1. Advanced Computer Graphics (Spring 2005) COMS 4162, Lecture 5: Image Processing 2: Warping Ravi Ramamoorthi http://www.cs.columbia.edu/~cs4162

2. To Do  Assignment 1, Due Feb 15.  All info in this and previous lecture  Enough to implement second half of ass  Questions/difficulties so far in doing assignment?

3. Outline  Image Warping  General ideas  Resampling filters  Antialiased shift and resize  Not well covered in textbook Many slides courtesy Tom Funkhouser

4. Image Warping

5. Example Mappings

8. Forward Warping/Mapping  Iterate over source, sending pixels to destination

9. Forward Warping: Problems  Iterate over source, sending pixels to destination  Same source pixel may map to multiple dest pixels  Some dest pixels may have no corresponding source  Holes in reconstruction  Must splat etc.

10. Inverse Warping/Mapping  Iterate destination, finding pixels from source

11. Inverse Warping/Mapping  Iterate over dest, finding pixels from source  Non-integer evaluation source image, resample  May oversample source  But no holes  Simpler, better than forward mapping

12. Outline  Image Warping  General ideas  Resampling filters  Antialiased shift and resize  Not well covered in textbook Many slides courtesy Tom Funkhouser

13. Filtering or Resampling  Compute weighted sum of pixel neighborhood  Weights are normalized values of kernel function  Equivalent to convolution at samples with kernel  Find good filters using ideas of last lecture

14. Filters for Assignment Implement 3 filters  Nearest neighbor or point sampling  Hat filter (linear or triangle)  Mitchell cubic filter (form in assigments). This is a good finite filter that approximates ideal sinc but without ringing or infinite width. Alternative is gaussian Construct 2D filters by multiplying 1D filters

15. Filtering Methods Comparison

16. Inverse Warping/Mapping  Iterate destination, finding pixels from source

17. Outline  Image Warping  General ideas  Resampling filters  Antialiased shift and resize  Not well covered in textbook Many slides courtesy Tom Funkhouser

18. Antialiased Shift Shift image based on (fractional) x and y  Check for integers, treat separately  Otherwise convolve/resample with kernel/filter:

19. Antialiased Scale Magnification Magnify image  Interpolate between orig. samples to evaluate frac vals  Do so by convolving/resampling with kernel/filter:

20. Antialiased Scale Minification checkerboard.bmp 300x300: point sample checkerboard.bmp 300x300: Mitchell

21. Antialiased Scale Minification Minify (reduce size of) image  Similar in some ways to mipmapping for texture maps  We use fat pixels of size 1/s, with new size sw (w is original size, s is scale factor < 1).  Each fat pixel must integrate over corresponding region in original image using the kernel f.

22. Antialiased Scale Minification Minify (reduce size of) image  Similar in some ways to mipmapping for texture maps  We use fat pixels of size 1/s, with new size sw (w is original size, s is scale factor < 1).  Each fat pixel must integrate over corresponding region in original image using the kernel f.  Hence, center filter at a/s, width 1/s times normal width  Caveats: floating point, parametric form for filter

23. To Do  Assignment 1, Due Feb 15.  All info in this and previous lecture  Enough to implement second half of ass  Questions/difficulties so far in doing assignment?