# Bilateral_filtering

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```					ICME 2005
IEEE International Conference on Multimedia & Expo
July 6-8, 2005, Amsterdam, The Netherlands

Separable bilateral filtering
for fast video preprocessing

Tuan Q. Pham           &   Lucas J. van Vliet

Delft University of Technology
The Netherlands
1

Quantitative Imaging Group
Faculty of Applied Sciences
Contents

1. Bilateral filtering:
 Edge-preserving filter
 High computational complexity

2. Separable implementation:
 Good approximation of the original filter
 Linear computational complexity

3. Application to video preprocessing:
 Noise reduction
 Better compressed video
Gaussian filtering revisited

 Gaussian filtering: weights depend on distance to the center pixel

*                             =
Noisy step edge              Gaussian weights         Gaussian filtered result

 Adaptive filtering: avoid filtering across edges

.                             =
Noisy step edge         Edge-preserving weights   Edge-presered filtered result
Gaussian vs bilateral filtering

 Gaussian filtering: weights depend on distance to the center pixel

*                              =
Noisy step edge              Gaussian weights         Gaussian filtered result

 Bilateral filtering: weights depend on both spatial closeness and
photometric similarity

.                             =
Noisy step edge         Edge-preserving weights   Edge-presered filtered result
How bilateral filtering works?

 Every sample is a weighted average of its neighbors:
1
O( s0 ) 
K
 w ( s, s ) . I ( s )
s
0

Local modes
 The weight w  ws . wt is product of two Gaussian                  weights: after
bilateral
  ( s  s0 ) 2                         filtering
 Spatial proximity: w s  exp                 
     2 s2 
   I ( s )  I ( s0 ) 2 
 Tonal similarity:      wt  exp                            
          2 t 2

                           
Example: Gaussian filtering

Noisy input: PSNR = 39.1 dB   Gaussian filtered: PSNR = 67.9 dB
Example: Bilateral filtering

Noisy input: PSNR = 39.1 dB   Bilateral filtered: PSNR = 41.6 dB
Computational complexity

 Bilateral filtering kernel is space-variant → complexity is:

O  Nm d 
where N: number of pixels in the image
m: size of filtering kernel (m ≈ 7 is good enough)
d: image dimensionality

 Previous attempt for fast bilateral filtering:
Piecewise-linear: approximate bilateral filtering with M Gaussian
filtering - Durand & Dorsey (SIGGRAPH 2002) → complexity is:

O  N log( N ) . M        M ≈ 17 for 8-bit images

 Our approach: separable bilateral filtering
O  Nmd 
Is bilateral filter separable?

 Gaussian filter is space-invariant and separable:

 w ( s  s ) . I ( s)
s       0
O( s )   s
*           =
 w (s  s )
0
s       0
s

ws ( s)  gauss( x ) . gauss( y)
Kernel center
 Bilateral filter is NOT separable:                      I(s0)
 Space-variant kernel due to local                      I(s)
intensity dependency

 However, even a highly non-linear filter like median filter is
approximately separable (Narendra – PAMI 1981)
Separable bilateral filtering result

 Separable bilateral filtering is a good approximation of full kernel filtering:

noisy Erika σnoise = 10   bilateral filtered in x-dimension followed by y-dimension filtering
Separable bilateral filtering result

 Separable bilateral filtering is a good approximation of full kernel filtering:

noisy Erika σnoise = 10            bilateral filtered in x-dimension followed by y-dimension filtering

Image size           Brute-force            Durrand 2002        Separable        Aniso. diffusion
256x256                     4.46                 0.37              0.21                2.76
512x512                    17.88                 1.59              0.89               26.02
61x61x61                    5.29                 4.20              0.45                5.25
256x256x212                56 min          Out of memory           50.3          Out of memory
How separable bilateral filtering works?

 Pros: extremely fast (fixed spatial weight + LUT for tonal weight)
 Cons: effective filtering kernel is a slightly distorted
Performance of separable bilateral filtering

 Very good approximation of    Almost linear execution time
the full-kernel filter         per pixel
MPEG-1 Foreman with bilateral preprocessing

16       without preprocessing
with 3x3x3 full-kernel
with 9x9x5 separable

15
RMSE

14

13
0        400              800         1200
bit-rate (K bits/s)

 Better RMSE is achieved with separable bilateral filtering compared to
full-kernel bilateral filtering with the same computation requirement
Less artifact with Bilateral preprocessing
Conclusions

 Separable implementation of bilateral filtering:
 Very good approximation of the original filter
 Much faster than the original or other approximations

 Applications in video preprocessing:
 Improved quality of compressed video
 Reduced processing time → real-time possibility
Literature

   C. Tomasi and R. Manduchi, Bilateral fitering for gray and color
images, Proc. of ICCV, USA, 1998, 839-846.

   T.Q. Pham and L.J. van Vliet, Separable bilateral filtering for fast
video preprocessing, Proc. of ICME’05.

   F. Durrand and J. Dorsey, Fast bilateral filtering for the display of high
dynamic range images, Proc. of SIGGRAPH’02, 2002, 844-847.

   P. Perona and J. Malik, Scale-space filtering and edge detection
using anisotropic diffusion, PAMI, vol. 12, no. 7, 1990, 629-639.

   R. v.d. Boomgaard and J. v.d. Weijer. On the equivalence of local-
mode finding, robust estimation and mean-shift analysis as used in
early vision tasks. In Proc. of ICPR, pages 927-930, 2002.

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