-error dissimilarity-based classifiers. 9.7. Final discussion -- 10. Combining. 10.1. Combining for one-class classification. 10.2. Combining for standard two-class classification. 10.3. Classifier projection space. 10.4. Summary -- 11. Representation review and recommendations. 11.1. Representation review. 11.2. Practical considerations -- 12. Conclusions and open problems. 12.1. Summary and contributions. 12.2. Extensions of dissimilarity representations. 12.3. Open questions Language: en
Pages: 514
Authors: Andrew R. Webb
Categories: Mathematics
Type: BOOK - Published: 2003-07-25 - Publisher: John Wiley & Sons

Statistical pattern recognition is a very active area of study and research, which has seen many advances in recent years. New and emerging applications - such as data mining, web searching, multimedia data retrieval, face recognition, and cursive handwriting recognition - require robust and efficient pattern recognition techniques. Statistical decision making and estimation are regarded as fundamental to the study of pattern recognition. Statistical Pattern Recognition, Second Edition has been fully updated with new methods, applications and references. It provides a comprehensive introduction to this vibrant area - with material drawn from engineering, statistics, computer science and the social sciences - and covers many application areas, such as database design, artificial neural networks, and decision support systems. * Provides a self-contained introduction to statistical pattern recognition. * Each technique described is illustrated by real examples. * Covers Bayesian methods, neural networks, support vector machines, and unsupervised classification. * Each section concludes with a description of the applications that have been addressed and with further developments of the theory. * Includes background material on dissimilarity, parameter estimation, data, linear algebra and probability. * Features a variety of exercises, from 'open-book' questions to more lengthy projects. The book is aimed primarily at senior undergraduate and graduate students studying statistical pattern recognition, pattern processing, neural networks, and data mining, in both statistics and engineering departments. It is also an excellent source of reference for technical professionals working in advanced information development environments. Language: en
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Categories: Computers
Type: BOOK - Published: 2012-12-06 - Publisher: Physica

Symbolic processing has limitations highlighted by the symbol grounding problem. Computational processing methods, like fuzzy logic, neural networks, and statistical methods have appeared to overcome these problems. However, they also suffer from drawbacks in that, for example, multi-stage inference is difficult to implement. Deep fusion of symbolic and computational processing is expected to open a new paradigm for intelligent systems. Symbolic processing and computational processing should interact at all abstract or computational levels. For this undertaking, attempts to combine, hybridize, and fuse these processing methods should be thoroughly investigated and the direction of novel fusion approaches should be clarified. This book contains the current status of this attempt and also discusses future directions. Language: en
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Type: BOOK - Published: 2019-08-30 - Publisher: SIAM

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Type: BOOK - Published: 2021-10-21 - Publisher: Academic Press

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