The document describes a streaming multigrid solver for solving Poisson's equation on large images. It develops a multigrid method using a B-spline finite element basis that can efficiently process images in a streaming fashion using only a small window of image rows in memory at a time. The method achieves accurate solutions to Poisson's equation on gigapixel images in only 2 V-cycles by leveraging the temporal locality of the multigrid algorithm.

1. Streaming Multigrid for Gradient-Domain Operations on Large Images Michael Kazhdan , Johns Hopkins University Hugues Hoppe , Microsoft Research SIGGRAPH 08

10. Standard multigrid V-cycle

17. Gradient-domain image processing  Poisson Equation

18. ▽ u x 0 x 1 x i x N u 0 u 1 u i u N h u i+1

19. Δ U x i+1 x i-1 x i x N u i-1 u N h u i+1 u i

20. Δ U= f

21. Lu=f U u 1,1 u 1,2 u 1,3 u 2,1 u 2,2 u 2,3 u 3,1 u 3,2 u 3,3

27. 2D stencils of the multigrid 2D Restriction 2D Prolongation 2D Relaxation (Laplacian)

29. Standard multigrid V-cycle f l-1 =R l l -r l u l =P l l-1 u P l-1 +u R l L lmin u lmin =f lmin Base solution

33. 2 nd order B-spline 1D case

34. 2 nd order B-spline 2D case

39. Full data pipeline for gradient-domain processing

43. (n, v) Plot of the rms and max errors vs. # of multigrid V-cycles 2 nd -order element give the fastest convergence !

44. (n, k=2, v=1)

50. Image stitching 19,588 x 4,457 (87 MB) panorama form 9 photos Copy the image gradients and solving the Poisson eq.

51. Image stitching SM: streaming multigrid solver QT: quadtree (AT) solver of Agarwala [07]

52. Tone-mapping (HDR  normal tone) before after

53. Tone-mapping (HDR  normal tone) Stream multigraid is the first one to solve Poisson eq in time that is linear on the # of pixels

54. GB stitching and tone-mapping May not capture the true scene contrast

55. GB stitching and tone-mapping