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Gaussian KD-Tree for Fast High-Dimensional Filtering

Gaussian KD-Tree for Fast High-Dimensional Filtering. A. Adams, N. Gelfand , J. Dolson , and M. Levoy , Stanford University, SIGGRAPH 2009. Edge-Preserving Filtering. Noise Suppression . Detail Enhancement . High Dynamic Range Imaging .

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Gaussian KD-Tree for Fast High-Dimensional Filtering

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  1. Gaussian KD-Tree for Fast High-Dimensional Filtering A. Adams, N. Gelfand, J. Dolson, and M. Levoy, Stanford University, SIGGRAPH 2009.

  2. Edge-Preserving Filtering Noise Suppression Detail Enhancement High Dynamic Range Imaging

  3. Edge-Preserving Filtering for Image Analysis Input Image Base Image Detail Image

  4. Edge-Preserving Vs. Edge-Blurring Input Image Edge-Preserving Base Image Edge-Blurring Base Image

  5. Edge-Preserving Vs. Edge-Blurring Halo Artifacts Edge-Preserving Enhanced Image Edge-Blurring Enhanced Image

  6. Gaussian Filtering

  7. Gaussian Filtering

  8. Bilateral Filtering Intensity y x Output Input Space Weight Range Weight Space Weight Range Weight

  9. Bilateral Filtering Intensity y x Output Input Bilateral Weight Space Weight Range Weight

  10. Bilateral Filtering Input Image Gaussian: σp = 12 Bilateral: σp = 12, σc = 0.15

  11. Computational Complexity of Bilateral Filtering • O(n2d) • Image Size: n • Maximum Filter Size: n • Dimension: d • High Computational Complexity Intensity y x Input

  12. Novel Methods • Bilateral Grid • J. Chen, S. Paris, and F. Durand, “Real-time edgeaware image processing with the bilateral grid,” ACM Transactions on Graphics (Proc. SIGGRAPH 07). • Gaussian KD-Tree • A. Adams, N. Gelfand, J. Dolson, and M. Levoy, “Gaussian KD-Trees for Fast High-Dimensional Filtering,” ACM Transactions on Graphics (Proc. SIGGRAPH 09).

  13. High-Dimensional Filtering Intensity y x

  14. A Two-Dimensional Example I I Space Space Range Gaussian Filtering Kernel Signal Signal Output Signal x x Bilateral Filtering I Output Signal Large Kernel Size  High Computational Complexity! Kernel x

  15. Bilateral Grid Downsampling Bilateral Grid Spatial Grid I I Signal Signal x x Traditional Spatial Downsampling Bilateral Grid I Bilateral Grid Signal Kernel I x x Bilateral Grid Downsampling

  16. intensity space Bilateral Filter on the Bilateral Grid Image scanline BilateralGrid

  17. intensity intensity space space Bilateral Filter on the Bilateral Grid Image scanline BilateralGrid Gaussian blur grid values Slice: query gridwith input image Filtered scanline

  18. Bilateral Filtering for Color Image Bilateral Filtering Based on Luminance Bilateral Filtering Based on Color

  19. Bilateral Grid for Color Image High-Dimensional Grid (5d grid) High Memory Usage Cost Image

  20. Gaussian KD-Tree • Low Computational Complexity • Low Memory Usage

  21. Gaussian KD-Tree • Building The Tree • Querying The Tree

  22. Building The Tree Intensity Longest Dimension, η1d Gaussian KD-Tree η1 η1cut η1max η1min Bounding Box Space

  23. Building The Tree Intensity Gaussian KD-Tree η1 η2 η2d η2 η2cut η2min η2max Space

  24. Building The Tree Intensity η3max Gaussian KD-Tree η3d η1 η3cut η3 η2 η3 η3min Space

  25. Building The Tree Intensity Gaussian KD-Tree η1 η3 η2 η4 η4cut η4d η4 η4min η4max Space

  26. Building The Tree Gaussian KD-Tree Intensity η1 η3 η2 …… …………. η4 • Inner Node • Cutting Dimension • Min, Max Bound • Left, Right Child Space

  27. Building The Tree Intensity • Leaf Node • Position Space

  28. Querying The Tree Gaussian KD-Tree η1 Image Pixel Querying η3 η2 High-Dimensional Space …… …………. η4

  29. Querying The Tree Gaussian KD Tree Image Pixel Inner Node Leaf Node Different Weighting to Leaf Nodes

  30. Splatting

  31. ηcut 1-D Example of Splatting Space ηcut Querying Position Sample Distribution Splatting Splatting Space Querying Position

  32. Space 1-D Example of Splatting ηcut ηcut Querying Position Sample Distribution Splatting Splatting Space Querying Position

  33. Correcting Weights for Splatting pi q

  34. Querying The Tree Image Pixel Gaussian KD Tree Samples Inner Node Leaf Node Sample Splitting to Leaf Nodes

  35. Blurring The Leaf Nodes

  36. Slicing

  37. Summary r,g,b y High-Dimensional Space Resolution Reduction Input Image x • Monte-Carlo Sampling • Weighted Importance Sampling Gaussian KD Tree

  38. Applications • Bilateral Filtering 5-D Bilateral Grid Naïve Bilateral Filtering

  39. KD-Tree 3-D Bilateral Grid

  40. Complexity and Performance Analysis Filter Size Large 5D Grid Gaussian KD-Tree Naïve Small

  41. Applications • Non-local Mean Filtering Input Image Output Image

  42. Non-local Mean Filtering Target Patch Searching Patches ….. Patch

  43. Non-local Mean Filtering with PCA Patch Examples 16 Leading Eigenvectors http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/denoising_nl_means/

  44. Non-local Mean Filtering Target Patch High-Dimensional Space Searching Patches ….. Patch

  45. Non-local Mean Filtering with Gaussian KD-Tree High-Dimensional Space Gaussian KD Tree Image Pixel Inner Node Leaf Node Different Weighting to Leaf Nodes

  46. Applications • Non-local Mean Filtering Input Image Output Image

  47. Applications • Geometry Filtering Input Model Output Model with Gaussian Filtering Output Model with Non-local Mean

  48. Conclusions • Novel methods of non-linear filter. • Bilateral grid and Gaussian kd-tree • High-dimensional non-linear filter. • Edge preserving smoothing • Computational Complexity Reduction • Importance sampling

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