1. Image Geometry and Geometric Transformation

2. Geometric distortion
Geometric distortion is a form of geometric transformation
Image may subject to geometric distortion due to imperfect imaging devices or viewing conditions

3. 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’)

4. 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

5. Nearest Neighbor Gray Level Interpolation

6. 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)

7. 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

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