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Digital Image Processing, 2nd ed.www.imageprocessingbook.com

2002 R. C. Gonzalez & R. E. Woods

Chapter 3

Image Enhancementin the Spatial Domain

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Image Enhancement in theSpatial Domain

The spatial domain:

The image plane

For a digital image is a Cartesian coordinate system of discrete rowsand columns. At the intersection of each row and column is a pixel.Each pixel has a value, which we will call intensity.

The frequency domain :

A (2-dimensional) discrete Fourier transform of the spatial domain

We will discuss it in chapter 4.

Enhancement :

To improve the usefulness of an image by using some transformationon the image.

Often the improvement is to help make the image better looking,such as increasing the intensity or contrast.

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Background

A mathematical representation ofspatial

domainenhancement:

wheref(x,y): the input image

g(x,y): the processed image

T: an operator onf, defined over some neighborhood of

(x,y)

)],([),( yxfTyxg

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Gray-level Transformation

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Some Basic Gray Level Transformations

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Image Negatives

rLs 1

Let the range of gray level be [0,L-1], then

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Log Transformations

)1log( rcs

where c : constant

0r

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Power-Law Transformation

crs

where c, : positive constants

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Example 1: Gamma Correction

4.0

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4.0 3.0

6.01

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0.3

0.50.4

1

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Piecewise-Linear Transformation FunctionsCase 1: Contrast Stretching

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Piecewise-Linear Transformation FunctionsCase 2:Gray-level Slicing

An image Result of using the transformation in (a)

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Piecewise-Linear Transformation FunctionsCase 3:Bit-plane Slicing

Bit-plane slicing:

It can highlight the contribution made to total image appearance by

specific bits.

Each pixel in an image represented by 8 bits.

Image is composed of eight 1-bit planes, ranging from bit-plane 0 forthe least significant bit to bit plane 7 for the most significant bit.

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Piecewise-Linear Transformation FunctionsBit-plane Slicing: A Fractal Image

l d d

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Piecewise-Linear Transformation FunctionsBit-plane Slicing: A Fractal Image

2

7 6

45 3

1 0

Di i l I P i 2 d d

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Histogram Processing

Di it l I P i 2 d d

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Histogram Processing

Di it l I P i 2 d d

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Histogram Equalization

Histogram equalization:

To improve the contrast of an image

To transform an image in such a way that the transformed image has a

nearly uniform distribution of pixel values

Transformation: Assume rhas been normalized to the interval [0,1], with r= 0

representing black and r= 1 representing white

The transformation function satisfies the following conditions: T(r) is single-valued and monotonically increasing in the interval

10 r

10for1)(0 rrT

10)( rrTs

Di it l I P i 2 d d

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For example:

Di it l I P i 2 d d

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Histogram equalization is based on a transformation of theprobability density function of a random variable.

Letpr(r) andps(s) denote theprobability density function ofrandom variable rands, respectively.

Ifpr(r) and T(r) are known, then the probability densityfunctionps(s) of the transformed variables can be obtained

Define a transformation functionwhere w is a dummy variable of integration

and the right side of this equation is the cumulative distributionfunction of random variable r.

r

r dwwprTs 0 )()(

ds

drrpsp rs )()(

Di it l I P i 2 d d

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Given transformation function T(r),

ps(s) now is a uniform probability density function.

T(r) depends onpr(r), but the resultingps(s) always is uniform.

101)(

1)()()( srp

rpdsdrrpsp

r

rrs

)()()(

0rpdwwp

dr

d

dr

rdT

ds

drr

r

r

r

r dwwprT0

)()(

Digital Image Processing 2nd ed

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In discrete version:

The probability of occurrence of gray level rkin an image is

n : the total number of pixels in the imagenk: the number of pixels that have gray level rkL : the total number of possible gray levels in the image

The transformation function is

Thus, an output image is obtained by mapping each pixel with level rkin the input image into a corresponding pixel with levelsk.

1,...,2,1,0)()(0 0

LknnrprTs

k

j

k

j

j

jrkk

1,...,2,1,0)( Lkn

nrp kr


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Transformation functions (1) through (4) were obtained form the

histograms of the images in Fig 3.17(1), using Eq. (3.3-8).


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Histogram Matching

Histogram matching is similar to histogram equalization,

except that instead of trying to make the output image have a

flat histogram, we would like it to have a histogram of a

specified shape, saypz(z).

We skip the details of implementation.


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Local Enhancement

The histogram processing methods discussed above are global,

in the sense that pixels are modified by a transformation

function based on the gray-level content of an entire image.

However, there are cases in which it is necessary to enhance

details oversmall areas in an image.

original global local


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Use of Histogram Statistics for Image Enhancement

Moments can be determined directly from a histogram much faster than

they can from the pixels directly.

Let rdenote a discrete random variable representing discrete gray-levels in

the range [0,L-1], andp(ri) denote the normalized histogram component

corresponding to the ith value ofr, thenthe nth moment ofrabout its mean

is defined as

where m is the mean value ofr

For example, the second moment (also the variance ofr) is

1

0

)(L

i

ii rprm

1

0

2

2 )()()(L

i

ii rpmrr

1

0

)()()(L

i

i

n

in rpmrr


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Use of Histogram Statistics for Image Enhancement

Two uses of the mean and variance for enhancement

purposes:

The global mean and variance (global means for the entire

image) are useful for adjusting overall contrast andintensity.

The mean and standard deviation for a local region are

useful for correcting for large-scale changes in intensity

and contrast. ( See equations 3.3-21 and 3.3-22.)


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Use of Histogram Statistics for Image EnhancementExample: Enhancement based on local statistics


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Digital Image Processing, 2nd ed.

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Enhancement Using Arithmetic/Logic Operations

Two images of the same size can be combined using

operations of addition, subtraction, multiplication, division,

logical AND, OR, XOR and NOT. Such operations are done

on pairs of theircorresponding pixels.

Often only one of the images is a real picture while the other is

a machine generated mask. The mask often is a binary image

consisting only of pixel values 0 and 1. Example: Figure 3.27


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Enhancement Using Arithmetic/Logic Operations

AND

OR


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Image SubtractionExample 1


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igital mage rocessing, nd ed.www.imageprocessingbook.com


Image SubtractionExample 2

When subtracting two images, negative pixel values can result.So, if you want to display the result it may be necessary toreadjust the dynamic range by scaling.

Digital Image Processing, 2nd ed.i i b k

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g g g,www.imageprocessingbook.com


Image Averaging

When taking pictures in reduced lighting (i.e., lowillumination), image noise becomes apparent.

A noisy imageg(x,y) can be defined by

wheref(x,y): an original image

: the addition of noise

One simple way to reduce this granular noise is to takeseveral identical pictures and average them, thussmoothing out the randomness.

),(),(),( yxyxfyxg

),( yx


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Noise Reduction by Image AveragingExample: Adding Gaussian Noise

Figure 3.30 (a): An image

of Galaxy Pair NGC3314.

Figure 3.30 (b): Image

corrupted by additive

Gaussian noise with zeromean and a standard

deviation of 64 gray

levels.

Figure 3.30 (c)-(f):

Results of averagingK=8,16,64, and 128 noisy

images.


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Noise Reduction by Image AveragingExample: Adding Gaussian Noise

Figure 3.31 (a):

From top to bottom:

Difference images

between Fig. 3.30 (a)

and the four images in

Figs. 3.30 (c) through

(f), respectively.

Figure 3.31 (b):Corresponding

histogram.

Digital Image Processing, 2nd ed.www imageprocessingbook com

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Basics of Spatial Filtering

In spatial filtering (vs. frequency domain filtering), the output image is

computed directly by simple calculations on the pixels of the input image.

Spatial filtering can be either linear or non-linear.

For each output pixel, some neighborhood of input pixels is used in the

computation.

In general, linear filtering of an imagefof sizeMXNwith a filter mask of

size mxn is given by

where a=(m-1)/2 and b=(n-1)/2

This concept called convolution. Filter masks are sometimes called

convolution masks orconvolution kernels.

a

as

b

bt

tysxftswyxg ),(),(),(


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g g gwww.imageprocessingbook.com




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Nonlinear spatial filtering usually uses a neighborhood too, but

some other mathematical operations are use. These can

include conditional operations (if , then), statistical

(sorting pixel values in the neighborhood), etc.

Because the neighborhood includes pixels on all sides of thecenter pixel, some special procedure must be used along the

top, bottom, left and right sides of the image so that the

processing does not try to use pixels that do not exist.



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Smoothing Spatial Filters

Smoothing linear filters

Averaging filters (Lowpass filters in Chapter 4))

Box filter

Weighted average filter

Box filter Weighted average


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Smoothing Spatial Filters

The general implementation for filtering anMXNimage with

a weighted averaging filter of size mxn is given by

where a=(m-1)/2 and b=(n-1)/2

a

as

b

bt

a

as

b

bt

tsw

tysxftsw

yxg

),(

),(),(

),(


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www.imageprocessingbook.com


Smoothing Spatial FiltersImage smoothing with masks of various sizes


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www.imageprocessingbook.com


Smoothing Spatial FiltersAnother Example


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g p g


Order-Statistic Filters

Order-statistic filters

Median filter: to reduce impulse noise (salt-and-

pepper noise)


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g p g


Sharpening Spatial Filters

Sharpening filters are based on computing spatial

derivatives of an image.

The first-order derivative of a one-dimensional

functionf(x) is

The second-order derivative of a one-dimensional

functionf(x) is

)()1( xfxfxf

)(2)1()1(2

2

xfxfxfxf


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g p g


Sharpening Spatial FiltersAn Example


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Use of Second Derivatives for EnhancementThe Laplacian

Development of the Laplacian method

The two dimensional Laplacian operator for continuous

functions:

The Laplacian is a linear operator.

2

2

2

22

y

f

x

ff

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

2

yxfyxfyxfx

f

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

2

yxfyxfyxfyf

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


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To sharpen an image, the Laplacian of the image is subtracted

from the original image.

Example: Figure 3.40

positive.ismaskLaplaciantheoftcoefficiencentertheif),(

negative.ismaskLaplaciantheoftcoefficiencentertheif),(),(

2

2

fyxf

fyxfyxg


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Use of Second Derivatives for EnhancementThe Laplacian: Simplifications

The g(x,y) mask

Not only f2


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Use of First Derivatives for EnhancementThe Gradient

Development of the Gradient method

The gradient of functionfat coordinates (x,y) is defined as

the two-dimensional column vector:

The magnitude of this vector is given by

y

fx

f

G

G

y

xf

2

1

22

2

122)(mag

y

f

x

fGGf yxf

yx GGf


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Use of First Derivatives for EnhancementThe Gradient

Roberts cross-gradient

operators

Sobel

operators


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Use of First Derivatives for EnhancementThe Gradient: Using Sobel Operators


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Combining SpatialEnhancement Methods


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Combining SpatialEnhancement Methods


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Chapter 03 - dcc

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