This document provides an introduction to bilateral filtering and its applications. It begins by explaining that bilateral filtering, unlike Gaussian blurring, does not average across edges in an image. The bilateral filter definition is presented as a weighted average of pixels, with an additional edge-preserving range weight term. Examples are given to illustrate how the bilateral filter works on 1D images and height fields. The influence of the space and range parameters is demonstrated, with the space parameter controlling the spatial extent of the kernel and range parameter defining the minimum amplitude of an edge. The document explores varying these parameters and how they should be set depending on the application. Some more advanced topics covered include how the bilateral filter can cross thin lines and be iterated or adapted

1. A Gentle Introduction
to Bilateral Filtering
and its Applications
“Fixing the Gaussian Blur”:
the Bilateral Filter
Sylvain Paris – MIT CSAIL

2. Blur Comes from
Averaging across Edges
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input output
Same Gaussian kernel everywhere.

3. Bilateral Filter
No Averaging across Edges
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The kernel shape depends on the image content.
[Aurich 95, Smith 97, Tomasi 98]

4. space weight
not new
range weight
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normalization
factor
new
Bilateral Filter Definition:
an Additional Edge Term
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Same idea: weighted average of pixels.

5. Illustration a 1D Image
• 1D image = line of pixels
• Better visualized as a plot
pixel
intensity
pixel position

6. space
Gaussian Blur and Bilateral Filter
space range
normalization
Gaussian blur
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Bilateral filter
[Aurich 95, Smith 97, Tomasi 98]
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7. q
p
Bilateral Filter on a Height Field
output input
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reproduced
from [Durand 02]

8. Space and Range Parameters
• space s : spatial extent of the kernel, size of
the considered neighborhood.
• range r : “minimum” amplitude of an edge
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9. Influence of Pixels
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Only pixels close in space and in range are considered.
space
range

10. s = 2
s = 6
s = 18
r = 0.1 r = 0.25
r = 
(Gaussian blur)
input
Exploring the Parameter Space

11. s = 2
s = 6
s = 18
r = 0.1 r = 0.25
r = 
(Gaussian blur)
input
Varying the Range Parameter

12. input

13. r = 0.1

14. r = 0.25

15. r = 
(Gaussian blur)

16. s = 2
s = 6
s = 18
r = 0.1 r = 0.25
r = 
(Gaussian blur)
input
Varying the Space Parameter

17. input

18. s = 2

19. s = 6

20. s = 18

21. How 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

22. A Few
More Advanced
Remarks

23. Bilateral Filter Crosses Thin Lines
• Bilateral filter averages across
features thinner than ~2s
• Desirable for smoothing: more pixels = more robust
• Different from diffusion that stops at thin lines
close-up kernel

24. Iterating the Bilateral Filter
• Generate more piecewise-flat images
• Often not needed in computational photo.
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25. input

26. 1 iteration

27. 2 iterations

28. 4 iterations

29. Bilateral Filtering Color Images
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For gray-level images
For color images
intensity difference
color difference
The bilateral filter is
extremely easy to adapt to your need.
scalar
3D vector
(RGB, Lab)
input
output

30. Hard to Compute
• Nonlinear
• Complex, spatially varying kernels
– Cannot be precomputed, no FFT…
• Brute-force implementation is slow > 10min
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31. Questions?