F01943024 Reference Yang Qingxiong Recursive bilateral filtering ECCV 2012 Deriche Rachid Recursively implementating the Gaussian and its derivatives ICIP 1993 2 ID: 173725
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Slide1
Recursive Bilateral Filtering
F01943024Slide2
Reference
Yang, Qingxiong. "Recursive bilateral filtering."
ECCV
2012
.Deriche, Rachid. "Recursively implementating the Gaussian and its derivatives." ICIP 1993.
2Slide3
Outline
Recap of Bilateral FilteringWhy Recursive Filtering?Recursive Bilateral Filtering
Complexity Analysis
Applications
Conclusion3Slide4
Recap of Bilateral Filtering
Naïve image smoothing: Gaussian filteringNot edge-preserving
Bilateral filtering
4Slide5
Recap of Bilateral Filtering
The bilateral filter is a robust edge-preserving filter introduced by Tomasi and
Manduchi
in 1998.
5Slide6
Recap of Bilateral Filtering
Being non-linear, the brute force implementations of the bilateral filter are slow when kernel is large.Pham and
Vliet
implemented the BF as a separable operation.
The cost is still high for large kernels. (In ICME 2005)By constraining the spatial filter kernel to box filter, Weiss showed that the result depends only on the histogram of the neighborhood. However, this method works efficiently only on grascale image. (In Siggraph
2006)
Paris and Durand presented a volumetric data structure called
bilateral grid
. The BF corresponds to convolving a grid with a 3D/5D Gaussian. However, the memory cost maybe unacceptable when filter kernel is small. (In
Siggraph
2007)
6Slide7
Recap of Bilateral Filtering
Porikli is the first to remove the dependency of the filter(kernel) size by integral histogram. He also proposed a Taylor series based solution to remove the box filter constraint. (In CVPR 2008)
Yang showed that Durand’s method can be implemented using recursive Gaussian filter so that its complexity will be independent of the filter size. (In CVPR 2009)
The state-of-the-art implementation is proposed by Adams.
Gaussian KD-trees, complexity O(Nlog(N)D), memory cost O(ND)Permutohedral
lattice, complexity O(ND
2
), memory cost much higher
This paper:
implementing bilateral filter in a recursive
fasion
with complexity O(ND), memory cost O(ND)
7Slide8
Why Recursive Filtering?
FIR filterIIR filter
8Slide9
Why Recursive Filtering?
1st
-order recursive filtering
2
nd-order recursive filtering
Ex:
[
Deriche
1992] demonstrated that Gaussian filter
can be computed using 2
nd
-order recursive filtering.
9Slide10
Why Recursive Filtering?
Deriche, R. Recursively implementing the Gaussian and its derivatives. In ICIP. 1992
causal
anticausal
10Slide11
Why Recursive Filtering?
So why?Filter size dependencyRecursive implementation is linear in the number of pixels.
11Slide12
Recursive Bilateral Filtering
Bilateral filtering
Modified range kernel
The proposed method measures the range distance by accumulating the color difference between every two neighboring pixels on the path between
k and i.
12Slide13
Recursive Bilateral Filtering
Claim: For any bilateral filter containing
the new range filter kernel
and
any spatial filter kernel that can be recursively implemented, an exact recursive implementation can be obtained by simply altering the coefficients of the recursive system defined by the spatial filter kernel at each pixel location.Recursive implementation of the spatial filter
Recursive bilateral filter
Con be proved by math induction:
http://www.cs.cityu.edu.hk/~qiyang/publications/eccv-12
/
13Slide14
Complexity Analysis
Recursive implementation of the spatial filter
Recursive
bilateral
filterNew range kernel can be computed recursively.
2n multiplication operations and 2n-1 addition/subtraction operations
Only 3n-2 additional multiplication operations
14Slide15
Applications
Non-Photorealistic Rendering
15Slide16
Applications
Tone Mapping
16Slide17
Applications
Stereo Matching (average 10x
faster)
17Slide18
Conclusion
A recursive implementation of the bilateral filter is proposed in this paper. Witha new range filter kernel
any spatial filter kernel that can be recursively implemented.
It’s the first bilateral whose computational and memory complexity are linear in both input size and dimensionality.
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