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lec06-resampling2.pptx
- 2. Detour: Time complexity of convolution β’ Image is w x h β’ Filter is k x k β’ Every entry takes O(k2) operations β’ Number of output entries: β’ (w+k-1)(h+k-1) for full β’ wh for same β’ Total time complexity: β’ O(whk2)
- 3. Optimization: separable filters β’ basic alg. is O(r2): large filters get expensive fast! β’ definition: w(x,y) is separable if it can be written as: β’ Write u as a k x 1 filter, and v as a 1 x k filter β’ Claim:
- 4. Separable filters u1 u2 u3 v1 v2 v3 * u1 u2 u3 u3 u2 u1 u1v1
- 5. Separable filters u1 u2 u3 v1 v2 v3 * u1 u2 u3 u3 u2 u1 u1v1 u1v2
- 6. Separable filters u1 u2 u3 v1 v2 v3 * u1 u2 u3 u3 u2 u1 u1v1 u1v2 u1v3
- 7. Separable filters u1 u2 u3 v1 v2 v3 * u1 u2 u3 u3 u2 u1 u1v1 u1v2 u1v3 u2v1
- 8. Separable filters u1 u2 u3 v1 v2 v3 * u1 u2 u3 u3 u2 u1 u1v1 u1v2 u1v3 u2v1 u2v2
- 9. Separable filters u1 u2 u3 v1 v2 v3 * u1 u2 u3 u3 u2 u1 u1v1 u1v2 u1v3 u2v1 u2v2 u2v3
- 10. Separable filters u1 u2 u3 v1 v2 v3 * u1 u2 u3 u3 u2 u1 u1v1 u1v2 u1v3 u2v1 u2v2 u2v3 u3v1
- 11. Separable filters u1 u2 u3 v1 v2 v3 * u1 u2 u3 u3 u2 u1 u1v1 u1v2 u1v3 u2v1 u2v2 u2v3 u3v1 u3v2
- 12. Separable filters u1 u2 u3 v1 v2 v3 * u1 u2 u3 u3 u2 u1 u1v1 u1v2 u1v3 u2v1 u2v2 u2v3 u3v1 u3v2 u3v3 w
- 13. Separable filters β’ Time complexity of original : O(whk2) β’ Time complexity of separable version : O(whk)
- 14. Image resizing
- 15. Why do we need to talk about resizing? β’ Need to zoom in to a region to get more details β’ Can we get more details? Louis Daguerre, 1838
- 16. Why do we need to talk about resizing? β’ Far away objects appear small, nearby objects appear larger β’ Need to recognize objects at multiple scales β’ Resizing images to same size helps recognition
- 17. Why is resizing hard? β’ E.g, consider reducing size by a factor of 2 β’ Simple solution: subsampling β’ Example: subsampling by a factor of 2
- 18. Why is resizing hard? β’ Dropping pixels causes problems
- 21. Aliasing in time
- 22. Aliasing in time
- 23. Why does aliasing happen? β’ We βmissβ things between samples β’ High frequency signals might appear as low frequency signals β’ Called βaliasingβ Β© Kavita Bala, Computer Science, Cornell University
- 24. Letβs look at it mathematically β’ Fourier basis signals β’ π΅π π = π π2πππ π , time period N/k, frequency k/N β’ Suppose we sample every P pixels. Then we are only looking at the values β’ π΅π π , π΅π 2π , β¦ , π΅π ππ , β¦ β’ Consider two basis vectors π΅π and π΅π+π/π β’ π΅π+π/π ππ = π π2π π+ π π ππ π = π π2ππππ π +π2ππ( π π )( π π ) = π π2ππππ π +π2ππ = π π2ππππ π = π΅π(ππ)
- 25. Letβs look at it mathematically β’ π΅π ππ = π΅π+ π π ππ β’ The two basis vectors look identical if we sample every P pixels β’ Our eyes see the lower frequency: aliasing β’ How do we avoid aliasing? β’ Remove high frequencies that may be aliased β’ Sample at a high enough rate (lower P)
- 26. Avoiding aliasing β’ Suppose image has limited high frequency components β’ Fourier transform of x is X β’ Suppose X is non-zero only for βππππ₯ to ππππ₯ 0 N/2 -N/2 -kmax kmax
- 27. Avoiding aliasing β’ When sampling once every P pixels, βππππ₯ is indistinguishable from β ππππ₯ + π/π β’ If image has both coefficients, they will get mixed β’ So we want that ππππ₯ < βππππ₯ + π π β 2ππππ₯ < π π 0 N/2 -N/2 -kmax kmax -kmax+N/P
- 28. Nyquist Sampling Theorem β’ 2ππππ₯ < π π β’ Basis π΅π has a time period of π π or frequency of π π β’ 2ππππ₯ < π π β 2ππππ₯ π < 1 P β 2ππππ₯ < ππ πππππ β’ Need to sample at at least twice the rate of the highest frequency component
- 29. Subsampling the image β’ First smooth the image to remove high frequency components β’ How should we smooth? β’ One simple answer: blur with Gaussian β’ Recall that Fourier transform of Gaussian is Gaussian β’ Choose sigma so that frequencies above ππ πππππ 2 get killed
- 30. Subsampling before and after smoothing Before After
- 31. Image sub-sampling 1/4 (2x zoom) 1/16 (4x zoom) Why does this look so crufty? Aliasing! 1/2 Source: S. Seitz
- 32. Subsampling images correctly G 1/4 G 1/8 Gaussian 1/2 β’ Solution: filter the image, then subsample Source: S. Seitz
- 33. Subsampling with Gaussian pre-filtering G 1/4 G 1/8 Gaussian 1/2 β’ Solution: filter the image, then subsample Source: S. Seitz
- 34. Compare with... 1/4 (2x zoom) 1/8 (4x zoom) 1/2 Source: S. Seitz
- 35. Low-pass filtering β’ Convolving with a Gaussian = remove high frequencies β’ βLow-passβ filtering: low frequencies βpassβ through filter, high frequencies donβt β’ Identity β Low-pass filtered image = βHigh-pass filteringβ
- 36. Gaussian pre-filtering β’ Solution: filter the image, then subsample blur F0 H * subsample blur subsample β¦ F1 F1 H * F2 F0
- 37. blur F0 H * subsample blur subsample β¦ F1 F1 H * F2 F0 Gaussian pyramid
- 38. Gaussian pyramids [Burt and Adelson, 1983] β’ In computer graphics, a mip map [Williams, 1983] Gaussian Pyramids have all sorts of applications in computer vision Source: S. Seitz
- 39. Gaussian pyramids - Searching over scales
- 40. Gaussian pyramids - Searching over scales
- 41. 2 ) * ( 2 3 ο― ο½ gaussian G G 1 G The Gaussian Pyramid High resolution Low resolution Image ο½ 0 G 2 ) * ( 0 1 ο― ο½ gaussian G G 2 ) * ( 1 2 ο― ο½ gaussian G G 2 ) * ( 3 4 ο― ο½ gaussian G G blur blur blur blur
- 42. Gaussian pyramid and stack Source: Forsyth
- 43. Memory Usage β’ What is the size of the pyramid? 4 3