Digital Image Processing - UGentsanja/ImageProcessingCourse/03b...Digital Image Processing Dr. ir....

Digital Image Processing

Dr. ir. Aleksandra PizuricaProf. Dr. Ir. Wilfried Philips

30 November 2006

Aleksandra.Pizurica @telin.UGent.be Tel: 09/264.3415

Telecommunicatie en Informatieverwerking

UNIVERSITEIT GENT

Telecommunicatie en Informatieverwerking

UNIVERSITEIT GENT

Image Enhancement

03b.3

version: 30/11/2006 © A. Pizurica, Universiteit Gent, 2006

Objectives of Image Enhacement

• Image enhancement means enhancement for visual interpretation

• This covers a wide variety of processing methods (e.g., making details better visible, improving contrast, making images sharper,…)

• The evaluation of the results is higly subjective (hence no general theory of image enhancement)

• The main objective of image enhancement is to process an image such that the result is more suitable for a specific application.

• Image enhancement trechniques are highly problem oriented (e.g., a method that enhances well an X-ray image may be unsuitable for a satellite image)

• A possible broad categorization:• Spatial domain enhancement techniques• Frequency domain enhancement techniques

03b.4


Spatial Domain Enhancement

Spatial domain –- aggregation of image pixels

g(x,y) = T( f(x,y) )

f(x,y) – input image

f(x,y) – output image

T – operator on f defined over some neighborhood of (x,y) image f(x,y)

x

y

origin

(x,y)

3x3 neighborhood of (x,y)

Simplest case: neighborhoud of size 1x1 (a single pixel)In this case T is intensity transformation

s = T( r ) r and s – grey values of f and g, resp., at any position (x,y)

03b.5


Intensity transformation

Contrast enhancementThe values of r below m are compressed into a narrow range of s towards black. The opposite efect for values above m.

Also: a narrow intensity range around m is transformed into a wider intensity range.

r s

80 180

255

r

s

s=T(r)

m

T(r)

rdark

dark

light

light

03b.6


Contrast enhancement

r s

80 180

255

r

s

r s

110

255

r

slimiting case: thresholding

03b.7


Basic gray-level transformations

negative

log

n-th root

n-th power

inverse log

Values of the transformation function are typically stored in a one dimensional array.

The mapping from r to s is implemented as a look-up-table

input intensity r

Out

put i

nten

sity

s = T

(r)

identity

Which transformation produces this enhancement?

03b.8


Image negative

input intensity r

Out

put i

nten

sity

s = T(r)

© 2002 R. C. Gonzalez & R. E. Woods. Courtesy of GE Medical Systems

s = L-1-r r and s – grey values of input and output image, resp., at any position

Input (and output) gray levels are in the range [0, L-1]

Example: enhancement of digital mammograms

original negative

03b.9


Log transformations

s = c log(1+r) r and s – grey values of input and output image, resp., at any position

Assuming r ≥0 (gray levels are in the range [0, L-1])

input intensity r

outp

ut in

tens

ity

s = T(r) A narow range of low gray-level values in the input image is mapped to a wider range of output values. The opposite is true for higher gray values.

Expands the values of dark pixels. Compresses the values of bright pixels.

Compresses the overall dynamic range. Allows visualisation of images with a large range of pixel values!

03b.10


Visualisation by Log transform

Log transform is commonly used for visualisation of data that have a big dynamic range, such as

• Fourier spectra• Ultrasound data• Optical Coherence Tomography (OCT) data• Radar images• …

Fourier spectrum After log transform

03b.11


0.040.1

0.20.4

0.7

1.52.5

510

25

input intensity r

Out

put i

nten

sity

1

Power low transformations

s = c rγ r and s – grey values of input and output image, resp., at any position

c, γ - positive constants

γ>1 enhancement of details in bright regions

γ<1 enhancement of details in dark regions

03b.12


…Power low transformations…

s = rγγ=0.4 γ=4

s = rγ

03b.13


…Power low transformations

monitor

γ-correction

monitor

03b.14


Histogram processing

kk nrh =)(

kr : k-th grey level in the image

kn : number of pixels having gray level kr

Normalized histogramnnrp kk /)( = , n: number of image pixels

p(rk) is an estimate of probability of occurence of gray level rk

• Histograms are simple to calculate in software and hardware

• Provide image statistics

• Useful in image compression, denoising, analysis

03b.15


Histogram equalization

Conclusion: to flatten (equalize) the histogram, transform the image with its cumulative histogram!

input intensity r

equalized Image

histogram histogram

∑∑==

===k

j

jk

jjkk n

nrprTs

00)()(

This mapping is cumulative histogram

s = T(r)

03b.16


local histogramequalization

( 7x7windows)

global histogram

equalization

Local processing

original image

It is often useful to apply intensity transformations locally (within local windows)

In some cases such processing creates artifacts! (e.g., can create non-existing structures in an almost flat background)

03b.17


∑Ω∈

++=),(

),(),(),(ts

tysxftsmyxg

1,0,1,:),( −==Ω tstssupport of the mask

Basics of spatial filtering

image f(x,y)

x

y

origin

m(-1,1) m(-1,0) m(-1,1)

m(0,-1) m(0,0) m(0,1)

m(1,1) m(1,0) m(1,1)

filtear mask

)1,1()1,1(...)1,1()1,1(),( ++++−−−−= yxfmyxfmyxg

03b.18


7 3 2

0 9 1

1 2 3

Smoothing spatial filtersInput image output image

3 5

0 2

1 1

1 1 2 2 2

1 3 3 2 2

4 33

),( yxb ),( yxg

x

y

2#Ω−1 calculationsper pixel!∑

Ω∈++=

),(),(),(),(

tstsmytxsgyxgCorrelation!:

De index set Ω is the support of m

1,0,1,:),( −==Ω yxyx Ω includes usually (0,0) but not always!

Filter mask),( yxm

1 2 1

2 4 2

1 2 1

116

x

y

© W. Philips, Universiteit Gent, 1998-2006

03b.19


Example: mean filterOriginal mean over 5x5 window

=

1111111111111111111111111

251M

This filter partly suppresses noise but blurs the edgesNote: this filter is separable


03b.20


Order statistics filters

Order statistics filters are nonlinear spatial filters.

Operation: (1) perform ordering (ranking) the pixels within a filter mask. (2) replace the value of the center pixel by the ranking result

Best known example in this class: median filter

03b.21


2

4 3 2

0 9 1

1 2 3

Input image Output image

3 5

0 2

1 1

1 1 2 2 2

1 3 3 2 2

),( yxb ),( yxg

x

y

sorteren 0 1 1 2 2 3 3 4 9

Order statistics filters: Median filter

Basic idea: remove “outliers”The median is a more robust statistical measure than mean


03b.22


2 9 9 9

Properties of the median filter...2 2 22 9 22 2 2

+removes isolated pulses+preserves uniform areas

2 2 2 2 2 2 2 2 9

-removes thin vertical, horizontal, diagonal lines

2 9 22 9 22 9 2

2 2 2 2 2 2 9 9 9

+preserves “soft” horizontal and vertical edges

+preserves sharp horizontal and vertical edges

2 2 922 2 9

9 9 99

9 9 9

2 3 5 92 2 322 2 3

5 9 99

5 9 9

2 2 92 2 92 2 9

9 9 99 9 99 9 9

2 2 32 2 32 2 3

5 9 95 9 95 9 9


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... Properties of the median filter...

+preserves sharp diagonal edges

2 9 92 2 92 2 2

9 9 99 9 99 9 9

2 2 2 2 9 9

2 92 2

9 99 9

-damages corners2 3 3

Some of these problems can be avoided by choosing not-rectangular windows

+preserves “soft” diagonal edges

2 3 52 2 32 2 2

9 9 95 9 93 5 9

2 2 2 2 3 5

2 32 2

5 93 5

2 2 22 2 22 2 3

2 2 22 2 23 3 3

2 2 3 3 3 322 2 2 2

2 3 3 3


03b.24


Removal of impuls noise

median over 3x3

Median filter removes isolated noise peaks,without blurring the image

Impuls noise


03b.25


... Reduction of impuls noise …

median over 3x3

Median filter removes isolated noise peaks,without blurring the image

Noise-free original

03b.26


... Reduction of impulse noise

mean over 3x3 median over 3x3

Linear filters do not take into account impulsive (isolated ) character of the noise and blur the image

The median filter performs in this case much better than the linear filters

03b.27


Median filter and reduction of white noise

median over 3x3

For not-isolated noise peaks (e.g., white Gaussian noise) median filter is not very efficient.

original

03b.28


Repeated application of the median filter

Iterating median filter can remove noise better, but flat blobs appear in the image

median over 3x3 median over 3x3, 8x applied

03b.29


Sharpening spatial filters

• Principal objective: highlight fine detail or enhance image detail

• Often based on the first-order and second-order derivatives

• The derivatives of digital functions are defined in terms of differences

)()1( xfxfxf −+≈∂∂

)(2)1()1(2

2xfxfxf

xf −−++≈

∂∂

• In most applications second-order derivative is better suited for image enhancement

• First-order derivatives are used for edge extraction but also have important use in image enhancement

03b.30


),(~yx ffH

Second order derivatives: Laplaciaan…

2

2

2

22

yf

xff

∂∂+

∂∂=∇

),(),(),(),(),(

11

141

++−+++−−≈

yxfyxfyxfyxyxf f

digital implementation

−

010141010

lineair filter with filter mask

Enhancement:),(),(),( 2 yxfyxfyxg ∇−=

03b.31


Enhancement:),(),(),( 2 yxfyxfyxg ∇−=

… Second order derivatives: Laplaciaan

(b) Laplacian filtered

(c) Image (b) Scaled for

visualisation

difference (a)-(b)

(a) original

03b.32


Unsharp masking

Subtract a blurred version of an image from image itself

A generalization: high-boost filtering),(),(),( yxfyxAfyxfhb −=

),(),(),( yxfyxfyxfs −=

blurred version of f(x,y)

This can be re-written as

),(),()1(),(),(),()1(),(

yxfyxfAyxfyxfyxfAyxf

s

hb

+−==−+−=

If we choose that the sharp image as:

),(),(),( 2 yxfyxAfyxfhb ∇−=

),(),( 2 yxfyxfs −∇=

03b.33


“Unsharp masking” – a usual formulationUnsharp masking : approximate inverse filter for Gaussian PSF with parameter σ :

),(2

),(),( 22

yxfyxfyxg ∇−=σ

This filter amplifies noise. The noise amplification increases when z0increases

2/20 σ=z with

Filter mask:

−−

010141010

000010000

0z

Note: if the noise amplification is a problem, choose z0<σ2/2⇒ blurring increases but not necessarily excessively⇒ less noise amplification

03b.34


Unsharp masking: example

−−

⇒

010141010

000010000

0zUnsharp masking with z0=0.5

If z0 is bigger (e.g. z0=1) the resulting image is extra sharp

Gaussian filtered (z0=0.5)

Test image 1

Gaussian filtered (z0=0.5) +ruis

Test image 2

03b.35


Unsharp masking and noise

Test image 1 test image 2

filtered; z0=1

originals

sharper

noisier

Homomorphic filtering

03b.37


Homomorfische filteringGoal: local contrast enhancement: compensation of spatially varying illumination

log(1+b’(x,y )) high pass filter

0.50.250

0.25

0.5

0.75

fx

H(fx,0)

exp(bl(x,y )-1)bl(x,y )

Illumination model: b’(x,y )=b(x,y )I(x,y ) met b(x,y ) the ideal image and I(x,y ) the not-uniform lighting ⇒ log(b’(x,y ))=log(b(x,y ))+log(I(x,y ))

Assumption: log(I(x,y )) varies slowly w.r.t log(b(x,y ))⇒ remove log(I(x,y )) by high pass filtering log(b’(x,y ))

avoid log(0)


03b.38


Example: homomorphic filter

Remark: •Homomorphic filter is here applied to Y-component; U and V were not changed

original image after filtering

•Truncation of RGB-values <0 en >255


03b.39


Example: homomorphic filter

Homomorphic filter followed by an extra intensitytransform to improve global contrast

original image after filtering

homomor. filter contrast adjust g

g’


03b.40


Filtering in the Fourier domain

2D-DFTof the mask

2D-DFT of an image

2D-DFT of the filtered

image⊗

filtered image

inverseDFT

lkB ,

lkH ,~


03b.41


Vergelijking van het aantal berekeningen

Filteren van een N × N beeld met een n × n masker:•Via convolutie/correlatie:

• scheidbaar filter: 4n-2 bewerkingen/pixel• niet-scheidbaar filter: 2n2-1 bewerkingen/pixel

Voorwaartse en inverse DFT (via FFT)

Vermenigvuldiging in frequentiedomein

1log4log422

22

2+=

+ NN

NNN bewerkingen/pixel

•Via de DFT (als we de DFT van het masker al kennen):

De DFT-implementatie is voordeliger voor grote maskers•Break-even punt (niet-scheidbaar): (b.v. N=256 ⇒ n=4)Nn 2log2≈


03b.42


Voorbeelden: laagdoorlaatfilters

[ ]121121

161

121242121

161

=

Binomiaal-filter (3x3)

[ ]111111

91

111111111

91

=

Uitmiddeling over 3x3 omgeving

zorgt ervoor dat ⇒behoudt gemiddelde grijswaarde

fy=2l/Nfx =2k/M

geen zijlobes!gewenst als men hoge

frequenties wil onderdrukkenlk

yx

H

ffH

,~

),(~

=100 =),(~H

100 =),(~H

100 =),(~H


03b.43


Filter in het fourierdomein: voorbeeldreconstructie met 25% van de coëfficiënten

reconstructie met 6.25% van de coëfficiënten

Invloed op het beeld: • “Ringing” (rimpels)• “Blurring” (wazig maken)

Filter: binnen het blauw vierkant en 0 erbuiten1~, =lkH

Dit komt erop neer de fouriercoëfficiënten bij hoge spatiale frequenties nul te maken ⇒laagdoorlaatfilter


Digital Image Processing - UGentsanja/ImageProcessingCourse/03b...Digital Image Processing Dr. ir....

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