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8.4. SOLUTION OF SYSTEMS OF LINEAR EQUATIONS

Lecture



Linear equation system

  8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS (8.4.1)

Where   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS - size matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS can be considered as a system of   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS equations with   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS unknown. Three cases are possible:

1. The system of equations has a unique solution.   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS , for which   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS .

2. The system of equations satisfies several solutions.

3. The system of equations does not have an exact solution.

If the system of equations has at least one solution, then it is called joint, and otherwise - incompatible. Systems of equations that do not have solutions are often obtained in the study of physical systems, when the vector   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS describes the sequence of measurements of observable quantities, which by assumption are the result of the impact of some driving force inaccessible for direct observation, represented by the vector   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS . Matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS is obtained by mathematical modeling of the characteristics of a real system for which the vector   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS is the output signal. When correcting (restoring) images vector   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS usually represents the original image, vector   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS - blurred image, and the matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS describes a discrete mathematical model of the process leading to image blurring. Since the matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS and vector   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS usually determined with some error, it may turn out that this vector does not correspond to any of the possible impact vectors   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS .

We now consider the question of the existence of solutions of the system of equations   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS . From the structure of the system of equations it is clear that a solution will exist if and only if the vector   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS can be obtained by a linear combination of matrix columns   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS . In this case, the vector is said to be   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS lies in the column space of the matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS . A more rigorous formulation of the condition for the existence of a solution is the following [4]: ​​system of equations   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS has a solution if and only if for the matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS there is a conditional inverse matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS satisfying the equation   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS .

So, for the existence of a solution, it is necessary that when mapping from the space of the observed images into the space of the original images (using the conventionally inverse matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS ) and back (using the matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS ) again obtained the vector of observable quantities   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS . If the system of equations is underdetermined (   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS ), then the solution exists when the rank of the matrix   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS equals   8.4.  SOLUTION OF SYSTEMS OF LINEAR EQUATIONS i.e. the total number of rows. In all other cases, including the overdetermined system of equations, the existence of a solution must be checked.


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Digital image processing

Terms: Digital image processing