Digital Image Processing Image Filtering - IRISApeople.irisa.fr/Olivier.Le_Meur/teaching/MiticVis_partII.pdf · Digital Image Processing Image Filtering ... rFom R.C. Gonzalez, R.E.

Table of Content

Digital Image ProcessingImage Filtering

O. Le [email protected]

Univ. of Rennes 1http://www.irisa.fr/temics/staff/lemeur/

January 2, 2011

1

mailto:[email protected]

http://www.irisa.fr/temics/staff/lemeur/

Table of Content

1 Introduction

2 Point-to-point transformation

3 Linear �ltering (neighborhoodoperator)

4 Non Linear �ltering

5 Conclusion

2

IntroductionPoint-to-point transformation

Linear �ltering (neighborhood operator)Non Linear �ltering

Conclusion

Introduction

1 Introduction




5 Conclusion

3



Conclusion

Introduction: image transformation

There exist 3 types of transformation:

Point to point transformation:The output value at a speci�c coordinateis dependent only on one input value butnot necessarily at the same coordinate;

Local to point transformation:The output value at a speci�c coordinateis dependent on the input values in theneighborhood of that same coordinate;

Global to point transformation:The output value at a speci�c coordinateis dependent on all the values in the inputimage.

Note that the complexity increases with the size of the considered neighborhood...

4



Conclusion

Spatial coordinates-based transformationsPixel values-based transformations

Point-to-point transformation

1 Introduction

2 Point-to-point transformationSpatial coordinates-basedtransformationsPixel values-based transformations



5 Conclusion

5



Conclusion


Spatial coordinates-based transformations

Remark:

This section is composed of several pictures extracted fromhttp://eeweb.poly.edu/~onur/lectures/lectures.html.

Let im[x, y ] be an input image of size N × N.

A spatial coordinates-based transformation, also called warping, aims at providing animage IM[k, l ] from the input image im[x, y ]:

IM[k, l ] = im[x(k, l), y(k, l)]

x(k, l) and y(k, l) are the transformations or the pixel warping functions. These functionsjust modify the spatial coordinates of a pixel not its value;

Special cases to take into consideration:

the new coordinates [x(k, l), y(k, l)] is out of the image IM. In this case,IM[k, l ] = 0;the new coordinates must be integers (Rounding operation, nearest integers...).

6

http://eeweb.poly.edu/~onur/lectures/lectures.html



Conclusion


Transpose

The transpose tansformation is given by

x(k, l) = l

y(k, l) = k

7



Conclusion


Vertical Flip

The vertical �ip tansformation is given by

x(k, l) = N − k

y(k, l) = l

The horizontal �ip tansformation is given by

x(k, l) = k

y(k, l) = N − l

8



Conclusion


Translation

The translation tansformation is given by

x(k, l) = k + Tk

y(k, l) = l + Tl

where Tk and Tl are the transalation values for the x-axis and the y-axis respectively.

In the example below, we have (Tl = −50)

IM[k, l ] = im[x(k, l), y(k, l)]

IM[k, l ] = im[k, l + Tl ]

Di�erent transformations can be obtained depending on Tk and Tl values.

9



Conclusion


Rotation

The rotation tansformation is given by

x(k, l) = (k − x0)cosθ − (l − y0)sinθ + x0

y(k, l) = (k − x0)sinθ + (l − y0)cosθ + y0

where [x0, y0] is the spatial coordinates of the center of the rotation and θ the angle.

Extracted from http://eeweb.poly.edu/ onur/lectures/lectures.html.

10



Conclusion


Wave

x(k, l) = k

y(k, l) = l + α× sin(β × l)

x(k, l) = k + α× sin(β × k)

y(k, l) = l

α and β can be used to strengthen the e�ect.

11



Conclusion


Warp and swirl

x(k, l) = k

y(k, l) =sign(l − y0)

y0× (l − y0)2 + y0

The swirl e�ect is a rotation but theangle of the rotation θ varies with thepixel distance from the center of theimage [x0, y0]:

d =√

(k − x0)2 + (l − y0)2

θ =π

512r

If r −→ 0, θ is small...

12



Conclusion


Glass e�ect

A glass e�ect is obtained by adding a small and random displacement to each pixel:

x(k, l) = k + (RAND(1, 1)− 1/2)× 10

y(k, l) = l + (RAND(1, 1)− 1/2)× 10

Extracted from http://eeweb.poly.edu/ onur/lectures/lectures.html.

13



Conclusion


Summary

All 2D linear transformations: [x

y

]=

[a b

c d

] [k

l

]Scale, rotation, mirror...

Properties:

Origin maps to origin;

Lines map to lines;

Ratios are preserved...

A�ne transformations (linear transf. + translation):xyw

=

a b c

d e f

0 0 1

klw

Properties:

Origin does not necessarily map to origin;

Lines map to lines;

Ratios are preserved...

14



Conclusion


Summary

A�ne transformations (linear transf. + translation):

Translation: xy1

=

1 0 tx0 1 ty0 0 1

kl1

Scale: xy

1

=

sx 0 00 sy 00 0 1

kl1

2D in-plane rotation: xy

1

=

cosθ −sinθ 0sinθ cosθ 00 0 1

kl1

15



Conclusion


Pixel values-based transformations

Let im[x , y ] be an input image of size N × N.

A pixel values-based transformation aims at providing an image IM[k, l ] from theinput image im[x , y ]:

IM[k, l ] = f (im[k, l ])

Noticed that the spatial coordinates of pixel are not modi�ed. The function f isused to modify the pixel values.

The simplest one is the identity function: f (p) = p:

IM[k, l ] = f (im[k, l ])

IM[k, l ] = im[k, l ]

16



Conclusion


Histogram

An image histogram is a graphical representation of the tonal distribution in a digitalimage. It plots the number of pixels for each tonal value.

Histogram gives information about the global distribution of an image.

Histogram plots the number of pixels in the image (vertical axis) with a particularbrightness value (horizontal axis).

(a) Original (b) Histogram (intensity)

17



Conclusion


Histogram

(a) Original (b) Histogram (intensity)

(c) Original (d) Histogram (intensity)

18



Conclusion


Histogram

High dynamic range in the last caseproviding the best quality.

From R.C. Gonzalez, R.E. Woods, Digital image processing.

19



Conclusion


Histogram equalization

The goal is to increase the global contrast of images, especially when the usable dataof the image is represented by close contrast values.

Consider a discrete grayscale image {x} and let ni be the number of occurrences ofgray level i .

The probability of an occurrence of a pixel of level i in the image ispx (i) = p(x = i) =

nin, 0 ≤ i < L, L being the total number of gray levels in the

image, n being the total number of pixels in the image.

Let us also de�ne the cumulative distribution function: cdfx (i) =∑i

j=0 px (j).

We want to produce an image {y}, such that cdfy (i) = iK .

y = cdfx (x)× (max {x} −min {x}) + min {x}

20



Conclusion


Negative image

The negative image is obtained by f (p) = 255− p (pixel values are coded on 8 bits).

IM[k, l ] = f (im[k, l ])

IM[k, l ] = 255− im[k, l ]

21



Conclusion


Piece continuous transformation

The objective of such transformation is:

to compress pre-determined ranges of values: Range compression;

to accentuate pre-determined ranges of values: Range stretching.

f (p) =

α1 × p 0 ≤ p < a1α2 × (p − a1) a1 ≤ p < a2...

αi (p − ai−1) +(∑i−1

j=1 αj (αj − αj−1)

ai−1 ≤ p < ai

...

Obviously, we have:

αi < 1, compression;

αi > 1, stretching.

22



Conclusion



Example of contrast stretching:

Let f a function de�ned on three pieces:

f (p) =

α1p, 0 ≤ p < a1α2(p − a1) + α1a1, a1 ≤ p < a2α3(p − a2) + (α2(a2 − a1) + α1a1), a2 ≤ p ≤ 255

23



Conclusion



Example of contrast stretching:

Let imaging that α1 and α2 are null. The �ltered image contains only grey levelbelonging to [a1, a2]. We just keep a slice of the image.

24



Conclusion



Binary thresholding:

IM[x , y ] =

{0 im[x,y]<T255 otherwise

Gamma correction:

IM[x , y ] = im[x , y ]γ

25



Conclusion

De�nitionLow-pass �lters in spatial domainHigh-pass �lters in spatial domainDi�erentiation �lter in spatial domainFrequency domain �ltering

Linear �ltering with neighborhoods operator

1 Introduction


3 Linear �ltering (neighborhoodoperator)De�nitionLow-pass �lters in spatial domainHigh-pass �lters in spatial domainDi�erentiation �lter in spatialdomainFrequency domain �ltering


5 Conclusion

26



Conclusion


De�nition

De�nition of neighborhood around a pixel of spatial coordinate (x , y). Theneighborhood is called V (x , y):

Two examples:

27



Conclusion


De�nition

2D Finite Impluse Response (FIR) �lter:

IM[x , y ] = (im ∗ h)[x , y ]

IM[x , y ]︸︷︷︸Output

=∑

k∈V (x,y)

∑l∈V (x,y)

h(k, l)︸︷︷︸Filter coe�.

im[x − k, y − l ]︸︷︷︸Input

with h is the 2D impulse response called also the Point Spread Function (PSF) or thekernel of the transform. It is composed of the �lter coe�cients (�nite length).

The gain of the �lter is equal to

g =∑

k∈V (x,y)

∑l∈V (x,y)

h(i , j)

28



Conclusion


De�nition


=∑

k∈V (x,y)

∑l∈V (x,y)



where H is the convolution kernel.

Example: the �lter support is (3× 5). The convolution kernel is:

H =

h(−2,−1) h(−1,−1) h(0,−1) h(1,−1) h(2,−1)h(−2, 0) h(−1, 0) h(0, 0) h(1, 0) h(2, 0)h(−2,−1) h(−1,−1) h(0,−1) h(1,−1) h(2,−1)

29



Conclusion


De�nition

Example for a neigborhood of size (2N + 1)× (2N + 1):

IM[x , y ] =N∑

k=−N

N∑l=−N

h(k, l)im[x−k, y−l ]

Number of multiplications per output point: O(N2)

N=2

30



Conclusion


Average �lter

The most simple low-pass �lter is the local averaging operation. The main e�ect of alow-pass �lter is a blurring. The size of the kernel is (2N + 1)× (2N + 1):

h(k, l) =

{1

(2N+1)2−N ≤ k, l ≤ −N

0 Otherwise

For N = 1, the convolution kernel is given by:

H =1

9

1 1 11 1 11 1 1

31



Conclusion


Average �lter

Three examples of averaging for di�erent sizes of kernel. From the left-hand side tothe right-side, N = {1, 3, 8}:

The amount of blur increases with the size of the kernel (the number of operation tooO(N2)). In order to �lter pixels located near the edges of the image, edge pixel valuesare replicated to give su�cient data (this is not the case here).

32



Conclusion


Gaussian �lter

The kernel h is given by the following function

h(x , y) =1

2πσ2exp

(−x2 + y2

2σ2

)

Each pixel's new value is set to a weighted average of that pixel's neighborhood. Theoriginal pixel's value receives the heaviest weight (having the highest Gaussian value)and neighboring pixels receive smaller weights as their distance to the original pixelincreases. This results in a blur that preserves boundaries and edges better than other,more uniform blurring �lters.Note that the �lter support is truncated...

33



Conclusion


Gaussian �lter

h(x , y) =1

2πσ2exp

(−x2 + y2

2σ2

)Example of kernel:

σ = 0.84089642, N = 3:Note that the center element (at [4, 4]) has the largest value, decreasingsymmetrically as distance from the center increases.

H =

0.00000067 0.00002292 0.00019117 0.00038771 0.00019117 0.00002292 0.000000670.00002292 0.00078633 0.00655965 0.01330373 0.00655965 0.00078633 0.000022920.00019117 0.00655965 0.05472157 0.11098164 0.05472157 0.00655965 0.000191170.00038771 0.01330373 0.11098164 0.22508352 0.11098164 0.01330373 0.000387710.00019117 0.00655965 0.05472157 0.11098164 0.05472157 0.00655965 0.000191170.00002292 0.00078633 0.00655965 0.01330373 0.00655965 0.00078633 0.000022920.00000067 0.00002292 0.00019117 0.00038771 0.00019117 0.00002292 0.00000067

Note that 0.22508352 (the central one) is 1177 times larger than 0.00019117which is just outside 3σ.

34



Conclusion


Gaussian �lter

h(x , y) =1

2πσ2exp

(−x2 + y2

2σ2

)Example of kernel:

σ = 0.6, N = 1:

H =1

16

1 2 12 4 21 2 1

σ = 1, N = 2:

H =1

1444

1 9 18 9 19 81 162 81 918 162 324 162 189 81 162 81 91 9 18 9 1

35



Conclusion


Gaussian �lter

Example of Gaussian �ltering with σ = 2:

36



Conclusion


Gaussian �lter

Increasing sigma increases the smoothing

37



Conclusion


Other low pass �lters

2D Pyramidal �lter:

H =1

81

1 2 3 2 12 4 6 4 23 6 9 6 32 4 6 4 21 2 3 2 1

Conic �lter:

H =1

25

0 0 1 0 00 2 2 2 01 2 5 2 10 2 2 2 00 0 1 0 0

38



Conclusion


High-pass �lter

The high-pass �ltered image can be thought of as the original image minus the lowpass �ltered image.

IM[x , y ] = im[x , y ]−N∑

k=−N

N∑l=−N

h(k, l)im[x − k, y − l ]

IM[x , y ] =N∑

k=−N

N∑l=−N

h(k, l)im[x − k, y − l ]

with h the convolution kernel.

H =

0 −1 0−1 5 −10 −1 0

H =

−1 −1 −1−1 9 −1−1 −1 −1

H =

1 −2 1−2 5 −21 −2 1

39



Conclusion


High-pass �lter

Three examples of high-pass �ltering for di�erent sizes of kernel. From the left-handside to the right-side, N = {1, 3, 8}:

When the kernel's size increases, the �ltering is more important and then the result isless noisy.

40



Conclusion


Di�erentiation �lter

f ′(x) = lim∆x→0

f (x + ∆x)− f (x)

∆x

Local variations of intensity are an important source of information in imageprocessing. These local variations are gradient (it measures the rate of change of thefunction):

∇im[k, l ] =

[∂im

∂x[k, l ],

∂im

∂y[k, l ]

]In the illustration below: GX = ∂im

∂x[k, l ] and GY = ∂im

∂y[k, l ].

41



Conclusion



GX = ∂im∂x

[k, l ]: Kernel = hx =[−1 1

]∂im

∂x[k, l ] ≈ im[k + 1, l ]− im[k, l ]

GY = ∂im∂y

[k, l ] Kernel = hy =[−1 1

]T∂im

∂y[k, l ] ≈ im[k, l + 1]− im[k, l ]

However, most of the time, we use the following kernel[−1 0 1

]and[

−1 0 1]T

(phase =zero). Below, from the left-hand side to the right: original,[−1 1

],[−1 0 1

].

42



Conclusion



However, these �lters are very sensitive to the noise. In order to enhance therobustness, these �lters are combined with a blurring �lter:

hx =

−1 0 1−2 0 2−1 0 1

hy =

−1 −2 −10 0 01 2 1

These kernel are the Sobel's kernel. The blurring �lter is

[1 2 1

].

IMx [k, l ] = (im ∗ hx )[k, l ]

IMy [k, l ] = (im ∗ hy )[k, l ]

In the same vein, Prewitt and Robert's �lters.

43



Conclusion



Norm of the gradient: ‖∇IM[k, l ]‖2 =√IMx [k, l ]2 + IMy [k, l ]2;

Its orientation: arg(∇IM[k, l ]) = arctan(IMy [k,l ]

IMx [k,l ]

)From left-hand side to the right: IMx , IMy and the norm.

44



Conclusion



The Laplacian of a picture is the second derivative:

∂2im

∂2x[k, l ] ≈ im[k + 1, l ] + im[k − 1, l ]− 2× im[k, l ]

∂2im

∂2y[k, l ] ≈ im[k, l + 1] + im[k, l − 1]− 2× im[k, l ]

∇2im[k, l ] =∂2im

∂2x[k, l ] +

∂2im

∂2y[k, l ]

∇2im[k, l ] ≈ im[k + 1, l ] + im[k − 1, l ] + im[k, l + 1] + im[k, l − 1]− 4× im[k, l ]

For a 4-neighborhood, the kernel is given by

h =

0 1 01 −4 10 1 0

We can extent this kernel to compute the laplacian in all directions (8-neighborhood):

h =

1 1 11 −8 11 1 1

45



Conclusion



The second order derivatives have a stronger response to �ne details (e.g. thin lines)than the �rst order derivatives.

46



Conclusion


Frequency domain �ltering

where H is the convolution kernel.

IM[x , y ] = (im ∗ h)[x , y ]

im1[x , y ] ∗ im2[x , y ]F→ IM1[u, v ]× IM2[u, v ]

im1[x , y ]× im2[x , y ]F→ IM1[u, v ] ∗ IM2[u, v ]

When the size of the kernel is large, it is better to apply the �lter in the frequency domain.

For more information:Digital Image Processing, by R. C. Gonzalez and R. E. Woods, 3rd edition, Pearson PrenticeHall, 2008.

47



Conclusion



We can spatially �lter an image by Fourier transforming and applying a frequency�lter:

IM[x , y ] = im[x , y ] ∗ h[x , y ]

˜IM[u, v ] = IM[u, v ]× H[u, v ]

where, H[u, v ] is the �lter in the frequency function.

48



Conclusion


Ideal low pass �lter

From the left-hand side to the right: Ideal low pass �lter transfert function, �lterdisplayed as an image, �lter radial cross section.

H(u, v) =

{1 D(u, v) ≤ D0

0 D(u, v) > D0

With D the euclidean distance from the spectrum center (N2, N2

).

49



Conclusion


Ideal low pass �lter

Low pass �ltering:

Ringing and blurring

50



Conclusion


Butterworth low pass �lter

H(u, v) =1

1 +(D(u,v)D0

)2n

51



Conclusion


Butterworth low pass �lter

Top: spatial representation of the �lter for di�erent orders;Bottom: intensity pro�les through the center of the �lters.

Butterworth low pass �ltering:

Smooth transition in blurring, no ringing is present.

52



Conclusion


Gaussian low pass �lter

H(u, v) = exp

(−D2(u, v)

2D20

)with D0 = σ. Gaussian low pass �ltering:

Smooth transition in blurring, no ringing is present.

53



Conclusion


Low pass �lter

Comparison between the ideal, the Butterworth and the gaussian low pass �lter:

Ideal

Butterworth

Gaussian

54



Conclusion


High pass �ltering in the frequency domain

HHP(u, v) = 1− HLP(u, v)

55



Conclusion


High pass �ltering in the frequency domain

Ideal high-pass �lters enhance edges but su�er from ringing artefacts, just likeIdeal LPF;

smoother results with the two others.

56



Conclusion



Example of two band-pass �lters:

57



Conclusion


Laplacian in the frequency domain

We remind (see previous lecture):

∇2im[k, l ] =∂2im

∂2x[k, l ] +

∂2im

∂2y[k, l ]

dnx(t)dtn

F−→ (j2πf )nX (f ).

From this, it follows that

∂2im

∂2x[k, l ] +

∂2im

∂2y[k, l ]

F−→ −(u2 + v2)IM[u, v ]

The Laplacian �lter is then implemented in the frequency domain by

H(u, v) = −(u2 + v2)

58



Conclusion


Laplacian in the frequency domain

Finally, to compute the Laplacian, we need :

1 to compute the Fourier transform of the picture;

2 to multiply the spectrum by −(u2 + v2);

3 to compute the inverse Fourier transform.

59



Conclusion


The cortex transform

The cortex transform is �rst described by A. Watson as the modeling of the neuralresponse of retinal cells to visual stimuli.

The cortex �lter in the frequency domain is

cortex[bi ,θi ](ρ, θ) = dombi× fanθi (θ)

Where,

bi and θi represent the frequency band and the index of orientation, respectively;

(ρ, θ) are polar coordinates.

The cortex transform decomposes the input image im[x , y ] into a set of subbandimages B[bi ,θi ][k, l ]:

B[bi ,θi ][k, l ] = F−1{cortex[bi ,θi ](ρ, θ)×F {im[x , y ]}

}

60



Conclusion


The cortex transform

Frequency responses of several cortex �lters (brightness represents gain for the givenspatial frequency).

Complete set of cortex �lters

61



Conclusion

De�nitionRank �lteringHomomorphic �lteringAdaptive �ltering

Introduction

1 Introduction



4 Non Linear �lteringDe�nitionRank �lteringHomomorphic �lteringAdaptive �ltering

Conditional meanAnisotropic Kuwahara �lteringBilateral �ltering

5 Conclusion

62



Conclusion


De�nition

The most important drawback of the linear �ltering is that all pixels in the image aremodi�ed by the �ltering process.

To overcome this problem, non linear �ltering is used. It aims, for instance, to protectsome parts of the picture having particular features (edges...) or to remove data

without blurring the whole image (impluse noise).

Three kinds of �lters:

Rank �ltering;

Adaptive linear �ltering;

Morphological operators.

63



Conclusion


Rank �ltering

Rank �lters are based on three steps:

1 Data selection, also calledwindowing;

2 Data ranking (in ascendant order);

3 1D linear weighting of the ordereddata list.

64



Conclusion


Special case of a generalized rank �lter

If all weights of the linear �lter are null, except one in the median position. This �lteris called a median �lter.

IM[x , y ] = MED (im[xi , yi ]|[xi , yi ] ∈ V [x , y ])

V [x , y ] is the neighborhood (a set of N samples).

if the size of the neighborhood is odd, the output value is the median value;

if the size is even, the output value is the average of the two middles values.

The median �lter is very e�cient in �ltering signals corrupted by impulsive noise but itis not very e�cient in gaussian noise environment.

65



Conclusion


Special case of a generalized rank �lter

However, when the number of the samples is large, the ordering procedure becomescumbersome.

Idea: the median �lter is taken over the outputs of several FIR substructures and thenumber of the substructures is much smaller than the number of the data samplesinside the �lter window.

IM[x , y ] = MED (y(1), . . . , y(m))

where, m is linear FIR �lters.

66



Conclusion


Homomorphic �ltering

Homomorphic �ltering is a generalized technique for signal and image processing, in-volving a nonlinear mapping to a di�erent domain in which linear �lter techniques areapplied, followed by mapping back to the original domain.

In many case, we want to remove shading e�ects from an image. The objective is then

to enhance high frequencies;

to attenuate low frequencies (but �ne details have to be preserved).

Consider the following model of image formation:

im[x , y ] = i(x , y)︸︷︷︸illumination

× r(x , y)︸︷︷︸re�ection

The illumination component varies slowly and then a�ects low frequencies mostly;

The re�ection component varies faster and then a�ects the high frequenciesmostly.

67



Conclusion



What is the solution to separate LF and HF?

im[x , y ] = i [x , y ]× r [x , y ]

Fourier transform

IM[u, v ] = I [u, v ] ∗ R[u, v ]

Due to the convolution in the frequency domain, LF and HF from i [x , y ] and r [x , y ]are mixed together...

Too complex to �lter LF and HF in such condtion.

68



Conclusion



What is the solution to separate LF and HF? We can take the log !

im[x, y ] = i [x, y ]× r [x, y ]

log(im[x, y ]) = log(i [x, y ]× r [x, y ])

log(im[x, y ]) = log(i [x, y ]) + log(r [x, y ])

1 Take the log and apply the Fourier tansform to the new signal:

log(im[x, y ]) = log(i [x, y ]) + log(r [x, y ])

F (log(im[x, y ])) = F (log(i [x, y ])) + F (log(r [x, y ]))

Z [u, v ] = Ilog [u, v ] + Rlog [u, v ]

2 Filtering in the frequency domain: H[u, v ]

Z [u, v ]× H[u, v ] = Ilog [u, v ]× H[u, v ] + Rlog [u, v ]× H[u, v ]

3 Take the inverse Fourier transform and apply the exponential function:

F−1 (Z [u, v ]× H[u, v ]) = F−1 (Ilog [u, v ]× H[u, v ]

)+ F−1 (

Rlog [u, v ]× H[u, v ])

z[x, y ] = i [x, y ] + r [x, y ]

exp (z[x, y ]) = exp(i [x, y ]

)+ exp (r [x, y ])

69



Conclusion



H[u, v ] = γL +γH

1 +

[D0√u2+v2

]2nwith,

γL a parameter that a�ects low frequencies;

γH a parameter that a�ects high frequencies;

70



Conclusion


Adaptive �ltering

An adaptive �lter is a �lter that self-adjusts its transfer function according to anoptimizing algorithm.

The goal is still to smooth the signal. However, we want to preserve edges...

Filtering by pixel grouping;

Conditional mean, Bilateral �ltering and mean shift �lter;

Di�usion.

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Conclusion


Conditional mean


=∑

k∈V (x,y)

∑l∈V (x,y)



Principle: pixels in a neighbourhood are averaged only if they di�er from the centralpixel by less than a given threshold.

h(k, l) =

{1 if |IM[x − k, y − k]− IM[k, l ]| < TH

0 Otherwise.

Example with a neighbourhood equal (2× 3 + 1)(2× 3 + 1),TH = 32:

72



Conclusion


Anisotropic Kuwahara �ltering

Method proposed by Kuwahara and adapted by Nagao in 1980.

Selection of the sub-domain thathas the minimum variance (9windows for Nagao);

Replace the value of the centralpixel by the average value of thesub-domain having the minimumvariance.

Example for a window 5× 5:

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Conclusion


Anisotropic Kuwahara �ltering

74



Conclusion


Bilateral �ltering

The idea is to use a weighted �ltering but with an outlier rejection.Pixels that are very di�erent in intensity from the central pixel are weighted less eventhough they may be in close proximity to the central pixel.

This is applied as two Gaussian �lters at a localized pixel neighborhood:

IM[x , y ] =1∑

k,l c[x,y ][k, l ]s[x,y ][k, l ]

∑k,l

im[k, l ] c[x,y ][k, l ]s[x,y ][k, l ]︸︷︷︸h[k,l ]

One in the spatial domain named the domain �lter:

c[x,y ][k, l ] = exp

(−d([x , y ], [k, l ])

2σd

)where d is Euclidean distance.

One in the intensity space named the range �lter:

s[x,y ][k, l ] = exp

(−φ(im[x , y ], im[k, l ])

2σr

)where φ is a suitable measure of distance in intensity space.

75



Conclusion



76



Conclusion



Low contrast texture has been removed and edges are well preserved.

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Conclusion

Conclusion

1 Introduction




5 Conclusion

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Conclusion

Conclusion

Image Transformation: global to point; point to point; local to point;

Histogram;

Linear Filtering;

Non linear �ltering.

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Digital Image Processing Image Filtering - IRISApeople.irisa.fr/Olivier.Le_Meur/teaching/MiticVis_partII.pdf · Digital Image Processing Image Filtering ... rFom R.C. Gonzalez, R.E.

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