Linear Algebra for Pattern Processing: Projection, Singular Value Decomposition, and Pseudoinverse

Kanatani, Kenichi

  • 出版商: Morgan & Claypool
  • 出版日期: 2021-04-30
  • 定價: $1,560
  • 售價: 9.0$1,404
  • 語言: 英文
  • 頁數: 155
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 1636391079
  • ISBN-13: 9781636391076
  • 相關分類: 線性代數 Linear-algebra
  • 立即出貨 (庫存=1)

商品描述

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.

商品描述(中文翻譯)

線性代數是許多科學領域中最基礎的基礎之一,大多數線性代數的教科書都是由數學家撰寫的。然而,這本書專門針對模式信息處理的學生和研究人員,分析信號(如圖像)並探索計算機視覺和計算機圖形應用。作者本身就是這個領域的研究人員。

這種模式信息處理涉及大量的數據,這些數據由高維向量和矩陣表示。在這裡,線性代數的作用不僅僅是對大規模向量和矩陣進行數值計算。事實上,數據處理通常伴隨著“幾何解釋”。例如,我們可以將一個數據集視為與另一個數據集“正交”,並定義它們之間的“距離”,或者引用幾何關係,如將某些數據“投影”到某個空間中。這些幾何概念不僅有助於我們在直觀的術語中對抽象的高維空間進行心理可視化,還能引導我們找到哪種處理方法適合哪種目標。

首先,我們介紹線性空間的“投影”概念,並以投影的術語描述“光譜分解”、“奇異值分解”和“伪逆”。作為它們的應用,我們討論了同時線性方程組的最小二乘解和不一定是正定的向量隨機變量的協方差矩陣。我們還討論了將子空間擬合到點數據和在高維度中對矩陣進行因式分解,與運動圖像分析相關。最後,我們介紹了計算機視覺中從三個攝像頭視圖重建一個點的三維位置的應用,以說明線性代數在處理帶有噪聲的數據中的作用。這本書預計能幫助模式信息處理的學生和研究人員深入理解線性代數的幾何概念。