Arithmetic and Geometric Transformations (Chapter 2) CS474/674 – Prof. Bebis

Addition Useful for combining information between two images: 0 <= α <= 1

Averaging (see Example 2.5 on page 75) Image quality can be improved by averaging several images together (e.g., very useful in astronomy) Note: images must be registered!

Subtraction Useful for “change” detection.

Geometric Transformations Transformation applied on the coordinates of the pixels (i.e., relocate pixels). A geometric transformation has the general form (x,y) = T{(v,w)} where (v,w) are the original pixel coordinates and (x,y) are the transformed pixel coordinates.

Geometric Transformations affine transformation y=v sinθ + w cosθ

Forward mapping Transformed pixel coordinates might not lie within the bounds of the image. Transformed pixel coordinates can be non-integer. Certain locations in the transformed image might not have a corresponding pixel in the input image. No one-to-one correspondence!

Forward mapping (cont’) An example of holes due to image rotation, implemented using the forward transformation.

Inverse Mapping To guarantee that a value is generated for every pixel in the output image, we must consider each output pixel in turn and use the inverse mapping to determine the position in the input image. To assign intensity values to these locations, we need to use some form of intensity interpolation.

Interpolation (cont’d) Zero-order interpolation: nearest-neighbor

Interpolation (cont’d) First-order interpolation: average

Interpolation (cont’d) Bilinear interpolation I(x,y) = ax + by + cxy + d The 4 unknowns (a,b,c,d) can be determined from 4 equations formed by the 4 nearest neighbors.

Interpolation (cont’d) Bilinear interpolation

Interpolation (cont’d) Bicubic interpolation It involves the 16 nearest neighbors of a point (i.e., 4x4 window). The 16 unknowns a ij can be determined from sixteen equations formed by the 16 nearest neighbors.

Examples: Interpolation

Image Registration Goal: align two or more images of the same scene. How: estimate a transformation that aligns the two images.

Image Registration (cont’d) Under certain assumptions, an affine transformation can be used to align two images. –There are 6 unknowns (i.e., t 11, t 12, t 21, t 22, t 31, t 32 ) –We need at least 6 equations. –Three correspondences are enough, more are better. Unknowns? Equations? Correspondences?