Lie Gonzalez Ch3

7/31/2019 Lie Gonzalez Ch3

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CCU, Taiwan

Wen-Nung Lie

Chapter 3 : ImageEnhancement in the Spatial

Domain


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Backgrounds

Spatial domain methods are procedures that operate

directly on pixels

Neighborhood about a point (x,y) Neighborhood = 11 point processing or contrast stretching Larger neighborhood mask processing

)],([),( yxfTyxg =


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Some basic point

transformations

Implemented by look-up

tables

Functions

Linear (negative andidentity)

Logarithmic (log and

inverse log)

Power-law (nth power and

nth root)

)1log( rcs +=

)(or +== rcscrs


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Log transform

Expand the dynamic range of low gray-levels values

Compress the dynamic range of images with large variations in

pixel values

Power-law transformation

Curves generated with have exactly the opposite effect as

those generated with

1>

1


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Gamma correction

A variety of devices used for image capture,printing, and display respond according to a powerlaw

The process used to correct the power-lawresponse phenomena is called gamma correction

CRT has an intensity-to-voltage response of produce images that are darker than intended preprocess the images with beforeinputting it into the monitor

Current image standards do not contain the value

of gamma with which an image was created

5.2~8.1=

4.05.21 =


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Power-law transformation for

contrast manipulation


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Piecewise-linear

transformation

The form of piecewise functions can be arbitrarily complex

Contrast stretching increase the dynamic range of images

being processed


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Graylevel slicing & bit-plan

slicing


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Bit-plane decomposition

1. The MSB plane contains the majority of information

2. The LSB plane are likely random


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Histogram processing

Histogram

A discrete function , where is the kthgraylevel and is the number of pixels in the imagehaving graylevel

A normalized histogram is given by

where n is the total number of image pixels

An image with low contrast has a narrow histogram

A high contrast image tends to occupy the entire rangeof graylevels and be distributed uniformly

kk nrh =)(

kn

kr

kr

nnrp kk /)( =


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Histogram equalization

Point transformation

Single-valued

Monotonically increasing

Modify the image histogram, via point

transformation r s, such that its shape isapproximately flat

Cdf(cumulative distribution function)

transformation

1r0),( = rTs


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Histogram equalization (cont.)

Discrete implementation

Each pixel with graylevel is mapped intoin the output image

1,...,1,0,)()(0 0

==== = = Lknn

rprTs

k

j

k

j

jjrkk

kr ks

ds

drrpsp

dwwprTs

rs

r

r

)()(

)()( 0

=

==

1s0,1)( =sps Uniform distribution


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Histogram equalization (cont.)

Histogram equalization is a nonlinear(non-uniform) stretching technique, itstretches the histogram as flat aspossible (approximating a linear cdffunction)


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Histogram matching

(specification)

Transform the image such that it has anapproximately desired histogram shape

Given and (input and desired output),

compute

for a value ofk, find a smallest h such that .

Then transform kinto h (0h,k255).

)( kr rp )( jz zp

=

=k

j

jrk rps0

)( =

=h

j

jzh zpv0

)(

kh sv

Histogram specification is a trial-and-error process, thereare no rules for specifying histograms

f


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result

Specified histogram


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Local enhancement

Global mean and variance

Local mean and variance

An example

enhance dark areas while leaving the light area asunchanged as possible

=

=1

0

)(L

i

ii rprm

=

=1

0

2

2 )()()(L

i

ii rpmrr

= xyxy Sts tstsSrprm

),(

,, )( = xy xyxy

Sts

tsStsS rpmr),(

,2

,2 )(][

=

),(

),(),(

yxf

yxfEyxg

otherwise

DkDkMkmif GSGGS xyxy 210 AND

4.0,02.0,4.0,0.4 210 ==== kkkE 3x3 local region


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original enhanced

Also causesome artifacts

Enhancementmask


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Enhancement using

arithmetic/logic operations

Arithmetic/logic operations between two or more

images

arithmetic : subtraction, addition

logic : AND, OR, NOT AND and OR are for masking, i.e., for selecting subimages or

for ROI processing


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Image subtraction

Operation :

Commercial applications

Mask mode radiography in medical imaging

injecting a contrast medium into the patients bloodstream andsubtracting the mask (i.e., the image without injected medium)

from a series of incoming images

showing how the contrast medium propagates through the

arteries

Change detection via image subtraction

tracking moving objects or vehicles

),(),(),( yxhyxfyxg =


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angiography

Change/motion detection

_=

1st frame 2nd frame motion detected


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Image averaging If the noise is uncorrelated and has zero average

value, we can average Kdifferent noisy images

and get

The images must be registered (aligned)accurately to avoid the introduction of blurring

Important application : astronomy, IR imaging

high noise often occurs in low signal case orinsufficient cooling of sensors

=

=K

i

i yxgK

yxg1

),(1

),(

2

),(

2

),(

1and),()},({ yxyxg

KyxfyxgE

==

),(),(),( yxyxfyxg +=

),( yxgi


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original noisy

K=8 K=16

K=64 K=128


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Spatial filtering

Filtering in the spatial domain by convolution

directly on the pixels

Need a filter mask (or the impulse response) for

operation Calculate the response at each point (x,y) (i.e., slide the

filter mask from point to point)

The mask is often of odd size (2a+1)(2b+1), e.g. 33 Filtering by FFT and manipulation in the

frequency domain


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= = ++=a

as

b

bttysxftswyxg ),(),(),(

Nonlinear spatial filtering :operations that can not be

represented by sum-of-products

Linear spatial filtering

How to manage the boundary problem ?


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Smoothing spatial filter Used for blurring and noise reduction

Also called averaging filter or lowpass filter

Two 33 examples The division scale factor is equal to sum of the coefficients

(avoiding overflow) and had better be a power of 2

weighted average

the weight decreases as the distance to the center pixel increases

Discrete Gaussian averaging with variance

1/9

1 1 1

1 1 1

1 1 1

1/16

1 2 1

2 4 2

1 2 1

2

= =

= =

++=

a

as

b

bt

a

as

b

bt

tsw

tysxftsw

yxg

),(

),(),(

),(


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More larger the window, more blurring effect it has

More flat the coefficients are, more blurring they result in


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Order-statistics (Median) filter Nonlinear spatial filter whose response is based on

ordering (ranking) the pixels contained in the window effective to eliminate impulse noise or salt-and-pepper noise,

with considerably less blurring than linear smoothing filter

eliminate noise less than half of the window size

1-D or 2-D window size

Original noisy Low-pass filter Median filter


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Sharpening spatial filter To highlight fine detail in an image

Image differentiation (1st and 2nd-order

derivatives) enhances edges and other

discontinuities digital 1st-order derivative of 1-D signal

we can expect a 2nd order derivative to enhance fine

detail much more than a 1st-order derivative

)(2)1()1(2

2

xfxfxfx

f

++=

)()1( xfxfx

f

+=


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2nd-order derivative --

Laplacian operator Isotropic filter -- whose response is independent of the

direction of the discontinuities in the image, or rotation-

invariant

The Laplacian is the simplest isotropic derivative operator

2

2

2

22

y

f

x

ff

+

=

),(2),1(),1(2

2

yxfyxfyxf

x

f++=

0 -1 0

-1 4 -1

0 -1 0

-1 -1 -1

-1 8 -1

-1 -1 -1

),(2)1,()1,(2

2

yxfyxfyxfy

f++=

Digital Laplacian

),(4)]1,()1,(),1(),1([2

yxfyxfyxfyxfyxff +++++=

U f L l i f


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Use of Laplacian for

sharpening enhancement Sharpening enhancement -- add the Laplacian response to

the original image

),(),(),( 2 yxfyxfyxg =

Sharpening -- Enhance the highfrequency component

)]1,()1,(

),1(),1([),(5

),(),(),( 2

+++

++=

+=

yxfyxf

yxfyxfyxf

yxfyxfyxg

0 -1 0

-1 5 -1

0 -1 0

-1 -1 -1

-1 9 -1

-1 -1 -1

U h ki & hi h


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Unsharp masking & high-

boost filtering Unsharp masking in publishing industry

High-boost filtering

),(),(),( yxfyxfyxfs = Sharpening reduce the lowfrequency component

),(),(

),(),()1(

1),(),(),(

2

yxfyxAf

yxfyxfA

AyxfyxAfyxf

s

hb

=

+==

0 -1 0

-1 A+4 -1

0 -1 0

-1 -1 -1

-1 A+8 -1

-1 -1 -1

The sharpeningeffect decreases as

A increases

If we select

),(),(),(2

yxfyxfyxfs =

A li ti f hi h b t


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Application of high-boost

filtering

Sharpen the image and simultaneouslybrighten it

A=1 A=1.7


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Use of 1st-order derivative Gradient vector

Gradient magnitude

Approximation

=

=

y

fx

f

GG

y

xf

21

21

])()[(

][)(

22

22

yf

xf

GGmagf yx

+=

+== f

yx

GGf +


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Popular gradient operators -1 0

0 1

0 -1

1 0

Roberts cross-gradient operators

Sobel gradient operator-1 -2 -1

0 0 0

1 2 1

-1 0 1

-2 0 2

-1 0 1

The coefficients sum to zero, toyield zero on flat areas

Laplacian operator Sobel operator

C bi i ti l


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Combining spatial

enhancement methods Use Laplacian to highlight fine detail

also produce noiser results than the gradient

Use gradient to enhance prominent edges The gradient has a stronger response in ramps and steps areas than

does the Laplacian

The response of the gradient to noise is lower than Laplacian

The response to noise can be lowered by smoothing the gradientwith an averaging filter

Combining Laplacian and gradient operators smooth the gradient and multiply it by the Laplacian image

(preserve details in the strong areas while reducing noise in the flatareas)

The above result is added to the original image

Smoothed gradient Product of Laplaciand h d di


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original Laplacian

sharpening Sobel gradient

by 5x5 window and smoothed gradient

final sharpening increase DRwith =0.5

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Lie Gonzalez Ch3