Mathematical Principles in image processing (Outline of Part1) and mathematical principles part1

Mathematical Principles in image processing (Outline of Part1) and mathematical principles part1

Mathematical Principles in image processing (Outline of Part1)

The development of digital image processing technology has a high requirement on the mathematical foundation. In some emerging new methods, the dazzling mathematical derivation has discouraged many people who want to study in depth. A regular science and engineering student generally has a mathematical foundation including calculus, linear algebra, and probability theory. However, when analyzing the principles of some image processing algorithms, it seems that there is no way to start. The problems actually involved can be attributed to the following reasons: 1) calculus, linear algebra, and probability theory are very important mathematical foundations, but it shows that not all of these courses are directly applied to image processing algorithms; 2) when you separate image processing from mathematics, in fact, they did not try to establish the relationship between them. 3) the foundation of some new methods or so-called high algorithms has exceeded the basic fields discussed in the above three mathematical courses, this involves partial differential equations, variational methods, complex variable functions, real-time variable functions, functional analysis, and so on. 4) if you are not a student of mathematics, you need to learn all the content mentioned above, the workload is too huge, so I'm afraid it's hard to take care of it.

In my spare time, I have summarized some mathematical principles necessary for Image Processing Based on my learning and practice. These contents mainly involve calculus, vector analysis, field theory, functional analysis, partial differential equations, complex variable functions, and variational methods. Linear Algebra and probability theory I think it is more basic, so I did not take its income. I have summarized, summarized, and extracted some of the knowledge points that are most likely to be known when studying image processing, then they were reorganized in a step-by-step manner. The application of these mathematical knowledge is discussed based on specific image processing algorithms. Thus establishing a bridge between mathematical knowledge and image processing.

This part of content is mainly a summary of my daily research and study, and I did not plan to publish it. After all, this Topic is still a little small and difficult. However, I have previously captured a small part of it and sent it to the Internet. Some readers have shown great interest and asked where to find other parts. So when talking about sharing, I will be more willing to post it on my blog gradually, and it is a systematic and ordered release. There is not much content in all. There are only six Chapters in total, about 200 or 300 pages. As the beginning of this part of content, the following describes the general principles ". I will gradually publish all the content to my blog for your reference. However, due to my limited personal energy, this work obviously cannot be achieved overnight. If you are interested in a sub-title, you can leave a message at the bottom of the blog. I will adjust the publishing priority as appropriate. Happy sharing and research!

Chapter 1 Essential mathematical Basics
1.1 limit and Its Application
1.1.1 limit of Series
1.1.2 convergence of Series
1.1.3 Function Limit
1.1.4 extreme applications
1.2 differential Mean Value Theorem
1.2.1 role Mean Theorem
1.2.2 Laplace Mean Value Theorem
1.2.3 cosine Mean Value Theorem
1.2.4 Taylor Formula
1.3 vector algebra and Field Theory
1.3.1 Newton-levenitz Formula
1.3.2 inner and outer Products
1.3.3 derivative and Gradient
1.3.4 curve credits
1.3.5 Green Formula
1.3.6 points and Path-independent conditions
1.3.7 curved points
1.3.8 Gaussian formula and divergence
1.3.9 stoks formula and Rotation
1.4 Fourier series expansion
1.4.1 concept of function term series
1.4.2 properties of function item Series
1.4.3 concept of Fourier Series
1.4.4 evolution of Fourier Transformation
1.4.5 convolution Theorem and Its proof
References in this Chapter

Chapter 2 further mathematical content
2.1 preliminary revariant Function Theory
2.1.1 parsing functions
2.1.2 reset points
2.1.3 basic theorem
2.1.4 series expansion
2.2 leberger points Theory
2.2.1 leberger measure of a point set
2.2.2 measurable functions and their properties
2.2.3 leberger points
2.2.4 limit theorem of Integral Sequence
2.3 Functional and abstract Space
2.3.1 Linear Space
2.3.2 distance space
2.3.3 fan Space
2.3.4 Panama Space
2.3.5 Inner Product Space
2.3.6 Hilbert Space
2.3.7 sobov Space
2.4 From function to Calculus
2.4.1 understand functional concepts
2.4.2 concept of variation
2.4.3 basic equations of the variational method
2.4.4 understand Hamilton Principle
2.4.5 variational under equality constraints
2.4.6 he fixed point theorem
2.4.7 bounded variance function space

References in this Chapter

Chapter 1 ubiquitous Gaussian distribution
3.1 Convolution Integral and neighbor Processing
3.1.1 concept of Convolution Integral
3.1.2 template and neighborhood Processing
3.1.3 Gaussian smoothing of images
3.2 edge detection and Differential Operators
3.2.1 Hamilton operator
3.2.2 LAPLACE OPERATOR
3.2.3 Gaussian-LAPLACE OPERATOR
3.2.4 Gaussian difference operator
3.3 smooth edge processing
3.3.1 bilateral Filtering Algorithm Application
3.3.2 heterosexual diffusion filtering
3.3.3 method based on total variation
3.4 Application of Mathematical Physical Equations
3.4.1 derivation of Poisson Equation
3.4.2 Poisson editing of an image
3.4.3 discretization Numerical Solution
3.5 multi-scale space and its construction
3.5.1 construction of Gaussian filtering and multi-scale space
3.5.2 Scale Space Based on the spread of the opposite sex
References in this Chapter



Chapter 2 theoretical basis of Image Encoding
4.1 rate distortion function
4.2 Shannon Bottom Border
4.3 non-memory Gaussian Source
4.4 Gaussian source with memory
The significance of Gaussian distribution in 4.5
References in this Chapter

Chapter 2 subband encoding and Wavelet Transformation
5.1 basic principles of sub-band Encoding
5.1.1 digital signal processing Basics
5.1.2 multi-sampling rate signal processing
5.1.3 Image Information sub-band decomposition
5.2 Hal functions and their transformations
5.2.1 Hal Function Definition
5.2.2 properties of Hal Functions
5.2.3 matrix and Transformation
5.2.4 two-dimensional discrete linear transformation
5.2.5 halky Functions
5.2.6 Hal Transformation
5.3 wavelet and its mathematical principles
5.3.1 history of Wavelet
5.3.2 understand the concept of Wavelet
5.3.3 multi-resolution analysis
5.3.4 construction of wavelet functions
5.3.5 wavelet sequence Expansion
5.3.6 Discrete Wavelet Transformation
5.3.7 Continuous Wavelet Transformation
5.3.8 allowable conditions and basic features of Wavelet
5.4 Fast Wavelet Transform Algorithm
5.4.1 fast wavelet positive transformation
5.4.2 fast wavelet Inverse Transformation
5.4.3 Image Wavelet Transformation
References in this Chapter


Chapter 2 orthogonal transformation and Image Compression
6.1 Fourier Transformation
6.1.1 Fourier transformation in signal processing
6.1.2 Fourier transformation of digital images
6.1.3 Fast Fourier Transform Algorithm
6.2 discrete cosine transformation
6.2.1 basic concepts and mathematical descriptions
6.2.2 fast algorithm for discrete cosine transformation
6.2.3 significance and Application of discrete cosine transformation
6.3 power change
6.3.1 multiplication function
6.3.2 Discrete Fourier transformation and its fast algorithm
6.3.3 application of wallx Transformation
6.4 Carol South-loy Transformation
6.4.1 some essential basic concepts
6.4.2 derivation of principal component Transformation
6.4.3 Implementation of principal component Transformation
6.4.4 Image Compression Based on K-L Transformation
References in this Chapter

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