Image Geometry and Geometric Transformation

Geometric distortion Geometric distortion is a form of geometric transformation Image may subject to geometric distortion due to imperfect imaging devices or viewing conditions http://ccrs.nrcan.gc.ca/resource/tutor/fundam/chapter2/10_e.php

Geometric Distortion Assume the original image f(x,y) is subject to geometric distortion yielding g(x’,y’) Coordination transformation function need 8 or more points to find {ci; 1  i  8} (x,y) (x’,y’)

Gray Level Interpolation Spatial transformations establish a correspondence between a point (x’, y’) in the distorted image g(x’,y’) and original image f(x,y). To correct the geometric transformation, one needs to estimate gray values of f(x,y), If x’ and y’ are integers, then If x’ and y’ are fraction numbers, but fall within the border of the original image, then interpolation will be needed to find

Nearest Neighbor Gray Level Interpolation

Bilinear Interpolation Estimate the value of (=g(x’,y’)) using four nearest neighbors when x’ and y’ are fractional numbers. Let Substitute g(x1,y1), g(x1,y2), g(x2,y1), g(x2,y2) into above equation and solve for a, b, c, d. It’s 4 equations and 4 unknowns. (x1,y2) (x2,y2) (x’,y’) (x1,y1) (x2,y1)

Example 5.16 An image with 25 regularly spaced tiepoints. Geometric distortion by rearranging the tiepoints Distorted image, nearest neighbor interpolation Restored image, NN Distorted image, bilinear transformation Restored image, BT

Another Example Original image Distorted image using bilinear transform Difference between a and b Geometrically restored image using bilinear transform for gray level interpolation