Image Warps and Halftoning

1. Image Warps and Halftoning

2. Image Warping • Scale • Rotate • Warp • Basically, we have to move image pixels • This is done by mapping the pixels from the source to the destination image (actually backwards) then resampling warp Source image Destination image

3. Image Warping • Overview • Mapping • Forward • Inverse • Resampling • Point sampling • Triangle filter (bilinear interpolation) • Gaussian filter

4. Mapping • Forward mapping: • Inverse mapping: v y u x

5. v y u x Mapping Examples • Scale (separately in horizontal and vertical)

6. v y u x Mapping Examples • Rotate counterclockwise θ degrees

7. Mapping Examples • General functions of u and v: swirl fisheye rain

8. Forward Mapping for (int u = 0; u < umax; u++) { for (int v = 0; v < vmax; v++) { float x = fx(u,v); float y = fy(u,v); dst (x,y) = src(u,v); } } source image destination image (u,v) f (x,y)

9. Multiple source pixels map to same destination No source pixels map to destination Forward Mapping • Iterate over source image • Need to resample destination • Generate one sample for each pixel • Complex to figure out where the nearest samples are for each pixel Rotate 30o

10. Inverse Mapping for (int x = 0; x < xmax; x++) { for (int y = 0; v < ymax; y++) { float u = fx-1(x,y); float v = fy-1(x,y); dst (x,y) = src(u,v); } } source image destination image (u,v) f-1 (x,y)

11. Inverse Mapping • Iterate over destination image • Need to resample source • Generate source sample for each destination pixel • May oversample source, but oversampling is simpler than in forward case Rotate -30o

12. Resampling • Need to evaluate source image at arbitrary (u,v) • (u,v) not generally integer coordinates • 3 methods • Point sampling • Triangle filtering • Gaussian filtering source image destination image (u,v) f-1 (x,y)

13. Point Sampling • Just find closest source pixel to each destination pixel int iu = round(u); int iv = round(v); dst(x,y) = src(iu, iv); • Simple, but causes aliasing Rotate -30o

14. (u1,v2) (u2,v2) a (u,v) b (u1,v1) (u2,v1) Triangle Filtering • Convolve four closest source pixels to desired (u,v) with triangle filter (bilinear interpolation) • Bilinear interpolation: • a = lerp(src(u1,v2),src(u2,v2)) • b = lerp(src(u1,v1),src(u2,v1)) • dst(x,y) = lerp(a,b)

15. Gaussian Filtering • Compute weighted sum of pixel neighborhood • Weights are normalized values of Gaussian function (u,v) w

16. Filter Comparison • Tradeoffs • Fast but aliased (point sampling) • Slow and blurry (Gaussian) Point sampling Bilinear (triangle) Gaussian

17. resample_src(u,v,w); w Image Warping Examples • Using reverse mapping and Gaussian filtering • for (int x = 0; x < xmax; x++) { for (int y = 0; v < ymax; y++) { float u = fx-1(x,y); float v = fy-1(x,y); dst (x,y) = resample_src(u,v,w); } } source image destination image (u,v) f-1 (x,y)

18. Scale • Scale(src,dst,sx,sy) • float w = max(1/sx,1/sy); • for (int x = 0; x < xmax; x++) { for (int y = 0; v < ymax; y++) { float u = x/sx ; float v = x/sy ; dst (x,y) = resample_src(u,v,w); } }

19. Rotate • Rotate(src,dst,θ) • for (int x = 0; x < xmax; x++) { for (int y = 0; v < ymax; y++) { float u = x*cos(-θ)- y*sin(-θ); float v = x*sin(-θ)+ y*cos(-θ); dst (x,y) = resample_src(u,v,w); } }

20. Swirl • Swirl(src,dst,θ,cx,cy) • for (int x = 0; x < xmax; x++) { • for (int y = 0; v < ymax; y++) • float u = (x-cx)*cos(-θ*|x-cx|)- (y-cy)*sin(-θ*|y-cy|)+cx; float v = (x-cx)*sin(-θ*|x-cx|)+ (y-cy)*cos(-θ*|y-cy|)+cy; dst (x,y) = resample_src(u,v,w); } }

21. Quantization Artifacts • Errors due to limited intensity resolution • Why? • Frame buffers only have so many bits per channel • Physical display devices have a limited dynamic range, especially hard copy devices

22. Uniform Quantization I(x) I I(x) P(x) x P(x) 4 bits/pixel

23. Uniform Quantization • Images with increasing bits per pixel 1 bit 2 bits 4 bits 8 bits

24. Dealing with Quantization • Halftoning techniques • Classical, or brute force halftoning • Dithering methods • Error Diffusion • Other techniques

25. Classical Halftoning • First Variant • First, use dots of varying sizes to represent intensities • Size of dot proportional to intensity I(x) P(x)

26. Classical Halftoning Newspaper image NYT 9/21/99

27. Classical Halftoning • What if your display device can’t display produce multiple dot sizes? • Use cluster of pixels • Number of “on” pixels in cluster proportional to intensity • Trades spatial resolution for intensity resolution

28. Dithering • Algorithms to distribute errors among pixels • Reduce objectionable regular artifacts • Replace them with noise Original – 8 bits/pixel Uniform quantization – 1 bit/pixel Floyd-Steinberg error diffusion – 1 bit/pixel

29. I I(x) P(x) x Random Dither • Randomize quantization error • Errors appear as noise I I(x) P(x) x

30. Random Dither Original – 8 bits/pixel Uniform quantization – 1 bit/pixel Random dither – 1 bit/pixel

31. Ordered Dither • Patterned errors that try to minimize regular artifacts • Recursively defined “Dither Matrix” stores pattern of thresholds

32. Ordered Dither

33. Ordered Dither Original – 8 bits/pixel Random dither – 1 bit/pixel Ordered dither – 1 bit/pixel

34. Floyd-Steinberg Error Diffusion • Quantization errors are spread over neighboring pixels to the right and below

35. Floyd-Steinberg Error Diffusion Original – 8 bits/pixel Random dither – 1 bit/pixel Ordered dither – 1 bit/pixel Floyd-Steinberg error diffusion – 1 bit/pixel