1. GEOMETRIC OPERATIONS

2. Transformations and directions Affine (linear) transformations Translation, rotation and scaling Non linear (Warping transformations) source to target (coordinate for each input image pixel) I(x,y)  O(x´,y´) target to source (coordinate for each output image pixel) O(x,y)  I(x´,y´) BETTER, WHY?

3. Translation Target to source x´ = x -T x y´ = y -T y

4. Translation

5. Rotation Target to source x´ = x cos(  - y sin(  ) y´ = x sin(  + y cos(  )

6. Rotation

7. Scaling Target to source x´ = x / S x y´ = y / S y

8. Scaling

9. General affine equations Linear equations in general form x´ = ( x cos(  + y sin(  )) S x +T x = ( S x cos(  x + (S x sin(  )) y +T x = a 2 x + a 1 y + a 0 y´ = (-x sin(  + y cos(  )) S y +T y = (-S y sin(  x + (S y cos(  )) y + T y = b 2 x + b 1 y + b 0

10. Warping Increasing higher order terms increases accuracy x 2 y 2 x 3 y 3

11. Warping and reference points Warping transformed Original with 4 reference points

12. Number of reference points 3 for first order warping 6 for second order warping 10 for third order warping

13. Number of reference points Example: correction of a lense degradation by a third order equation x´= a 9 x 3 + a 8 y 3 + a 7 x 2 y + a 6 y 2 x + a 5 x 2 + a 4 y 2 + a 3 x + a 2 y + a 1 xy + a 0 y´= b 9 x 3 + b 8 y 3 + b 7 x 2 y + b 6 y 2 x + b 5 x 2 + b 4 y 2 + b 3 x + b 2 y +b 1 xy +b 0

14. Lens distortion filter

15. Curve bend filter

16. Sampling and resizing In downsizing, first low pass filtering to avoid aliasing (Nyquist theorem) Interpolation in resizing (up- and downsizing )

17. Low pass filtering and resizing

18. Group discussion Discuss the application areas of geometric operations

19. Interpolation methods Nearest neighbor Bilinear Bicubic

20. Interpolation Nearest neighbor * 5 6 4545 Address: 5.3, 4.4 Select pixel: column 5, row 4

21. Interpolation, Bilinear * 5 6 4545 a b c d Address: 5.3, 4.4 Weighted average of neighbour pixels: cd = (1-0.3)*c + 0.3 *d ab = (1-0.3)*a + 0.3 *b abcd= (1-0.4)ab + 0.4*cd abcd is the final pixel value

22. Interpolation, Bicubic Second order interpolation In 16*16 neighbourhood, sometimes 64*64 Optimizing moving images

23. Nearest neighbour and bilinear Nearest neighborBilinear

24. Anti-aliasing filter

25. Extra information abour geometric corrections in remote sensing http://www.geo- informatie.nl/courses/grs20306/ course/Schedule/Geometric- correction-RS-new.pdf