2 Point Operations

7/31/2019 2 Point Operations

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Bernd Girod: EE368 Digital Image Processing! Point Operations no. 1

Chapter Overview: Point Operations

Relating gray values to brightness!

Gray value quantization!

Webers Law!

Gamma characteristic!

Adjusting brightness and contrast!

Gray-level histograms and histogram equalization!

Point operations for combining images! Averaging!

Subtraction!

The need for image registration!


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Quantization: how many bits per pixel?

8 bits 5 bits 4 bits

3 bits 2 bits 1 bit

Contouring


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How many gray levels are required?

Contouring is most visible for a ramp!

Digital images typically are quantized to 256 gray levels. !

32 levels!

64 levels!

128 levels!

256 levels!


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130!

129!

128!

127!

126!

125!

Brightness discrimination experiment

Can you see the circle?!

Visibility threshold!

I

I+ I

I I KWeber

12%Weber fraction!Webers Law!

Note: I is luminance,

measured in cd m2


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Contrast ratio without contouring

Luminance ratio between two successive quantizationlevels at visibility threshold!

For!

Typical display contrast ratio!

Modern flat panel display in dark room 1000:1!

Cathode ray tube 100:1!

Print on paper 10:1!

Suggests uniform perception in the log(I) domain(Fechners Law)!

Imax

Imin

= 1+ KWeber( )

N1

K

Weber= 0.010.02 N= 256

Imax

Imin

= 13156


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Cathode ray tubes (CRTs) are nonlinear!

Cameras contain -predistortion circuit!

Gamma characteristic

= 2.0 . . . 2.3!

Luminance!

I!

Voltage U!

I ~U

U~ I1


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log vs. -predistortion

Similar enough for most practical applications!

U

I

U ~ I1

U ~ log(I)

Imax

Imin

=100


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Photographic film

measures film contrast! General purpose films: = -0.7 . . . -1.0! High-contrast films: = -1.5 . . . -10!

Lower speed films tend to have higher absolute !

slope -!

logEEis exposure

densityd

shoulder!

toe!

linear region!

I = I0 10d

= I0 10 logE+d0( )

= I0 10d0

E

Luminance!

Hurter & Driffield curve (H&D curve)!for photographic negative!

d0

0

2.0

1.0


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Brightness adjustment by intensity scaling

Original image! Scaled image!

f x,y[ ] a f x,y[ ]

Scaling in the -domain is equivalent to scaling in the linear luminance domain

. . . same effect as adjusting camera exposure time.

I~ a f x,y[ ]( )

= a

f x,y[ ]( )


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Contrast adjustment by changing

Original image! increased by 50%!

f x,y[ ] a f x,y[ ]( )

with = 1.5

. . . same effect as using a different photographic film . . .!


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Contrast adjustment by changing

Original ramp 0!

Scaled ramp 20!

Scaled ramp 0.50!

Scaling chosen to!

approximately preserve!brightness of mid-gray!


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Gray level histograms

Distribution of gray levels can be judged by measuring a

histogram:!

For B-bit image, initialize 2B

counters with 0 !

Loop over all pixelsx,y!

When encountering gray levelf[x,y]=i, increment counter #

Normalized histogram may be thought of as anempirical probability distribution.!

You can also use fewer, larger bins to trade off amplitude

resolution against sample size.!


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Example histogram

gray level!

#pixels

Cameramanimage


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Example histogram

gray level!

#pixels

Poutimage


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Histogram equalization

Idea: find a non-linear transformation!

!to be applied to each pixel of the input imagef[x,y], suchthat a uniform distribution of gray levels in the entire range

results for the output image g[x,y]. Analyse ideal, continuous case first, assuming!

T(f)is strictly monotonically increasing, hence, there

exists!

Goal: pdfpg(g)= 1 over the range !

0 f 1

f = T1

g( )

g = T f( )

0 g 1

0 g 1

0 g 1


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Histogram equalization for continuous case

From basic probability theory!

Consider the transformation function!

Then . . . !

pg g( ) = pf f( )df

dg

f=T1 g( )

g = T f( ) = pf ( )0

f

d 0 f 1

dg

df= pf f( )

pg g( ) = pf f( )df

dg

f=T

1 g( )

= pf f( )1

pf f( )

f=T

1 g( )

=1 0 g 1


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Histogram equalization for discrete case

Now,fonly assumes discrete amplitude values !!with empirical probabilities!

Discrete approximation of!

The resulting values gkare in the range [0,1] and need tobe scaled and rounded appropriately. !

f0 , f1,, fL1

P0 =n0

n P

1=

n1

n P

L1 =nL1

nwhere n = n

l

l=0

L1

g = T f( ) = pf ( )0

f

d

gk =

T fk( ) = Pi

i=0

k

for k= 0,1...,L 1


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Histogram equalization example

Original image Pout Poutafter histogram equalization


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Original image Pout . . . after histogram equalization

gray level!

#pixels

gray level!

#pixels


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Original image

CameramanCameraman

after histogram equalization


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Original image Cameraman . . . after histogram equalization

gray level!

#pixels

gray level!

#pixels


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Original image Moon Moon

after histogram equalization


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Original image Moon . . . after histogram equalization

gray level!

#

pixels

gray level!

#

pixels


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Contrast-limited histogram equalization

gray level!

Histogram Clipping!

Input gray level!

Output

graylevel!

Equalization with!

clipped histogram!

Equalization with!original histogram!

Input gray level!

#

pixels


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Adaptive histogram equalization

Apply histogram equalization based on a histogram obtainedfrom a portion of the image!

Must limit contrast expansion in flat regions of the image,e.g. by clipping individual histogram values to a maximum(CLAHE - Contrast-limited adaptive histogram equalization)!

Sliding window approach:different histogram (and mapping)

for every pixel!

Tiling approach:subdivide into overlapping regions,

mitigate blocking effect by smooth blending

between neighboring tiles!

[Pizer, Amburn et al. 1987]!


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Original! Global histogram! Tiling

8x8 histograms!

Tiling

32x32 histograms!


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Original image Dental Xray

Dental Xrayafterequalization ofglobal histogram!

Adaptive histogram equalization, 16x16 tiles


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Original image Globalhistogram

equalization

Adaptive histogramequalization

8x8 tiles


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Point operations for combining images

Image averaging for noise reduction!

Combination of different exposures forhigh-dynamic range imaging!

Image subtraction for change detection!

Accurate alignment is always a requirement!


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Image averaging for noise reduction

http://www.cambridgeincolour.com/tutorials/image-averaging-noise.htm!

1 image 2 images 4 images!


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High-dynamic range imaging

16 exposures, one f-stop (2X) apart!

Combined image![Debevec, Malik, 1997]!


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Image subtraction

Find differences/changes between 2 mostly identical images!

Example: digital subtraction angiography!

- +

Contrastenhancement

http://www.isi.uu.nl/Research/Gallery/DSA/!


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Video background subtraction

New Frame Background Frame

-!

+!

Abs(.) > ?

-!

+!

Abs(.) > ?

New Frame Background Frame

Update: Background[t] := Background[t-1] + (1- ) New[t]


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Image subtraction example from IC manufacturing:

die-to-die comparison of photomasks


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Where is the defect?

Image A (no defect)! Image B (w/ defect)!


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Absolute difference between two images

w/o alignment! w/ alignment and thresholding!

2 Point Operations

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