Bilateral Filtering for Gray and Color Images

C. Tomasi R. ManduchiIEEE ICCV, 1998

Presented by Jan HellerReading Group, 18.12.2008

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 1 / 15

Outline

1 Bilateral FilterProperties of Bilateral FilterImplementation

2 Relationship with Other Frameworks

3 Applications

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 2 / 15

Bilateral Filter Properties of Bilateral Filter

Properties of Bilateral Filter

h(x) =

∫∞−∞

∫∞−∞ f(ξ)c(ξ,x)s(f(ξ), f(x)) dξ∫∞

−∞∫∞−∞ c(ξ,x)s(f(ξ), f(x)) dξ

Eqs. (5) + (6)

Low-pass filter with edge preservationNonlinear ⇒ hard to compute, complex spatially varying kernelsIterating In+1 = h[In] leads to piecewise linear approximation oforiginal image, Figures 7(a,b,c)Crosses lines thiner than ∼ 2σs

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 3 / 15

Bilateral Filter Properties of Bilateral Filter

Bilateral Filter

h(x)︸ ︷︷ ︸

= k−1(x)∞∫

−∞

∞∫

−∞c(ξ,x)︸ ︷︷ ︸

s(f(ξ), f(x))︸ ︷︷ ︸

f(ξ)︸ ︷︷ ︸

dξ

Output Input

*=

Figure: Equation (5) – the Gaussian Case. The figures are reproduced from: Fast bilateral

filtering for the display of high-dynamic-range images, Durand and Dorsey ACM SIGGRAPH conference (c) 2002,

Association for Computing Machinery, Inc. http://doi.acm.org/10.1145/566570.566574

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 4 / 15

Bilateral Filter Properties of Bilateral Filter

Related Work

Yaroslavsky, L. PDigital Picture Processing. An Introduction.Berlin-Heidelberg-New York, Springer-Verlag, 1985.

s(f(ξ), f(x)) = e−

“|f(ξ)−f(x)|

σr

”2

, c(ξ,x) = χB(x,σd)(ξ).

S. M. Smith and J. M. BradySUSAN - A New Approach to Low Level Image ProcessingInternational Journal of Computer Vision, 1997.

s(f(ξ), f(x)) = e−

“|f(ξ)−f(x)|

σr

”2

, c(ξ,x) = e−

“‖ξ−x‖σd

”2

.

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 5 / 15

Bilateral Filter Implementation

Implementation I

Twice as expensive as a nonseparable domain filter of the same sizeThe range component depends nonlinearly on the image, and isnonseperable

Brute Force

O(|S|2), limiting the influence of spatial Gaussian to pixel in distance< 2σs,O(|S|σ2

s).

Approximation by separable kernels

T.Q. Pham and L.J. van VlietSeparable bilateral filtering for fast video preprocessingICME, 2005.

O(|S|σs)Produces axis-aligned artefacts

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 6 / 15

Bilateral Filter Implementation

Implementation II

Local Histograms

Ben WeissFast median and bilateral filtering, ACM Trans. Graph., 2006

O(|S| log σs), spatial weight is a square box function

Bilateral Grid

Sylvain Paris and Frédo DurA fast approximation of the BF using a signal processing approach,ECCV 2006

O(|S|+ |S|σs

|R|σr

), reformulation of the BF in a higher-dimensionalhomogenous space, Γ : S ×R→ R2

Γ(px, py, r) =

{(I(px, py), 1) if r = I(px, py)(0, 0) otherwise

⇒ Gσs,σr ∗ Γ(px, py, r)

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 7 / 15

Relationship with Other Frameworks

Relationship with Other Frameworks I

PDE Based Filters

Frédo Durand and Julie DorseyFast bilateral filtering for the display of HDR images, SIGGRAPH 2002

BF restricted to the 4 adjacent neighbors of each pixel actuallycorresponds to a discrete version of Perona-Malik model.

Danny Barash and Dorin ComaniciuA common framework for nonlinear diffusion, adaptive smoothing,bilateral filtering and mean shift, Image and Video Computing 2004

Bilateral filter as the sum of several Perona-Malik filters at differentscales.

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 8 / 15

Relationship with Other Frameworks

Relationship with Other Frameworks II

Robust Statistics

Frédo Durand and Julie DorseyFast bilateral filtering for the display of HDR images, SIGGRAPH 2002

The range weight can be seen as a robust metric, that is, itdifferentiates between inliers and outliers.

Figure: Images reproduced from Durand and Dorsey [2002].C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 9 / 15

Applications

Denoising

(a) (b) (c)

(d) (e) (f)

Figure: (a) noisy input, (b) Median filter, 3x3 window, (c) Median filter, 5x5 window, (d) Bilateral filter, 7x7

window, (e) Bilateral filter, lower sigma, (f) Bilateral filter, higher sigma. Reproduced from

http://people.csail.mit.edu/sparis/bf_course/slides/04_applications_simple_bf.pdf.

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 10 / 15

Applications

Picture Simplification

Holger Winnemoller, Sven C. Olsen, Bruce GoochReal-Time Video Abstraction, SIGGRAPH 2006

Figure: Reproduced from Real-Time Video Abstraction

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 11 / 15

Applications

Contrast Reduction

Figure: Reproduced from

http://people.csail.mit.edu/sparis/bf_course/slides/04_applications_simple_bf.pdf

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Applications

Mesh Smoothing

Jones et al.Non-Iterative, Feature-Preserving Mesh Smoothing, SIGGRAPH 2003

Figure: Reproduced from Non-Iterative, Feature-Preserving Mesh Smoothing

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 13 / 15

Appendix For Further Reading

For Further Reading

Sylvain Paris, Pierre Kornprobst, Jack Tumblin, and Frédo DurandA Gentle Introduction to Bilateral Filtering and its Applications.A class at ACM SIGGRAPH 2008.

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 14 / 15

Appendix Any Questions?

Any Questions?

C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 15 / 15