Recursive Bilateral Filtering
F01943024
Reference
• Yang, Qingxiong. "Recursive bilateral filtering." ECCV 2012.
• Deriche, Rachid. "Recursively implementating the Gaussian
and its derivatives." ICIP 1993.
2
Outline
•
•
•
•
•
•
Recap of Bilateral Filtering
Why Recursive Filtering?
Recursive Bilateral Filtering
Complexity Analysis
Applications
Conclusion
3
Recap of Bilateral Filtering
• Naïve image smoothing: Gaussian filtering
– Not edge-preserving
• Bilateral filtering
4
Recap of Bilateral Filtering
• The bilateral filter is a robust edge-preserving filter introduced by Tomasi
and Manduchi in 1998.
5
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)
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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(ND2), memory cost much higher
• This paper: implementing bilateral filter in a recursive fasion
with complexity O(ND), memory cost O(ND)
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Why Recursive Filtering?
• FIR filter
n
yi 
a
l
 xil
l 1
• IIR filter
n 1
yi 
 (a
l0
n
l
 x i  l )   (bk  y i  k )
k 1
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Why Recursive Filtering?
n 1
yi 
 (a
l0
n
l
 x i  l )   (bk  y i  k )
k 1
• 1st-order recursive filtering
y i  a 0  x i  b1  y i 1
Ex:
y i  (1  a )  x i  a  y i 1
• 2nd-order recursive filtering
y i  a 0  x i  a1  x i 1  b1  y i 1  b 2  y i  2
[Deriche 1992] demonstrated that Gaussian filter
can be computed using 2nd-order recursive filtering.
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Why Recursive Filtering?
• Deriche, R. Recursively implementing the Gaussian and its
derivatives. In ICIP. 1992
G (i ) 
causal
1
2
exp( 
2
i
2
2
2
)
y i  a 0  x i  a1  x i 1  b1  y i 1  b 2  y i  2
a
a
anticausal y i  a 2  x i  1  a 3  x i  2  b1  y i  1  b 2  y i  2
a
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Why Recursive Filtering?
• So why?
– Filter size dependency
– Recursive implementation is linear in the number of pixels.
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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.
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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
n 1
yi 
 (a
l0
n
l
 x i  l )   (bk  y i  k )
k 1
• Recursive bilateral filter
n 1
yi 
 (R
l0
n
i ,i  l
 a l  x i  l )   ( R i ,i  k  b k  y i  k )
k 1
Con be proved by math induction: http://www.cs.cityu.edu.hk/~qiyang/publications/eccv-12/
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Complexity Analysis
• Recursive implementation of the spatial filter
n 1
yi 
 (a
l0
n
l
 x i  l )   (bk  y i  k )
k 1
2n multiplication operations and 2n-1 addition/subtraction operations
• Recursive bilateral filter
– New range kernel can be computed recursively.
R i , i  k  R i , i  ( k 1 )  R i  ( k 1 ), i  k
n 1
yi 
 (R
l0
n
i ,i  l
 a l  x i  l )   ( R i ,i  k  b k  y i  k )
k 1
Only 3n-2 additional multiplication operations
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Applications
• Non-Photorealistic Rendering
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Applications
• Tone Mapping
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Applications
• Stereo Matching (average 10x faster)
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Conclusion
• A recursive implementation of the bilateral filter is
proposed in this paper. With
– a 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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