1

Tuan Q. Pham & Lucas J. van Vliet

Delft University of TechnologyThe Netherlands

Quantitative Imaging Group

Faculty of Applied Sciences

Separable bilateral filteringfor fast video preprocessing

ICME 2005IEEE International Conference on Multimedia & ExpoJuly 6-8,  2005,  Amsterdam, The Netherlands

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

* =

Noisy step edge

. =

Adaptive filtering: avoid filtering across edges

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

* =

Noisy step edge Edge-preserving weights Edge-presered filtered result

. =

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

.

How bilateral filtering works?

The weight is product of two Gaussian weights:

Spatial proximity:

Tonal similarity:

Local modesafter

bilateral filtering

Every sample is a weighted average of its neighbors:

0 0

1( ) ( , ) . ( )

s

O s w s s I sK

202

( ) ( )exp

2

tt

I s I sw

. s tw w w2

02

( )exp

2

ss

s sw

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:

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:

Our approach: separable bilateral filtering

dO Nm

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

O Nmd

Is bilateral filter separable?

Gaussian filter is space-invariant and separable:

=*

x

g(x)

y

g(y)

Bilateral filter is NOT separable: Space-variant kernel due to local

intensity dependency

I(s0)

I(s)

Kernel center

0

00

( ) . ( )( )

( )

s

s

ss

w s s I sO s

w s s

( ) gauss( ) . gauss( )sw s x y

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

Separable bilateral filtering result

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

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

Separable bilateral filtering result

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

Separable bilateral filtering is a good approximation of full kernel 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

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

How separable bilateral filtering works?

Performance of separable bilateral filtering

Very good approximation of the full-kernel filter

Almost linear execution time per pixel

Better RMSE is achieved with separable bilateral filtering compared to full-kernel bilateral filtering with the same computation requirement

MPEG-1 Foreman with bilateral preprocessing

0 400 800 120013

14

15

16

bit-rate (K bits/s)

RM

SE

without preprocessingwith 3x3x3 full-kernelwith 9x9x5 separable

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.