Introduction to Digital Image Processing 1051 · Bilateral Filtering Noise Reduction I Noise typically presents itself in imagery as a random, high-frequency, uncorrelated modulation

Bilateral Filtering

Introduction to Digital Image Processing1051.782

Carl Salvaggio

Chester F. Carlson Center for Imaging ScienceCollege of Science

Rochester Institute of Technology

Spring Quarter 2009-2010

Carl Salvaggio Introduction to Digital Image Processing

Bilateral Filtering

Noise Reduction

I Noise typically presents itself in imagery as a random,high-frequency, uncorrelated modulation in the signal level

I Typical methods for noise removal areI Spatial filtering with a blur kernelI Frequency domain filtering with a lowpass filterI Nonlinear filtering using a statistical filter such as the median

filter

I All of these method reduce the appearance of noise in theimage, but at the expense of other high-frequencyinformation, most notably edge detail


Bilateral Filtering

Traditional Approaches

Original Image

Original Image + N(0, 10) Gaussian Filter (σ = 5) Median Filter (10x10)


Bilateral Filtering

Bilateral Filtering for Gray and Color Images

“Bilateral filtering smooths images while preserving edges, bymeans of a nonlinear combination of nearby image values. Themethod is noniterative, local, and simple. It combines gray levelsor colors based on both their geometric closeness and theirphotometric similarity, and prefers near values to distant values inboth domain and range. In contrast with filters that operate onthe three bands of a color image separately, a bilateral filter canenforce the perceptual metric underlying the CIE-Lab color space,and smooth colors and preserve edges in a way that is tuned tohuman perception. Also, in contrast with standard filtering,bilateral filtering produces no phantom colors along edges in colorimages, and reduces phantom colors where they appear in theoriginal image.”[Tomasi & Manduchi, 1998]


Bilateral Filtering

The Idea - Spatial Domain

A lowpass spatial domain filter applied to a multiband image f(x)produces a multiband response image as follows

h(x) =1

kd(x)

∫ ∞−∞

∫ ∞−∞

f(ξ)c(ξ, x)dξ (1)

where c(ξ, x) is a measure of geometric closeness between theneighborhood center x and a nearby point ξ. The normalizationfactor kd(x) is given by

kd(x) =

∫ ∞−∞

∫ ∞−∞

c(ξ, x)dξ (2)

If the filter is shift-invariant, c(ξ, x) is only a function of the vectordifference ξ − x and kd(x) is a constant.


Bilateral Filtering

The Idea - Range Domain

Range (brightness) domain filtering is carried out similarly as

h(x) =1

kr (x)

∫ ∞−∞

∫ ∞−∞

f(ξ)s(f(ξ), f(x))dξ (3)

where s(f(ξ), f(x)) measures the photometric (brightness)similarity between the pixel at the neighborhood center and that ofa nearby point ξ.

The normalization factor in this case is

kr (x) =

∫ ∞−∞

∫ ∞−∞

s(f(ξ), f(x))dξ (4)

Note that the normalization factor in the range filtering case isdependent on the image f and as such nonlinear.


Bilateral Filtering

The Idea - Bilateral Filter

Bilateral filtering simultaneously combines the spatial and rangedomain filters

h(x) =1

k(x)

∫ ∞−∞

∫ ∞−∞

f(ξ)c(ξ, x)s(f(ξ), f(x))dξ (5)

where the normalization factor is

k(x) =

∫ ∞−∞

∫ ∞−∞

c(ξ, x)s(f(ξ), f(x))dξ (6)

This process replaces the pixel at x with an average ofphotometric/radiometric similar and nearby pixel values.


Bilateral Filtering

The Gaussian Case

The most common implementation of bilateral filtering utilizes theshift-invariant Gaussian filter for both the closeness, c(ξ, x), andthe similarity, s(f(ξ), f(x)) functions. These filters are Gaussianfunctions of the Euclidean distance between their arguments.

For the closeness function we have

c(ξ, x) = e− 1

2

“d(ξ,x)σd

”2

(7)

where d(ξ, x) is the Euclidean distance between x and ξ

d(ξ, x) = d(ξ − x) = ‖ξ − x‖ (8)


Bilateral Filtering

The Gaussian Case (continued)

Analogously we have the similarity function

s(ξ, x) = e− 1

2

“δ(f(ξ),f(x))

σr

”2

(9)

where δ(φ, f) is the Euclidean distance between two suitableintensity measures φ and f, namely

δ(φ, f) = δ(φ− f) = ‖φ− f‖ (10)

which in the greyscale image case could simply involve imageintensity values.


Bilateral Filtering

The Gaussian Case (continued)

Current Neighborhood Closeness Filter Similarity Filter Bilateral Filter

f(ξ) c(ξ, x) = e− 1

2

„d(ξ,x)σd

«2

s(ξ, x) = e− 1

2

„δ(f(ξ),f(x))

σr

«2

c(ξ, x)s(ξ, x)

σd = 5, σr = 50


Bilateral Filtering

Bilateral Filter Visualization

Row 192Row 128Row 127


http://www.cis.rit.edu/~cnspci/courses/1051-782/media/bilateral/127/movie.mpg



Bilateral Filtering

σd = 1

σd = 3

σd = 5

σd = 10

σr = 10 σr = 30 σr = 50 σr = 100 σr = 300


Bilateral Filtering

Color Image Filtering

When filtering true color images, the range/similarity filter is bestdefined in a CIE-Lab color space where Euclidean distance isproportional to perceptible color changes.


Bilateral Filtering

RGB to CIE-Lab Conversion

This is a two-step process. First the RGB color triplet needs to beconverted to tristimulus values as

R =

{ (R+0.055

1.055

)2.4if R > 0.04045

R12.92 otherwise

G =

{ (G+0.055

1.055

)2.4if G > 0.04045

G12.92 otherwise

B =

{ (B+0.055

1.055

)2.4if B > 0.04045

B12.92 otherwise

(11)


Bilateral Filtering

RGB to CIE-Lab Conversion (continued)

and XYZ

=

0.4124564 0.3575761 0.18043750.2126729 0.7151522 0.07217500.0193339 0.1191920 0.9503041

R

G

B

(12)


Bilateral Filtering


Second, the tristimulus values need to be converted to CIE-Labspace (with reference to a particular illuminant) as

L = 116f

(Y

Yreference

)− 16

a = 500

[f

(X

Xreference

)− f

(Y

Yreference

)]b = 200

[f

(Y

Yreference

)− f

(Z

Zreference

)](13)

where

f (x) =

{x

13 if x > 0.008856

7.787x + 16116 otherwise

(14)


Bilateral Filtering


and the white reference tristimulus values for several differentstandard illuminants are given in the following table

Illuminant Xreference Yreference Zreference

D65 (daylight)* 95.0430 100.0000 108.8801A (tungsten) 109.8490 100.0000 35.5825

F2 (cool white fluorescent) 99.1858 100.0000 67.3938F11 (narrow band fluorescent) 100.9610 100.0000 64.3506

F7 (daylight fluorescent) 95.0416 100.0000 108.7489F8 (daylight fluorescent) 96.4274 100.0000 82.4211

* typically used for RGB values on an standard display


Bilateral Filtering

Example

Original σd = 10, σr = 1 (CIE L*a*b*)


Bilateral Filtering

Example - Iterative Application

Original σd = [5, 5, 5], σr = [1, 1, 1] (CIE L*a*b*)


Bilateral Filtering

References

Tomasi, C. & Manduchi, R. (1998).In Proceedings of the 1998 IEEE International Conferenceon Computer Vision IEEE, Bombay, India.


Introduction to Digital Image Processing 1051 · Bilateral Filtering Noise Reduction I Noise typically presents itself in imagery as a random, high-frequency, uncorrelated modulation

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