Geometric Transformation

Geometric transformations are widely used for image registration and the removal of geometric distortion. Common applications include construction of mosaics, geographical mapping, stereo and video.

In special transformation each point [x,y] is mapped to new point [x’,y’] in new coordinate system. Mapping from (x, y) to (u, v) coordinates. A digital image array has an implicit grid that is mapped to discrete points in the new domain. These points may not fall on grid points in the new domain.

An affine transformation is any transformation that preserves linearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation).

In general, an affine transformation is a composition of rotations, translations, magnifications, and shears.

This can be expressed as a matrix notation, corresponding of 3 dimensional vector. For this purpose we are adding a scale dimension to original x and y dimensions of input point.

q = Tp

Where p is input pixel coordination, T is transformation matrix (3×3) and q is new point in space.

To handle the mismatch of point grid, with computed position, we will have to calculate new position with either the nearest neighbor or interpolation. Simplest is to pick closes pixel to calculated position. In other cases, you can use bilinear or bicubic interpolation. Both can be found on the internet.

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    Examples of geometric transformations

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