overview of recursive bilateral filtering

1. Introduction
• The Bilateral filter is a robust edge-preserving
filter introduced by Tomasi and Manduchi.
• Bilateral filter can be implemented recursively
as long as spatial filter kernel can be
implemented recursively and range filter
kernel can be decomposed into a recursive
product.

2. BLOCK DIAGRAM
Input xi output

3. Recursive Filtering:
• Let x denote the one-dimensional (1D) input
signal of a causal recursive system of order n,
and y denote the output, then
• 1st-order recursive filtering
• 2nd-order recursive filtering
 



 
n
k
kik
n
l
lili ybxay
1
1
0
)()(
110  iii ybxay 1)1(  iii yaxayEx:
2211110   iiiii ybybxaxay
3

4. This recursive system is then characterized by
the following transfer function
where {ha
k} denote the impulse response of the
recursive system whose Z-transform is Ha(Z).

5. • Deriche proposed Recursively implementing the Gaussian and
its derivatives.
)
2
exp(
2
1
)( 2
2
2 
i
iG 
2211110   iiiii ybybxaxayCausal:
Anti causal:
5
2nd order recursive implementation:
a
i
a
iii
a
i ybybxaxay 22112312  
where,

6. input

7. Gaussian blur

8. Spatial Parameter
small  large 
input
limited smoothing strong smoothing
)
2
exp(
2
1
)( 2
2
2 
i
iG 

9. How to set s:
• Depends on the application.
• Common strategy of s: proportional to image
size
–e.g. 2% of the image diagonal
–property: independent of image resolution



10. • Bilateral filtering
where Rk,i = R(xk, xi) is the range filter kernel
for measuring the range similarity of pixel k
and i and Sk,i = S(k, i) is the spatial filter kernel
for measuring their spatial similarity.

11. 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.

12. The new range filter kernel Rk,i measures the
range distance between pixel k and i by
accumulating the range distance between every
two neighboring pixels on the path between k
and i.
The range filtering kernel is often Gaussian
where |xj − xj+1|2 denotes the range cost of
traveling from pixel j to j + 1
(or from j + 1 to j) and Rj,j = 1, then

13. Using the new range filtering kernel, a recursive
implementation of the bi-lateral filter can be
obtained with a small modification of the
coeffcients (al and bk) of the recursive system
defined by the spatial filter kernel at each pixel
location.
where n ≥ 1

14. • The output of this modified recursive system is
then
• with the initial condition that y0 = a0x0, and xi = 0
when i < 0. Apparently, this is a bilateral filter
where Ri,k is the range filter kernel and
Si,k =∑ λi−m−kam (m=0 to n-1)is the spatial filter
kernel.
where

15. 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
 



 
n
k
kik
n
l
lili ybxay
1
1
0
)()(
 



 
n
k
kikkii
n
l
lilliii ybRxaRy
1
,
1
0
, )()(

16. Complexity Analysis:
• Recursive implementation of the spatial filter
2n multiplication operations and 2n-1
addition and subtraction operations are required.
• Recursive bilateral filter
–New range kernel can be computed recursively.
 



 
n
k
kik
n
l
lili ybxay
1
1
0
)()(
 



 
n
k
kikkii
n
l
lilliii ybRxaRy
1
,
1
0
, )()(
kikikiikii RRR   ),1()1(,,

17. • Only 3n-2 additional multiplication operations
and n operations for measuring the range
distance between two neighboring pixels.
• Recursive implementation method will be
independent of kernel size and only depends
on the number of pixels in an image

18. 2D Recursive Bilateral Filtering:
• Performing the proposed 1D recursive bilateral
filter both horizontally and vertically extends
the 1D filter to 2D.
The horizontal pass is performed first, the
vertical pass will be applied to the result
produced by the horizontal one (and vice-
versa).

19. APPLICATIONS:
Non-photorealistic Rendering:

20. Tone mapping:

21. Software
• MATLAB(7.1)
 FDA TOOL

22. Reference
• Yang, Qingxiong. "Recursive bilateral
filtering." ECCV 2012.
• Deriche, Rachid. "Recursively implementating
the Gaussian and its derivatives." ICIP 1993.
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