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.
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)
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Digital image processing
Terms: Digital image processing