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

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

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Edge-Preserving Filtering for Image Analysis Input Image Base ImageDetail Image

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

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

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Gaussian Filtering

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

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

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Bilateral Filtering Input ImageGaussian: p = 12 Bilateral: p = 12, c = 0.15

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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

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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).

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High-Dimensional Filtering x y Intensity

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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!

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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

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

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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

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

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

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Gaussian KD-Tree Low Computational Complexity Low Memory Usage

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Gaussian KD-Tree Building The Tree Querying The Tree

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Building The Tree Space Intensity Bounding Box Longest Dimension, 1 d 1 min 1 max 1 cut 11 Gaussian KD-Tree

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Building The Tree Space Intensity 2d2d 2 min 2 max 2 cut 11 Gaussian KD-Tree 22 22

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Building The Tree Space Intensity 3d3d 3 min 3 max 3 cut 11 Gaussian KD-Tree 22 33 33

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Building The Tree Space Intensity 4d4d 4 min 4 max 4 cut 11 Gaussian KD-Tree 22 44 33 44

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Building The Tree Space Intensity Inner Node Cutting Dimension Min, Max Bound Left, Right Child 11 Gaussian KD-Tree 22 33 44 .

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Building The Tree Space Intensity Leaf Node Position

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Querying The Tree 11 Gaussian KD-Tree 22 33 44 . High-Dimensional Space Image Pixel Querying

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

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Splatting

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

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

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Correcting Weights for Splatting q pi

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

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Blurring The Leaf Nodes

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Slicing

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

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Applications Bilateral Filtering Nave Bilateral Filtering 5-D Bilateral Grid

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3-D Bilateral Grid KD-Tree

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Complexity and Performance Analysis Filter Size Large Small 5D Grid Gaussian KD-Tree Nave

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Applications Non-local Mean Filtering Input ImageOutput Image

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

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

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

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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

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Applications Non-local Mean Filtering Input ImageOutput Image

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Applications Geometry Filtering Input ModelOutput Model with Gaussian Filtering Output Model with Non-local Mean

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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