Chapter 5. MATHEMATICAL DESCRIPTION OF DISCRETE IMAGES

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 size is a collection of elements where arranged in a vertical column

(5.1.1)

Line vector size is an ordered collection of elements where arranged as a horizontal line

(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:

(5.1.3)

Matrix

Matrix size is a collection of elements Where and arranged in the form of rows and columns of a two-dimensional table

(5.1.4)

Symbol denotes a zero matrix, all elements of which are equal to zero. The diagonal matrix is ​​a square matrix (when ), all elements of which are not lying on the main diagonal are zero, i.e. , if a . The unit matrix denoted by the symbol , 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; denotes a unit size matrix . Matrix can be divided into blocks (submatrices) :

. (5.1.5)

Addition of matrices

The sum of two matrices defined only in the case when both matrices have the same size. Matrix - the sum of matrices and has dimensions and its elements .

Matrix multiplication

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

(5.1.6)

When multiplying the matrix on the scalar matrix is ​​obtained whose elements .

Matrix inversion

If a is a square matrix, the inverse matrix and denoted by , has the following properties: and . If matrix exists, then the matrix 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.

(5.1.7)

If matrices and non-singular then

(5.1.8)

If the matrix non-singular, but a scalar then

. (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

, (5.1.10)

can be represented as

(5.1.11)

provided that the matrices and are not special.

Matrix transposition

When transposing a matrix size size matrix is ​​formed which is denoted by . Matrix rows match the columns, and the columns match the rows of the matrix . For any matrix

. (5.1.12)

If a then the matrix called symmetric. For any matrices and

(5.1.13)

If the matrix non-singular, then the matrix also non-singular and

. (5.1.14)

Direct matrix product

Left direct product of the matrix size to matrix size is a size matrix

. (5.1.15)

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

, (5.1.16)

, (5.1.17)

, (5.1.18)

, (5.1.18)

Trace matrix

Square matrix trace size equal to the sum of its diagonal elements and is denoted as

. (5.1.20)

If a and - square matrices, then

. (5.1.21)

The trace of the direct product of two matrices is

. (5.1.22)

Vector norm

Euclidean norm of a vector size called a scalar, defined as

. (5.1.23)

Matrix norm

Euclidean norm matrix size called a scalar, defined as follows:

. (5.1.24)

Rank matrix

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

. (5.1.25)

Rank matrices and satisfies inequalities

, (5.1.26a)

. (5.1.26b)

Rank of matrix sum and satisfies inequality

. (5.1.27)

Dot product of vectors

Scalar product of vectors and size is a scalar

(5.1.28)

or

. (5.1.29)

Matrix product of vectors

Matrix product of vector size by vector size is the matrix

(5.1.30)

Where .

Quadratic form

Quadratic vector shape size is a scalar

, (5.1.31)

Where - size matrix . Often the matrix take symmetrical.

Vector derivative

Derivative from scalar product by there is

, (5.1.32)

and the derivative of the scalar product by vector equals

. (5.1.33)

Derivative of a quadratic form by there is

. (5.1.34)