accelerating spatially varying gaussian filters
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Accelerating Spatially Varying Gaussian Filters
Jongmin Baek and David E. JacobsStanford University
Motivation
Input
GaussianFilter
SpatiallyVaryingGaussianFilter
1) Accelerating Spatially Varying Gaussian Filters
2) Accelerating Spatially Varying Gaussian Filters
3) Accelerating Spatially Varying Gaussian Filters
4) Applications
Roadmap
Gaussian Filters
Given pairs as input,
𝑣ˆ
𝑖=∑𝑗𝑣 𝑗⋅ exp (− ∥𝑝𝑖−𝑝 𝑗∥2
2 σ2 )
Position Value
Gaussian Filters
Each output value …
𝑣ˆ
𝑖=∑𝑗𝑣 𝑗⋅ exp (− ∥𝑝𝑖−𝑝 𝑗∥2
2 σ2 )
Gaussian Filters
… is a weighted sum of input values …
𝑣ˆ
𝑖=∑𝑗𝑣 𝑗⋅ exp (− ∥𝑝𝑖−𝑝 𝑗∥2
2 σ2 )
Gaussian Filters
… whose weight is a Gaussian …
𝑣ˆ
𝑖=∑𝑗𝑣 𝑗⋅ exp (− ∥𝑝𝑖−𝑝 𝑗∥2
2 σ2 )
Gaussian Filters
… in the space of the associated positions.
𝑣ˆ
𝑖=∑𝑗𝑣 𝑗⋅ exp (− ∥𝑝𝑖−𝑝 𝑗∥2
2 σ2 )
Gaussian Blur
Gaussian Filters: Uses
𝒑 𝒊=(𝒙 𝒊 , 𝒚 𝒊 ) ,𝒗 𝒊=(𝒓 𝒊 ,𝒈𝒊 ,𝒃𝒊)
Bilateral Filter
Gaussian Filters: Uses
𝒑 𝒊=(𝒙 𝒊 , 𝒚 𝒊 ,𝒓 𝒊 ,𝒈𝒊 ,𝒃𝒊 ) ,𝒗 𝒊=(𝒓 𝒊 ,𝒈𝒊 ,𝒃𝒊)
Non-local Means Filter
Gaussian Filters: Uses
𝒑 𝒊=(𝒙 𝒊 , 𝒚 𝒊 , )𝒗 𝒊=(𝒓 𝒊 ,𝒈𝒊 ,𝒃𝒊)
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
𝑣ˆ
𝑖=∑𝑗𝑣 𝑗⋅N (𝑝𝑖−𝑝 𝑗 ;𝜎
2 𝐼 )
𝑣ˆ
𝑖=∑𝑗𝑣 𝑗⋅N (𝑝𝑖−𝑝 𝑗 ;𝐾 𝑖 )
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 Kernel1 1 12 3 4
filtered w/ 1filtered w/ 2filtered w/ 3filtered w/ 4
1 1 12
3Output Signal4
In practice, the # of kernels can be very large.
Challenge #1
Pixel Location x
Desired Kernel K(x)
Range ofKernels needed
Sample a few kernels and interpolate.
Solution #1
Desired Kernel K(x)
Sampledkernels
Interpolate result!
Pixel Location x
K1
K2
K3
Interpolation needs an extra assumption to work:
The covariance matrix Ʃi 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
Sampledkernels
K1
K2
K3
In this example, x needs to be in the support of K1 & K2.
Defining the Support
Desired Kernel K(x)
Pixel Location x
K1
K2
K3
Dilating the Support
Desired Kernel K(x)
Pixel Location x
K1
K2
K3
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.
K1
K2
K3
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.
K1
K2
K3
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.
K1
K2
K3
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.
K1
K2
K3
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.
K1
K2
K3
Applications
HDR Tone-mapping
Joint Range Data Upsampling
Application #1: HDR Tone-mapping
Input HDR
Detail
BaseFil
ter 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 Image Simulated Sensor Data
Synthetic Example : Result
Kernel SamplingBilateral Filter
Synthetic Example : Relative Error
Bilateral Filter Kernel Sampling
2.41% Mean Relative Error 0.95% 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
KernelSampling
Performance
Kernel Sampling
Choudhury and Tumblin (2003) Naïve
Tonemap1
5.10 s 41.54 s 312.70 s
Tonemap2
6.30 s 88.08 s 528.99 sKernel
SamplingKernel Sampling
(No segmentation)Depth1 3.71 s 57.90 sDepth2 9.18 s 131.68 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
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