Transformation method

In the task four linear and three nonlinear methods of transformation are realized. For a linear transformation, in contrast to non-linear there is no nonlinear distortion of the coordinates, ie straight line remains straight.

The unknown coefficients are determined by solving the system of normal equations. If the number of points greater than the required minimum, the system of equations is solved by the method of least squares.

Offset.

It concerns to linear type of transformation. For calculation of transformation parameters it is enough one ground control point. Following formulas for transformation of coordinates are used:

Offset, rotate.

It concerns to linear type of transformation. For calculation of transformation parameters it is necessary minimum 2 points. Following formulas for transformation of coordinates are used:

Offset, rotate, scale

It concerns to linear type of transformation. For calculation of transformation parameters it is necessary a minimum of 2 points. Following formulas for transformation of coordinates are used:

Affine transformation

It concerns to linear type of transformation. For calculation of transformation parameters it is necessary minimum 3 points. Following formulas for transformation of coordinates are used:

Polynomial transformation

It concerns to nonlinear type of transformation. Following formulas for transformation of coordinates are used:

n – a degree of a polynomial

In the task may be done implementation of a polynomial transformation with number of coefficients from 4 to 21. The minimum number of points equal to the number of polynomial coefficients.

When choosing the type of transformation «Polynomial (automatic tuning)» the number of polynomial coefficients will be equal to the number of measured points.

Number of coefficients in the mode «Polynomial (manual setting)» is set in the bookmark «Advanced».

Polynomial surface of distortions has no sharp changes, however significant deformations of a surface of distortions are possible, especially outside location of ground control points. Therefore, it is not recommended to use polynomial transformation, if ground control points are located non-uniformly or the region of ground control points does not cover completely area of transformation.

Linear, non-linear rubber sheet

Unlike previous methods the transformed image will be exactly combined in the locations of measured points.

It concerns to nonlinear type of transformation. To calculate parameters of transformation must be at least 3 points. Transformation is carried out on the parameters defined for each triangle of Delaunay triangulation. For linear transformation - this are the coefficients of affine transformation, for nonlinear - the coefficients of polynomial transformations. Transformation by type of a linear rubber sheet is carried out more quickly than the nonlinear, but sometimes has changes of a surface of distortions on border of triangles.

In the case when the location of ground control points does not fully cover the area of transformation, there are areas in which the coefficients of transformation are not defined. For the solving this problem virtual points on border of the transformation area are created, values of nonlinear distortions in which are set equal to nonlinear distortions of the nearest real ground control point. Management of using virtual points is located on a bookmark Advanced.