A Gentle Introductionto Bilateral Filteringand its Applications
A Gentle Introductionto Bilateral Filteringand its Applications
“Fixing the Gaussian Blur”: the Bilateral Filter
Sylvain Paris – MIT CSAIL
Blur Comes from Averaging across EdgesBlur Comes from Averaging across Edges
*
*
*
input output
Same Gaussian kernel everywhere.
Bilateral FilterNo Averaging across EdgesBilateral FilterNo Averaging across Edges
*
*
*
input output
The kernel shape depends on the image content.
[Aurich 95, Smith 97, Tomasi 98]
space weight
not new
range weight
I
new
normalizationfactor
new
Bilateral Filter Definition:an Additional Edge TermBilateral Filter Definition:an Additional Edge Term
( ) ( )−− IIGG qpqp ||||||rs σσ∑
∈
=SW qp
1 IIBF qp][
Same idea: weighted average of pixels.
Illustration a 1D ImageIllustration a 1D Image
• 1D image = line of pixels
• Better visualized as a plot
pixelintensity
pixel position
space
Gaussian Blur and Bilateral FilterGaussian Blur and Bilateral Filter
space rangenormalization
Gaussian blur
( ) ( )∑∈
−−=S
IIIGGW
IBFq
qqpp
p qp ||||||1][rs σσ
Bilateral filter[Aurich 95, Smith 97, Tomasi 98]
space
spacerange
p
p
q
q
( )∑∈
−=S
IGIGBq
qp qp ||||][ σ
qqpp
Bilateral Filter on a Height FieldBilateral Filter on a Height Field
output input
( ) ( )∑∈
−−=S
IIIGGW
IBFq
qqpp
p qp ||||||1][rs σσ
pp
reproducedfrom [Durand 02]
Space and Range ParametersSpace and Range Parameters
• space σs : spatial extent of the kernel, size of the considered neighborhood.
• range σr : “minimum” amplitude of an edge
( ) ( )∑∈
−−=S
IIIGGW
IBFq
qqpp
p qp ||||||1][rs σσ
Influence of PixelsInfluence of Pixels
pp
Only pixels close in space and in range are considered.
space
range
σs = 2
σr = 0.1 σr = 0.25σr = ∞
(Gaussian blur)
σs = 6
σs = 18
input
Exploring the Parameter SpaceExploring the Parameter Space
σs = 2
σr = 0.1 σr = 0.25σr = ∞
(Gaussian blur)
σs = 6
σs = 18
input
Varying the Range ParameterVarying the Range Parameter
input
σr = 0.1
σr = 0.25
σr = ∞(Gaussian blur)
σs = 2
σs = 6
σs = 18
σr = 0.1 σr = 0.25σr = ∞
(Gaussian blur)
input
Varying the Space ParameterVarying the Space Parameter
input
σs = 2
σs = 6
σs = 18
How to Set the ParametersHow to Set the Parameters
Depends on the application. For instance:
• space parameter: proportional to image size– e.g., 2% of image diagonal
• range parameter: proportional to edge amplitude– e.g., mean or median of image gradients
• independent of resolution and exposure
A FewMore Advanced
Remarks
A FewMore Advanced
Remarks
Bilateral Filter Crosses Thin LinesBilateral Filter Crosses Thin Lines• Bilateral filter averages across
features thinner than ~2σs
• Desirable for smoothing: more pixels = more robust• Different from diffusion that stops at thin lines
close-up kernel
Iterating the Bilateral FilterIterating the Bilateral Filter
• Generate more piecewise-flat images
• Often not needed in computational photo.
][ )()1( nn IBFI =+
input
1 iteration
2 iterations
4 iterations
Bilateral Filtering Color ImagesBilateral Filtering Color Images
( ) ( )∑∈
−−=S
IIIGGW
IBFq
qqpp
p qp ||||||1][rs σσ
( ) ( )∑∈
−−=S
GGW
IBFq
qqpp
p CCCqp ||||||||1][rs σσ
For color images color difference
For gray-level images intensity difference
The bilateral filter isThe bilateral filter isextremely easy to adapt to your need.extremely easy to adapt to your need.
scalar
3D vector (RGB, Lab)
input
output
Hard to ComputeHard to Compute
• Nonlinear
• Complex, spatially varying kernels– Cannot be precomputed, no FFT…
• Brute-force implementation is slow > 10min
( ) ( )∑∈
−−=S
IIIGGW
IBFq
qqpp
p qp ||||||1][rs σσ
Questions?Questions?
