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

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Motivation Input Gaussian Filter Spatially Varying Gaussian Filter

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

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Gaussian Filters Position Value

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Gaussian Filters Each output value …

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Gaussian Filters … is a weighted sum of input values …

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Gaussian Filters … whose weight is a Gaussian …

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Gaussian Filters … in the space of the associated positions.

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Gaussian Blur Gaussian Filters: Uses

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Bilateral Filter Gaussian Filters: Uses

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Non-local Means Filter Gaussian Filters: Uses

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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)

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

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Spatially Varying Gaussian Filters Spatially varying covariance matrix Spatially Invariant

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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

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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

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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

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Naïve Approach (Toy Example) I LOST THE GAME Input Signal Desired Kernel 1 11234 filtered w/ 1 filtered w/ 2 filtered w/ 3 filtered w/ 4 111 2 3 Output Signal 4

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

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

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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

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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

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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

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

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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.

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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

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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

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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

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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

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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

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Applications HDR Tone-mapping Joint Range Data Upsampling

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

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Tone-mapping Example Bilateral Filter Kernel Sampling

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

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Synthetic Example Scene Image Ground Truth Depth

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Synthetic Example Scene ImageSimulated Sensor Data

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Synthetic Example : Result Kernel Sampling Bilateral Filter

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

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

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Real-World Example: Result Input Bilateral Naive Kernel Sampling

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Performance Kernel Sampling Choudhury and Tumblin (2003) Naïve Tonemap1 5.10 s41.54 s312.70 s Tonemap2 6.30 s88.08 s528.99 s Kernel Sampling (No segmentation) Depth1 3.71 s57.90 s Depth2 9.18 s131.68 s

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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

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