Accelerating Spatially Varying Gaussian Filters Jongmin Baek and David E. Jacobs Stanford University

Motivation Input Gaussian Filter Spatially Varying Gaussian Filter

1) Accelerating Spatially Varying Gaussian Filters 2) Accelerating Spatially Varying Gaussian Filters 3) Accelerating Spatially Varying Gaussian Filters 4) Applications Roadmap

Gaussian Filters Position Value

Gaussian Filters Each output value …

Gaussian Filters … is a weighted sum of input values …

Gaussian Filters … whose weight is a Gaussian …

Gaussian Filters … in the space of the associated positions.

Gaussian Blur Gaussian Filters: Uses

Bilateral Filter Gaussian Filters: Uses

Non-local Means Filter Gaussian Filters: Uses

Applications  Denoising images and meshes  Data fusion and upsampling  Abstraction / Stylization  Tone-mapping ... Gaussian Filters: Summary Previous work on fast Gaussian Filters  Bilateral Grid (Chen, Paris, Durand; 2007)  Gaussian KD-Tree (Adams et al.; 2009)  Permutohedral Lattice (Adams, Baek, Davis; 2010)

Summary of Previous Implementations:  A separable blur flanked by resampling operations.  Exploit the separability of the Gaussian kernel. Gaussian Filters: Implementations

Spatially Varying Gaussian Filters Spatially varying covariance matrix Spatially Invariant

Trilateral Filter (Choudhury and Tumblin, 2003)  Tilt the kernel of a bilateral filter along the image gradient.  “Piecewise linear” instead of “Piecewise constant” model. Spatial Variance in Previous Work

Spatially Varying Gaussian Filters: Tradeoff Benefits:  Can adapt the kernel spatially.  Better filtering performance. Cost:  No longer separable.  No existing acceleration schemes. Input Bilateral-filtered Trilateral-filtered

Problem:  Spatially varying (thus non-separable) Gaussian filter Existing Tool:  Fast algorithms for spatially invariant Gaussian filters Solution:  Re-formulate the problem to fit the tool.  Need to obey the “piecewise-constant” assumption Acceleration

Naïve Approach (Toy Example) I LOST THE GAME Input Signal Desired Kernel filtered w/ 1 filtered w/ 2 filtered w/ 3 filtered w/ Output Signal 4

In practice, the # of kernels can be very large. Challenge #1 Pixel Location x Desired Kernel K(x) Range of Kernels needed

Sample a few kernels and interpolate. Solution #1 Desired Kernel K(x) Sampled kernels Interpolate result! Pixel Location x K1K1 K2K2 K3K3

Interpolation needs an extra assumption to work:  The covariance matrix Ʃ i is either piecewise- constant, or smoothly varying.  Kernel is spatially varying, but locally spatially invariant. Assumptions

Runtime scales with the # of sampled kernels. Challenge #2 Desired Kernel K(x) Filter only some regions of the image with each kernel. (“support”) Pixel Location x Sampled kernels K1K1 K2K2 K3K3

In this example, x needs to be in the support of K 1 & K 2. Defining the Support Desired Kernel K(x) Pixel Location x K1K1 K2K2 K3K3

Dilating the Support Desired Kernel K(x) Pixel Location x K1K1 K2K2 K3K3

Algorithm 1) Identify kernels to sample. 2) For each kernel, compute the support needed. 3) Dilate each support. 4) Filter each dilated support with its kernel. 5) Interpolate from the filtered results.

Algorithm 1) Identify kernels to sample. 2) For each kernel, compute the support needed. 3) Dilate each support. 4) Filter each dilated support with its kernel. 5) Interpolate from the filtered results. K1K1 K2K2 K3K3

Algorithm 1) Identify kernels to sample. 2) For each kernel, compute the support needed. 3) Dilate each support. 4) Filter each dilated support with its kernel. 5) Interpolate from the filtered results. K1K1 K2K2 K3K3

Algorithm 1) Identify kernels to sample. 2) For each kernel, compute the support needed. 3) Dilate each support. 4) Filter each dilated support with its kernel. 5) Interpolate from the filtered results. K1K1 K2K2 K3K3

Algorithm 1) Identify kernels to sample. 2) For each kernel, compute the support needed. 3) Dilate each support. 4) Filter each dilated support with its kernel. 5) Interpolate from the filtered results. K1K1 K2K2 K3K3

Algorithm 1) Identify kernels to sample. 2) For each kernel, compute the support needed. 3) Dilate each support. 4) Filter each dilated support with its kernel. 5) Interpolate from the filtered results. K1K1 K2K2 K3K3

Applications  HDR Tone-mapping  Joint Range Data Upsampling

Application #1: HDR Tone-mapping Input HDR Detail Base Filter Output Attenuate

Tone-mapping Example Bilateral Filter Kernel Sampling

Application #2: Joint Range Data Upsampling Range Finder Data  Sparse  Unstructured  Noisy Scene Image Output Filter

Synthetic Example Scene Image Ground Truth Depth

Synthetic Example Scene ImageSimulated Sensor Data

Synthetic Example : Result Kernel Sampling Bilateral Filter

Synthetic Example : Relative Error Bilateral Filter Kernel Sampling 2.41% Mean Relative Error0.95% Mean Relative Error

Real-World Example Scene Image Range Finder Data *Dataset courtesy of Jennifer Dolson, Stanford University

Real-World Example: Result Input Bilateral Naive Kernel Sampling

Performance Kernel Sampling Choudhury and Tumblin (2003) Naïve Tonemap s41.54 s s Tonemap s88.08 s s Kernel Sampling (No segmentation) Depth s57.90 s Depth s s

1.A generalization of Gaussian filters Spatially varying kernels Lose the piecewise-constant assumption. 2.Acceleration via Kernel Sampling Filter only necessary pixels (and their support) and interpolate. 3.Applications Conclusion