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Generic, possibly nonlinear, pointwise operator intensity mapping, gray-level transformation: Image Enhancement in the Spatial Domain: Gray-level transforms Image Enhancement in the Spat

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Generic, possibly nonlinear, pointwise operator (intensity mapping,

gray-level transformation):

Image Enhancement in the Spatial Domain:

Gray-level transforms

Basic gray-level transformations:

Negative:

Generic log:

Power law:

γ

r c s

r l s

=

+

=

−

=

) 1 ln(

1

Gianni Ramponi

University of Trieste

http://www.units.it/ramponi

Chapter 3 Image Enhancement in the Spatial Domain:

Negative

Gianni Ramponi University of Trieste http://www.units.it/ramponi

Nonlinear mapping

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Nonlinear mapping

Gamma correction

1) Monitor response can

"compensate" for Weber-law sensitivity of HVS:

dp = k dL/L p = k log(L) higher sensit in dark areas dark transitions can be compressed with power law

L = x^gamma

2) Beware of nonlinearities

that are already included in image data (e.g., JPEG)

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Nonlinear mapping

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Note: stretching is formally useless if the image has to be thresholded)

Piecewise-linear contrast stretching

Gray-level slicing

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Bit-plane slicing

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Bit-plane slicing

4, 8, 16 gray levels respectively

Reconstruction: Sum_n [ bit-plane_n * 2^(n-1) ]

May be useful for data compression

Histogram properties

Histogram: normalized frequency (y) of gray level values (x).

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Histogram processing via gray mapping

(can be inverted and preserves gray-level ordering)

Histogram equalization

Let the gray levels in an image be represented as random variables r in

the range (0,1), with a probability density function (pdf):

Let be a monotonic, invertible transformation on r;

)

(r

pr

)

(r

T

s=

All the pixels below the curve in the interval are mapped to pixels below in

i.e., the two areas are equal:

Let us take the particular transformation

which is monotonic and invertible, since

it is the cumulative distribution function

(cdf) of r

)

(r

pr

) , (r r+dr

)

(s

ps (s,s+ds)

dr r p ds s

p s( ) = r( )

∫

=

) (r p w dw T

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The derivative of this function is of course

Substituting in

i.e the transformed variable has an exactly uniform pdf.

In a practical discrete case:

i.e., mapping each gray level into the value given above yields a

uniform pdf for the output image.

In general, only an approximately uniform distribution will be

obtained.

Note: no parameters are needed; the processing is automatic and

straightforward.

) ( /dr p r

1 ) ( )

( ) (s ds=p r dr → p s =

n n r p r T s

k

j j k

j j r k

0 0

∑

=

k s k

r

Example (continuous case):

Equalization is obtained via the transformation:

The transformed variable has a uniform pdf Indeed:

1 0 2 2 ) (r =− r+ ≤r≤

p r

∫ − + =− +

=

s

0

2 2 )

2 2 ( ) (

S Das, IIT Madras, Course on Computer Vision

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Example (discrete case):

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Histogram specification

Remember that the mapping

yields a (approx.) uniformly distributed output Another variable z,

with a different, known and desired pdf pz, will satisfy the same

equation:

substituting:

i.e., mapping each gray level rk into the zk value given above yields

the desired histogram (pdf) for the output image.

)) ( ( )

1

k k

j j z

z

= 0 ) ( ) (

n n r p r T s

k

j j k

j j r k

0 0

∑

=

sk: uniformly

distributed image

G: determined as cdf of

the desired pdf pz

zk: image with desired

histogram

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Then determine T(r) and G(z) (cdf’s of the histograms) :

T(r)

G(z)

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)) ( ( )

(

)

distributions: original target obtained

n’ k 0 0 0 790 1023 850 656+329 245+122+81

p’(z k) 0 0 0 0.19 0.25 0.21 0.24 0.11

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Local Histogram modification

At each location the local histogram is computed, the required mapping is determined, and the pixel is mapped (At the next step, it is convenient to update the histogram rather than to re-calculate it from scratch)

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Enhancement based on local statistics

Local values can be estimated for different image statistics, and

used to locally control a gray-level modification function.

E.g.: local mean and variance in the neighborhood Sxy:

Enhancement example: pixels in medium-variance, low-mean

areas are scaled by a positive factor:

Mg and Dg respectively are the global average and s.d of the

image; they are used to make the operator more robust.

∑

∈

−

=

Sxy t s

Sxy Sxy

Sxy

t

s

m

,

2 2

,

)]

, ( [ ] ) , ( [ )]

, ( [ ) ,





 < < <

=

otherwise y

x

f

Dg k Dg

k Mg k m if y x f

E

y

x

) , (

&

) , ( )

,

2 1

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Image subtraction

Assume an image is formed as:

where n(x,y) is i.i.d zero-mean noise If we can average K

acquisitions of the image, the variance of the noise is reduced by the

factor K:

This approach is useful when the sensor noise is relatively high:

poorly illuminated (static) scenes, astronomical images, …

∑

=

+

=

k k K

k

K y x f y x g K y x g

1 1

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

) , ( ) , ( ) , (x y f x y n x y

Image averaging

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Image averaging

Fig.3.30 A) Ideal B) Noise added (s.d.=64) C) K=8 D) K=16

Local operators

Generic, possibly nonlinear, neighborhood-based operator:

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

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∑

−

b t a

a s

b

b t a

a s

t s w

t y s x f t s w y

x g

) , (

) , ( ) , ( )

, (

The coefficients mask can be used in different ways, the

simplest of which is linear

filtering via the normalized convolution sum:

Note: if the output image is required to be the same size as the input image, the latter must

be suitably padded.

Local operators

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Local operators

[dipum]

Matlab implementation using ‘imfilter’

Local operators

[dipum]

Matlab:

correlation or convolution

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Local operators

[dipum]

Matlab: image padding

Linear lowpass filters

Both masks have power-of-two coefficients, which are simple to implement In the second one even the sum of the coefficients is a power of two.

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… A first elementary result in image segmentation!

Linear and nonlinear “lowpass” filters

Let Sxy be an mxn neighborhood of (x,y); define the Median filter:

)}

, ( { median )

, (

) ,

y x f

Sxy t

s ∈

=

Sort the pixel values in Sxy and take the one in position (mn+1)/2.

Note: mn should be odd; if it is even one can take as output the

average of the values in positions mn/2 and mn/2+1 The formal

statistical properties of the filter change.

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Define a 1-D digital derivative (other definitions are possible):

First-order: Note: it is not “centered”

Second-order:

2-D case:

Gradient:

Laplacian:

) ( ) 1 (x f x f

x

∂

) ( 2 ) 1 ( ) 1 ( )]

1 ( ) ( [ )]

( ) 1 ( [ 2

2

x f x f x f x f x f x f x f x

∂









∂

=

∇

y

f x

f

,

∂

=









∂

∂ +













∂

=

x

f y

f x

f

/ tan

;

|

2 2

α

f

2 2 2

2 2

y

f x

f f

∂

∂ +

∂

=

∇

Sharpening operators

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Measuring the derivatives of a signal:

Beware: all such definitions can be found in the literature

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