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Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES

Lecture



In ch. 2 addressed issues related to the mathematical description of continuous images. This chapter provides methods for the formal presentation of discrete images using both deterministic and statistical models.

5.1. ACTIONS WITH VECTORS AND MATRIXES

This section briefly discusses the mathematical operations occurring in the text that are performed with vectors and matrices. The rigorous conclusion and proofs of the theorems and propositions given below can be found in the literature [1–5].

Vector

Column vector Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is a collection of elements Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES where Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES arranged in a vertical column

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.1)

Line vector Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is an ordered collection of elements Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES where Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES arranged as a horizontal line

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.2)

In the book, bold lowercase letters will usually denote column vectors. The row vector will be denoted as a transposed column vector:

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.3)

Matrix

Matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is a collection of elements Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES Where Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES arranged in the form of rows and columns of a two-dimensional table

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.4)

Symbol Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES denotes a zero matrix, all elements of which are equal to zero. The diagonal matrix is ​​a square matrix (when Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES ), all elements of which are not lying on the main diagonal are zero, i.e. Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , if a Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . The unit matrix denoted by the symbol Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , there is a diagonal matrix, all the diagonal elements of which are equal to one. The index at the symbol of the unit matrix indicates its dimensions; Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES denotes a unit size matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . Matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES can be divided into blocks (submatrices) Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES :

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.5)

Addition of matrices

The sum of two matrices Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES defined only in the case when both matrices have the same size. Matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES - the sum of matrices Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES has dimensions Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and its elements Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES .

Matrix multiplication

The product of two matrices Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is defined only when the number of columns of the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES equal to the number of rows of the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . When multiplying the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES to matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES matrix is ​​obtained Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES whose elements are determined by equality

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.6)

When multiplying the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES on the scalar Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES matrix is ​​obtained Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES whose elements Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES .

Matrix inversion

If a Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is a square matrix, the inverse matrix and denoted by Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , has the following properties: Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . If matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES exists, then the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is called non-singular (non-degenerate). Otherwise, it is called special (degenerate). If some matrix has an inverse, then this inverse matrix is ​​unique. The inverse of the relative inverse matrix coincides with the original matrix, i.e.

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.7)

If matrices Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES non-singular then

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.8)

If the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES non-singular, but a scalar Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES then

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.9)

The inversion of singular square matrices and non-square matrices will be considered in Ch. 8. Inverse matrix relative to block square matrix

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , (5.1.10)

can be represented as

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.11)

provided that the matrices Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES are not special.

Matrix transposition

When transposing a matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size matrix is ​​formed Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES which is denoted by Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . Matrix rows Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES match the columns, and the columns match the rows of the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . For any matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.12)

If a Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES then the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES called symmetric. For any matrices Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.13)

If the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES non-singular, then the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES also non-singular and

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.14)

Direct matrix product

Left direct product of the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES to matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is a size matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.15)

Similarly, you can define the right direct product. This book will use only the left direct product. Direct works Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES may vary among themselves. Below are the properties of the operations of multiplication, addition, transposition and inversion of the direct product of matrices:

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , (5.1.16)

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , (5.1.17)

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , (5.1.18)

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , (5.1.18)

Trace matrix

Square matrix trace Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES equal to the sum of its diagonal elements and is denoted as

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.20)

If a Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES - square matrices, then

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.21)

The trace of the direct product of two matrices is

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.22)

Vector norm

Euclidean norm of a vector Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES called a scalar, defined as

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.23)

Matrix norm

Euclidean norm matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES called a scalar, defined as follows:

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.24)

Rank matrix

Matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES has rank Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES if the largest of all its square non-invalid blocks has size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . The concept of rank is used when reversing matrices. If matrices Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES non-singular, and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is an arbitrary matrix, then

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.25)

Rank matrices Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES satisfies inequalities

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , (5.1.26a)

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.26b)

Rank of matrix sum Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES satisfies inequality

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.27)

Dot product of vectors

Scalar product of vectors Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES and Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is a scalar

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.28)

or

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.29)

Matrix product of vectors

Matrix product of vector Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES by vector Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is the matrix

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES (5.1.30)

Where Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES .

Quadratic form

Quadratic vector shape Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES size Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES is a scalar

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , (5.1.31)

Where Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES - size matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . Often the matrix Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES take symmetrical.

Vector derivative

Derivative from scalar product Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES by Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES there is

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES , (5.1.32)

and the derivative of the scalar product Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES by vector Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES equals

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.33)

Derivative of a quadratic form Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES by Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES there is

Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES . (5.1.34)


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