1. Advanced Computer Graphics (Spring 2006) COMS 4162, Lecture 5: Image Processing 2: Warping Ravi Ramamoorthi http://www.cs.columbia.edu/~cs4162

2. To Do  Assignment 1, Due Feb 16.  All info in this and previous lecture  Enough to implement second half of assn  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. A note on notation  This lecture uses (u,v) for warped location in the source image (or old coordinates) and (u', v') for integer coordinates, and (x,y) for new coordinates  Most of the homework assignment uses (x,y) for old integer coordinates and (a,b) for new coordinates. The warped location is not written explicitly, but is implicit in the evaluation of the filter f

6. Example Mappings

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

10. 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.

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

12. 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

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

14. 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 (normalized) filters h using ideas of last lecture (u,v) width=2 s=0 ; for (u ' = u-width ; u ' <= u+width; u ' ++) for (v' = v-width ; v ' <= v+width; v ' ++) s += h (u ' - u, v' - v) src (u ', v ') ;

15. Inverse Warping/Mapping  Iterate destination, finding pixels from source Filter is really square with width w, not circle

16. A note on notation  This lecture uses (u,v) for warped location in the source image (or old coordinates) and (u', v') for integer coordinates, and (x,y) for new coordinates  Most of the homework assignment uses (x,y) for old integer coordinates and (a,b) for new coordinates. The warped location is not written explicitly, but is implicit in the evaluation of the filter  This lecture uses h for filter (so as not to confuse with mapping f), while assignment uses f  Assignment has some notational misprints, corrected here.

17. 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

18. Filtering Methods Comparison

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

20. Antialiased Shift Shift image based on (fractional) s x and s y  Check for integers, treat separately  Otherwise convolve/resample with kernel/filter: In notation of assignment (note slight misprint RHS should be (x,y))

21. Antialiased Scale Magnification Magnify image (scale s or γ > 1) [read s as γ in assn]  Interpolate between orig. samples to evaluate frac vals  Do so by convolving/resampling with kernel/filter:  Treat the two image dimensions independently (diff scales) In notation of assignment

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

23. Antialiased Scale Minification Minify (reduce size of) image  Similar in some ways to mipmapping for texture maps  We use fat pixels of size 1/γ, with new size γ*orig size (γ is scale factor < 1).  Each fat pixel must integrate over corresponding region in original image using the filter kernel. In notation of assignment

24. Antialiased Scale Minification Minify (reduce size of) image  Similar in some ways to mipmapping for texture maps  We use fat pixels of size 1/γ, with new size γ*orig size (γ is scale factor < 1).  Each fat pixel must integrate over corresponding region in original image using the filter kernel.  Hence, filter width 1/ γ normal width  Requires renormalization to work properly  Caveats: floating point, parametric form for filter

25. To Do  Assignment 1, Due Feb 16.  All info in this and previous lecture  Enough to implement second half of ass  Questions/difficulties so far in doing assignment?