Chapter 3 Image Enhancement in the Spatial Domain

Chapter 3Image Enhancement in the Spatial Domain

國立雲林科技大學資訊工程所張傳育 (Chuan-Yu Chang ) 博士Office: EB212TEL: 05-5342601 ext. 4337E-mail: [email protected]:MIPL.yuntech.edu.tw

2Medical Image Processing & Neural Networks Laboratory

Preview

The principal objective of enhancement is to process an image so that the result is more suitable than the original image for a specific application.

Image enhancement approaches Spatial domain methods

Based on direct manipulation of pixels in an image. Frequency domain methods

Based on modifying the Fourier transform of an image.


Spatial domain process willbe denoted by g(x,y)=T[f(x,y)]where f(x,y): input image g(x,y): processed image T: an operator

maskfilterkerneltemplatewindows

Background Spatial domain

Refers to the aggregate of pixels composing an image.

Operate directly on these pixels


Gray-Level (intensity) transformation Function s=T(r) where T is gray-level transformation function

Processing technologies: Point processing

Enhancement at any point in an image depends only on the gray level at that point.

Mask processing or filtering Use a function of the values of f in a predefined neighborhood of (x,y) to

determine the value of g at (x,y)

Background (cont.)

Contrast stretching

thresholding


Some basic Gray Level Transforms s = T(r) r : the gray level value before

process s: the gray level value after

process

Values of the transformation function typically are stored in a one-dimensional array and the mapping from r to s are implemented via table lookups.

Some Basic Gray Level Transforms


Image Negatives Reversing the intensity levels of an image Photographic Negative s=L-1-r Suited for enhancing white or gray detail embedded in dark

regions of an image

Some Basic Gray Level Transforms (cont.)


Log Transformations s=c log (1+r) Maps a narrow range of low gray-level values in the input

image into a wider range of output levels. To expand the values of dark pixels in an image while

compressing the higher-level values


A Fourier spectrum with values in the

range 0 to 1.5x106.

c=1, the range of values : 0 to 6.2.


Power-Law Transformations s=cr

s= c (r + )r

where c and ` are positive constants

Power-law curves with fractional values of r map a narrow range of dark input values into a wider range of output values, with the opposite being true for higher values of input levels.


To account for an offset


Some Basic Gray Level Transforms (cont.) Gamma Correction

The process used to correct this power-law response phenomena


Some Basic Gray Level Transforms (cont.) Example 3.1

MR image of fractured human spine Contrast

manipulation

c=1, =0.4

c=1, =0.3

c=1, =0.6

Fracturedislocation

褪色 (Washed-out)

The best enhancementin terms of contrast and discernable detail wasobtained.



Washed-out appearance

c=1, =3.0

c=1, =5.0

c=1, =4.0


Picewise-Linear Transformation FunctionPicewise-Linear Transformation Function

Some Basic Gray Level Transforms (cont.) Contrast Stretching

To increase the dynamic range of the gray levels in the image being processed.

Linear function If r1=s1 and r2=s2

Thresholding If r1=r2, s1=0 and s2=L-1

Control points


Picewise-Linear Transformation FunctionPicewise-Linear Transformation Function

Gray-level SlicingHighlighting a specific range of gray levels in an image.

To display a high value for all gray levels in the range of interest and a low value for all other gray levels.Brightens the desired range of gray levels but preserves the background and gray-level in the image.



Some Basic Gray Level Transforms (cont.) Bit-plane Slicing

Highlighting the contribution made to total image appearance by specific bits.

Separating a digital image into its bit planes is useful for analyzing the relative importance played each bit of the image. Determining the adequacy of the number of bits used to quantize

each pixel. Image compression.


Some Basic Gray Level Transforms (cont.) An 8-bit fractal image


Some Basic Gray Level Transforms (cont.) The eight bit planes of the image in Fig. 3.13


Histogramh(rk)= nk

rk is the kth gray-level

nk is the number of pixels in the

image having gray-level k Normalized Histogramp(rk)=nk/n

Histogram Processing


Histogram Processing (cont.) Histogram Equalization

10)( rrTs

Assume that the transformation function T(r) satisfies the follows(a) T(r) is a single-valued and monotonically increasing(b) 0<=T(r)<=1 for 0<=r <=1


Histogram Processing (cont.) The probability of occurrence of gray level rk in an image is

approximated by

The discrete version of the transformation function given as

A processed image is obtained by mapping each pixel with level rk in the input image into a corresponding pixel with level sk in the output image.

1,...,2,1,0

)()(

0

0

Lkn

n

rprTs

k

j

j

k

jjrkk

Histogram equalization automatically determines a transformationfunction that seeks to produce an output image that has a uniform histogram.

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

nrp kkr


Histogram Processing (cont.) Example 3.3Histogram equalization


Histogram Processing (cont.)



Histogram matching (Specification) Let s be a random variable with the property

Define a random variable z with the property

Assume that G(z)=T(r), therefore, that z must satisfy the condition

r

r dwwprTs0

)(

z

z dttpzG0

rTGsGz 11

(3.3-10)

(3.3-11)

(3.3-12)


Histogram matching (Specification) To specify the shape of the histogram that we wish the

processed image to have.

1,...,2,1,0

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

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

1,...,2,1,0

)()(

1

1

0

0

0

LksGz

LkrTGz

LkszpzGv

Lkn

n

rprTs

kk

kk

k

k

iizkk

k

j

j

k

jjrkk


(3.3-13)

(3.3-14)

(3.3-15)

(3.3-16)


Histogram Processing (cont.)1. 對原圖進行 histogram equalization

2. 對給予的 histogram, 計算轉換函數 G(z)根據手繪函數 G(z) 求出每個 Zq 所對應的 vq

3. 對每個 sk ，求對應的 Zk


Procedure for histogram matching1. Obtain the histogram of the given image

2. Use E.q.(3.3-13) to precompute a mapped level sk for each level rk

3. Obtain the transformation function G(z) from the given pz(z) using Eq.(3.3-14)

4. Precompute zk for each value of sk using the scheme defined in Eq(3.3-17)

5. Use the value from step (2) and step (4), mapping rk to its corresponding level sk, then map level sk into the final level zk.


Histogram Processing (cont.) Example 3.4 Comparison between histogram

equalization and histogram matching

火星的衛星影像，有大區域的深色區域，由其 histogram 觀察，會以為 histogram equalization 會有不錯的結果？



由於圖 3.20 中的histogram,gray level 0 及其附近有大量的值，因此根據 Eq(3.3-8) ，s0 會接近 190 。

直接以圖 (a) 進行equalization 會有褪色的感覺。



手繪的 histogram

手繪的 histogram 之轉換函數 G(z)

轉換後的結果

轉換後的 histogram


Local Enhancement

Global enhancement The pixels are modified by a transformation function based on

the gray-level content of an entire image. Local enhancement

To design transformation functions based on the gray-level distribution in the neighborhood of every pixel in the image.

Local enhancement Procedure Step 1: Define a square neighborhood Step 2: Move the center of this area from pixel by pixel

Calculate the histogram of the points in the neighborhood. Apply the histogram equalization or specification Assign new gray level to the center pixel

Step 3: Moved to an adjacent pixel location. Repeat Step 2 until end of the image



Example 3.5 Enhancement using local histograms

Original imageResult of global

histogram equalization

Result of local histogram equalization


Use of Histogram Statistics for Image Enhancement The global mean and variance

The mean is a measure of average gray level in an image The variance is a measure of average contrast in an

image. Let r denote discrete gray-levels in the range p(ri) denote the normalized histogram component

corresponding to the ith value of r. The nth moment of r is defined as

where m is the mean value of r

]1,0[ Lr

1

0

)()(L

ii

nin rpmrr

1

0

)(L

iii rprm

(3.3-18)

(3.3-19)


Use of Histogram Statistics for Image Enhancement

根據 (3.3-18) 及 (3.3-19)0=1, 1=0 The second moment is obtained by

(3.3-20) 為 r 的 variance 。 Standard deviation 定義為 variance 的平方根 (square root) 。

The global mean and variance are measured over an entire image and are useful primarily gross adjustments of overall intensity and contrast.

)(

)()(

2

1

0

22

r

rpmrrL

iii

(3.3-20)


Histogram Statistics for Image Enhancement

The local mean and variance The local mean is a measure of average gray level in

neighborhood Sxy

The variance is a measure of contrast in the neighborhood

xy

xyxyS

xy

xy

StstsSts

StststsS

rpmr

rprm

),(,

2,

2

),(,,

)(

)(

(3.3-21)

(3.3-22)


Example 3.6鎢絲 1( 清楚 )

鎢絲 2( 不清楚 )


Example 3.6

The problem is to enhance dark areas while leaving the light area as unchanged as possible. Consider the pixel as a point (x,y) as a candidate for

processing

21120

211

2

0

),(

),(

))((),(),(

,)3(

)2(

)1(

kkDk

otherwiseyxf

DkMkmifyxfEyxg

Thus

kkDkif

Dkif

Mkmif

xyxyxy

xy

xy

xy

SGGSGS

SG

GS

GS



對影像取 localmean average

對影像取 localstandard deviation

採用 3x3 local region

三個條件判別後的結果

白色部分為 E ，用來對原影像相乘，以得到強化的結果。




Enhancement using Arithmetic/Logic Operations Arithmetic/Logic operations are performed on a

pixel-by-pixel basis. Arithmetic operations: subtraction, addition,

division, multiplication. Logic operations: AND, OR, NOT When dealing with logic operations on gray-scale

images, pixel values are processed as strings of binary numbers.

Enhancement using Arithmetic/Logic Operations


Enhancement using Arithmetic/Logic Operations (cont.)




Image Subtraction The enhancement of difference between images

The difference between two images f(x,y) and h(x,y)

),(),(),( yxhyxfyxg


(3.4-1)


Enhancement using Arithmetic/Logic Operations (cont.) Most images are displayed using 8 bits.

Thus, we expect image values not to be outside the range from 0 to 255.

The value in a difference image can range from a minimum of -255 to a maximum of 255.

How to solve this problem? Solution 1: g’(x,y)=[g(x,y)+255]/2 Solution 2:

g’(x,y)=g(x.y)-min(g(x,y))g’’(x,y)=[g’(x,y)*255]/max(g’(x,y))


Image averaging Noisy image g(x,y) formed by the addition of noise (x,y) to

an original image f(x,y)

Assume that at every pair of coordinates (x,y) the noise is uncorrelated and has zero average value.

Averaging K different noisy images

To reduce the noise content by adding a set of noisy images The standard deviation at any point in the average image is

),(),(),( yxyxfyxg

K

ii yxg

Kyxg

1

),(1

),(

),(1

),(yx

Kyxg

Enhancement using Arithmetic Operations (cont.)

(3.4-2)

(3.4-3)

(3.4-6)

As K increases, Eq(3.4-6) indicates that the noise of the pixel values at each location (x,y) decreases.


Enhancement using Arithmetic/Logic Operations (cont.) Example 3.8 Noise reduction

by image averaging The images gi(x,y) must be

registered in order to avoid the introduction of blurring and other artifacts.

(a) Image of Galaxy pair NGC 3314.

(b) Image corrupted by additive Gaussian noise with zero mean and a standard deviation of 64 gray levels.

(c-f) Result of averaging K=8, 16, 64 and 128 noisy images.



In the histograms, the mean and standard deviation of the difference images decrease as K increases.


Basic of spatial filtering

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

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

yxfwyxfwyxfw

yxfwyxfwR

a

as

b

bt

tysxftswyxg ),(),(),(

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

If image size M×N, mask size m×n

Convolving a mask with an image

Basic of spatial filtering


mn

iii

mnmn

zw

zwzwzwR

1

2211 ...

9

1

992211 ...

iii zw

zwzwzwR

Basic of spatial filtering (cont.)


Smoothing Spatial Filter

Smoothing filters are used for blurring and for noise reduction. Smoothing Linear Filter

Sometimes are called averaging filter , lowpass filter Box filter

A spatial averaging filter in which all coefficients are equal

Weighted average Pixels are multiplied at different coefficient

9

19

1

iizR

a

as

b

bt

a

as

b

bt

tsw

tysxftswyxg

),(

),(),(),(


Smoothing Spatial Filter (cont.) Example 3.9

Image smoothing with masks of various sizes (a) Original image of size

500x500 (b-f) Results of

smoothing with square averaging filter masks of sizes n=3, 5, 9, 15, and 35.


Smoothing Spatial Filter (cont.) Spatial averaging is to blur an image for the purpose

getting a gross representation of objects of interest. The intensity of smaller objects blends with the background

and larger objects become “bloblike” and easy to detect.


Order-Statistic Filters Order-Statistic Filters (Nonlinear spatial filters)

Based on ordering the pixels contained in the image area encompassed by the filter. And then replacing the value of the center pixel with the value determined by the ranking result.

Median filter Particularly effective in the presence of impulse noise (salt-

pepper noise) Algorithm:

Step 1: sort the value of the pixels encompassed by the filter.Step 2: determine their median.Step 3: assign the median to the center pixel.


Order-Statistic Filters Max filter Min filter


Sharpening Spatial Filters Objectives:

To highlight fine detail in an image To enhance detail that has been blurred

The derivatives of a digital function are defined in terms of differences

First derivative Must be zero in flat segment Must be nonzero at the onset of a gray-level step or ramp Must be nonzero along ramps

)()1( xfxfx

f


Sharpening Spatial Filters

Second derivative Must be zero in flat areas Must be nonzero at the onset and the end of gray-level

step or ramp. Must be zero along ramps of constant slope

)(2)1()1(

)1()()()1(2

2

xfxfxf

xfxfxfxfx

f


Sharpening Spatial Filters (cont.)


Sharpening Spatial Filters (cont.) Summary

First-order derivatives generally produce thicker edges in an image.

Second-order derivatives have a stronger response to fine detail

First-order derivatives generally have a stronger response to a gray-level step

Second-order derivatives produce a double response at step changes in gray level.


Use of Second Derivatives for Enhancement- The Laplacian

Isotropic filter (rotation invariant) Whose response is independent of the direction of the

discontinuities in the image. Laplacian

),(4)1,()1,(),1(),1(

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

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

2

2

2

2

2

2

2

2

22

yxfyxfyxfyxfyxff

yxfyxfyxfy

f

yxfyxfyxfx

f

y

f

x

ff


Use of Second Derivatives for Enhancement- The Laplacian


Use of Second Derivatives for Enhancement- The Laplacian Image enhancement

positiveistcoefficiencentertheif

negativeistcoefficiencentertheif

yxfyxf

yxfyxfyxg

),(),(

),(),(),(

2

2

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

),(4)1,()1,(),1(),1(),(),(

yxfyxfyxfyxfyxf

yxfyxfyxfyxfyxfyxfyxg


Use of Second Derivatives for Enhancement- The Laplacian (cont.) Example 3.11

Imaging sharpening with the Laplacian.



Image enhancement using a composite Laplacian mask


Use of Second Derivatives for Enhancement- The Laplacian (cont.) Unsharp masking and high-boost filtering

Used in publishing industry Unsharp masking: To sharpen images consist of subtracting

a blurred version of an image from the image itself.

),(),(),( yxfyxfyxf s



Image enhancement with a high-boost filter


Use of First Derivatives for Enhancement -The Gradient The gradient of f at 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

2/122

2/122

)(

y

f

x

f

GG

magf

yx

f


Use of First Derivatives for Enhancement -The Gradient (cont.)


Use of First Derivatives for Enhancement -The Gradient Example 3.14

Use of the gradient for edge enhancement.


Chapter 3Image Enhancement in the

Spatial Domain


Spatial Domain



Spatial Domain


Spatial Domain

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Chapter 3 Image Enhancement in the Spatial Domain