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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
C. Tomasi, R. Manduchi () Bilateral Filtering RG, 18.12.2008 12 / 15
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