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 ImageDetail Image

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

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

6. Gaussian Filtering

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

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

10. Bilateral Filtering Input ImageGaussian: σ p = 12 Bilateral: σ p = 12, σ c = 0.15

11. Computational Complexity of Bilateral Filtering O(n 2 d) – Image Size: n – Maximum Filter Size: n – Dimension: d High Computational Complexity Input x y Intensity

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 x y Intensity

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

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

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

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

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

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

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

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

22. Building The Tree Space Intensity Bounding Box Longest Dimension, η 1 d η 1 min η 1 max η 1 cut η1η1 Gaussian KD-Tree

23. Building The Tree Space Intensity η2dη2d η 2 min η 2 max η 2 cut η1η1 Gaussian KD-Tree η2η2 η2η2

24. Building The Tree Space Intensity η3dη3d η 3 min η 3 max η 3 cut η1η1 Gaussian KD-Tree η2η2 η3η3 η3η3

25. Building The Tree Space Intensity η4dη4d η 4 min η 4 max η 4 cut η1η1 Gaussian KD-Tree η2η2 η4η4 η3η3 η4η4

26. Building The Tree Space Intensity Inner Node Cutting Dimension Min, Max Bound Left, Right Child η1η1 Gaussian KD-Tree η2η2 η3η3 η4η4 …… ………….

27. Building The Tree Space Intensity Leaf Node Position

28. Querying The Tree η1η1 Gaussian KD-Tree η2η2 η3η3 η4η4 …… …………. High-Dimensional Space Image Pixel Querying

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

30. Splatting

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

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

33. Correcting Weights for Splatting q pi

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

35. Blurring The Leaf Nodes

36. Slicing

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

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

39. 3-D Bilateral Grid KD-Tree

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

41. Applications Non-local Mean Filtering Input ImageOutput 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 Searching Patches ….. Patch High-Dimensional Space

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

46. Applications Non-local Mean Filtering Input ImageOutput Image

47. Applications Geometry Filtering Input ModelOutput 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