3 image enhancement in the spatial domain.ppt [相容模 …140.115.154.40/vclab/teacher/2012dip/3...

Image Enhancement in the Image Enhancement in the Spatial DomainSpatial Domain

Angela Chih Wei Tang (唐之瑋)Angela Chih-Wei Tang (唐之瑋)Department of Communication Engineering

National Central UniversityNational Central UniversityJhongLi, Taiwan

2012 Spring

Outline

Gray level transformations Histogram processings og a p ocess g Enhancement using arithmetic operations

S ti l filt i Spatial filtering

C.E., NCU, Taiwan Angela Chih-Wei Tang, 2012 2

Background

)],([),( yxfTyxg Spatial domain methods

operate directly on the pixelsT t i hb h d f ( ) T operates over some neighborhood of (x,y) neighborhood shape: square & rectangular arrays are

the most predominant due to the ease of implementation mask processing/filtering masks/filters/kernels/templates/windowsmasks/filters/kernels/templates/windows e.g., image sharpening

the center of the window f i l t i lmoves from pixel to pixel

the simplest form: gray-level transformationg y s=T(r)

T can operate on a set of input images e g sum of input images for e.g., sum of input images for

noise reduction3C.E., NCU, Taiwan Angela Chih-Wei Tang, 2012

Point Processing Examples

contrast stretching: produce an image of higher contrast contrast stretching: produce an image of higher contrast than the original darken the levels below m & brighten the levels above m

the limiting case of contrast stretching: thresholding function

Contrast Stretching Thresholding Function


Some Basic Gray Level Transformations

5C.E., NCU, Taiwan

Gray Level Transformations –I N tiImage Negatives

Re e se the intensit le els rLs 1 Reverse the intensity levels Enhance white/gray detail embedded in dark regions of an

image, especially when the black areas are dominant in size

rLs 1

g , p y

Original image Negative image6C.E., NCU, Taiwan Angela Chih-Wei Tang, 2012

Gray Level Transformations –L T f tiLog Transformations

s=clog(1+r), c: constant, r≥0 Map a narrow range of low gray-level values in the input

image into a wider range of output levels Compress the higher-level values Compress the higher level values An example: Fourier spectrum of wide range ( )

a significant degree of detail would lost in the display

610~0

Display in a 8 b8-bit display

Fourier spectrum( )

After log transformation(c=1 s:0~6.2)6105.1~0 7C.E., NCU, Taiwan 7

Gray Level Transformations –P L T f tiPower-Law Transformations

c & : positive crs , c & : positive constants fractional : map a

narrow range of dark

crs

narrow range of dark input values into a wider range of output valuesvalues

a family of possible transformation curves: varying varying

A variety of devices used for image capture,

i ti & di l printing & display respond according to a power law

d i d d device-dependent value of gamma

8C.E., NCU, Taiwan Angela Chih-Wei Tang, 2012

Example #1 of Power-Law Transformations Gamma CorrectionTransformations – Gamma Correction Correct the power-law response phenomena

an example: The intensity-to-voltage response of CRT an example: The intensity to voltage response of CRT devices is a power function (exponents : 1.8~2.5) darker than intended

5.2

4.05.2/1

9C.E., NCU, Taiwan

Example #2 of Power-Law Transformations Contrast ManipulationTransformations – Contrast Manipulation

Original image c=1, =0.6

c=1, =0.4 c=1, =0.3

Magnetic resonance (MR) image of a fractured human spine10

Example #3 of Power-Law Transformations Contrast ManipulationTransformations – Contrast Manipulation

Original image c=1, =3.0

c=1, =4.0 c=1, =5.0

A Washed-out appearance: a compression of gray levels is desirable11

Piecewise-Linear Transformation F nctionsFunctions

Advantage: arbitrarily Advantage: arbitrarily complex

Disadvantage: the specification requires specification requires considerably more user inputs

Example: contrast-stretching transformation increase the dynamic increase the dynamic

range of the gray levels of the input image

r1=r2=m (Thresholding function)

(r1,s1)=(rmin,0)(r2,s2)= (rmax,L-1) 12

Gray-Level Slicing

Highlight a specific range f l l i ft of gray levels is often

desired Various ways to y

accomplish this highlight some range

and reduce all others to and reduce all others to a constant level

highlight some range but preserve all other levels

O i i l i T f d Original image Transformed image by (a)

13C.E., NCU, Taiwan

Bit-plane Slicingp g

Instead of highlighting More

significantMore

significant Instead of highlighting

gray-level ranges, highlighting the contribution made to

gg

contribution made to total image appearance by specific bits

Higher-order bits: the majority of the visually significant datag

Lower-order bits: subtle detailsObt i bit l i Obtain bit-plane images thresholding gray-level

transformation function


OutlineOutline

Gray level transformationsHi t i Histogram processing

Enhancement using arithmetic operations Spatial filtering


Histogram Processing

Hi t A l t H )( Histogram: A plot vs. : kth gray level : number of pixels in the

kr kk nrH )(krkn p

image having gray level k=0,1,..,L-1

Normalized histogram

kkr

nnrp kk /)( Normalized histogram Purposes

image enhancementimage compression

nnrp kk /)(

image compression segmentation

Simple to calculate economic hardware implementation proper for real-time image

processing


Histogram Equalization/Histogram Linearization (I)

Having gray-level values automatically cover the entire gray scale controlling the probability density function (pdf) of its controlling the probability density function (pdf) of its

gray levels via the transformation function T(r), where

k k

jjrkk Lk

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

Continuous transformation S=T(r): assume T(r) satisfies (a.1) singled-valued: guarantee that the inverse

j j

j n0 0

transformation will exist (a.2) monotonically increasing: preserves the increasing

order from black to white in the output imagep g not monotonically increasing: at least a section of the

intensity range being inverted (b) : the output gray levels will be in the 10for10 rT (b) : the output gray levels will be in the

same range as the input levels10for 10 rTr


Histogram Equalization (II)g q ( )

According to an elementary probability theory, if and T(r) are known and satisfies condition (a), then the

b b l d f f h f d bl)(

)(rpr)(1 sT

probability density function of the transformed variable s can be obtained by

dr

)(sps

dsdrrpsp rs )()(

The probability density function of the transformed variable s is determined by the gray-level pdf of the input image and by the chosen transformation functionp g y


Histogram Equalization (III)g q ( ) Consider the transformation function

r

r dwwprTs0

)()(

)( drdTds r

)(])([)()(

0rpdwwp

drd

rdrdT

drds

rr

r

1s01]1[1)()(

rpsp 1s0,1]1[)(

)()()(

)(1

1

sTrsTrr

rs rprpsp

1. Performing the transformation gyields a random variable scharacterized by a uniform probability density function.probability density function.

2. T(r) depends on , but the resulting always is uniform, independent of

)(rpr)(sps

)(rp


independent of .)(rpr

Results of Histogram

Equalization (1/2)Equalization (1/2)

Transformation functions


Results of Histogram Equalization (2/2)


Histogram Matching (Hi t S ifi ti ) (1/2)(Histogram Specification) (1/2)

Motivation sometimes, it is useful to specify the shape of the

histogram of the processed image instead of uniform histogramhistogram

the original image & histogram specification are given

C ti l l & Histogram

l

r

r dwwprTs0

)()(z

)()( zTzG

Obtain T(r)Give input image r,

Continuous gray level r & z Equalization

sdttpzGz

z 0 )()(, define

)]([)( 11 rTGsGz

)()(Obtain G(z))(zpzGive

)]([)(

Discrete gray level

1210)(1 LksGz kk 1,...,2,1,0),( LksGz kk


Histogram Matching (2/2)

Inverse mapping from sto the corresponding z


Local Enhancement Motivation

to enhance details over small areas in an imageth t ti f l b l t f ti d t the computation of a global transformation does not guarantee the desired local enhancement

Solution define a square/rectangular neighborhood move the center of this area from pixel to pixel histogram equalization in each neighborhood region

OriginalImage

After GlobalEnhancement

After LocalEnhancement 24

Use of Histogram Statistics for Image g gEnhancement

P id i l hil 1L

nth moment of r

Provide simple while powerful image enhancement

1

0)()()(

L

ii

nin rpmrr

Histogram statistics global mean

f

1

0)(

L

iii rprm

a measure of average gray level in an image

0i

1

0

22 )()()(

L

iii rpmrr

global variance a measure of average

contrast

xy

xySts

tstss rprrm),(

,, )()(

contrast local mean local variance

xy

xyxySts

tsStsS rpmr),(

,2

,2 )(][


OutlineOutline




Enhancement Using Arithmetic O ti I S bt tiOperations – Image Subtraction

Original imageOriginal image

Zero the 4 lower-order

bit plane

Zero the 4 lower-order

bit plane

The enhancement

Original imageOriginal image bit planebit plane

),(),(),( yxhyxfyxg

of the differences between images to bring out more to bring out more

details, contrast stretching can be performedperformed histogram

equalization power-law

transformation

27C.E., NCU, Taiwan

Image Subtraction –Mask Mode RadiographMask Mode Radiography

The mask is an X-ray image of a region of a patient’s body The image to be subtracted is taken after injection of a The image to be subtracted is taken after injection of a

contrast medium into the bloodstream The bright arterial paths carrying the medium are

k bl h dunmistakably enhanced The overall background is much darken than the mask image

Mask imageMask image After inject of a contrast medium into the bloodstreamAfter inject of a contrast medium into the bloodstream28C.E., NCU, Taiwan 28Angela Chih-Wei Tang, 2012

The Range of the Image Values After Image Subtraction

F 8 bi i 255 255 For 8-bit image: -255~255 Two principle ways to scale a different image

add 255 to every pixel & divide by 2 add 255 to every pixel & divide by 2 limitation: the full range of the display may not be

utilized the value of the minimum difference is obtained & its

negative added to all the pixels in the difference image scale to the interval [0 255] scale to the interval [0, 255] more complex


Image Averagingg g g

N i i )()()( f Noisy image assumption: at every pair of coordinates (x,y), the noise

is uncorrelated & has zero average value

),(),(),( yxyxfyxg

s u co e a ed & as e o a e age a ue uncorrelated: covariance

if an image is formed by averaging K different nois images

0)])([( jjii mxmxE),( yxg

noisy images

K

ii yxg

Kyxg

1),(1),(

),(),( yxfyxgE 2

),(2

),(1

yxyxg K

as K increases: the variability (noise) of the pixel values at each location (x y) decreases

),(),( yxyxg K


at each location (x,y) decreases

An Example of I A iImage Averaging An important application of

image averaging: astronomy motivation: imaging with motivation: imaging with

very low light levels is routine, causing sensor noise frequently to render noise frequently to render single images virtually useless for analysis

solution: image averaging

31C.E., NCU, Taiwan

An Example of Image Averaging –Difference ImagesDifference Images

Th ti l l The vertical scale: number of pixels

The horizontal scale: gray l l [0 255]

]106.2,0[ 4

level [0, 255] The mean & standard

deviation of the difference images decrease as K increases the average image g g

should approach the original as K increases

The effect of a decreasing mean: the difference images become darker as Kincreases

32Angela Chih-Wei Tang, 2012

OutlineOutline




Basics of Spatial Filtering (1/2)

The concept of filtering f th f th comes from the use of the

Fourier transform for signal processing in the frequency domainfrequency domain.

At each point (x,y), the response of the filter at th t i t i l l t d that point is calculated using a predefined relationship.

l f l h Linear spatial filtering: the response is the sum of products

3434

Basics of Spatial Filtering (2/2)p g ( )

Linear spatial filtering often is referred to as “convolving a mask with an image.” filtering mask / convolution mask / convolution kernel

Nonlinear spatial filtering Nonlinear spatial filtering the filtering operation is based conditionally on the

values of the pixels in the neighborhood under consideration, but not explicitly use coefficients in the sum-of-products manner

e.g., median filter for noise reduction: compute the e.g., median filter for noise reduction: compute the median gray-level value in the neighborhood


What Happens When the Center of the Filter A h th B d f th I ?Approaches the Border of the Image ?

One or more rows/columns of the mask will be located One or more rows/columns of the mask will be located outside the image plane

Solution limit the excursions of the center of the mask to be at a

distance no less than (n-1)/2 pixels from the border the resulting filtered image is smaller than the original the resulting filtered image is smaller than the original

partial filter mask: filter all pixels only with the section of the mask that is fully contained in the image

Padding add rows & columns of 0’s (or other constant gray

level) or replicate rows/columnslevel), or replicate rows/columns the padding is stripped off at the end of the process side effect: an effect near the edges that becomes

more prevalent as the size of the mask increases36C.E., NCU, Taiwan Angela Chih-Wei Tang, 2012 36

Smoothing Linear Filters (1/2)g ( )

A i fil /l fil l h l f Averaging filters/low pass filters: replace the value of every pixel in an image by the average of the gray levels of the pixels contained in the neighborhood of the filter mask

Reduce sharp transitions in gray levels side effect: blur edgesR d i l t d t il i i d t Reduce irrelevant detail in an image and get a gross representation of objects of interest irrelevant: pixel regions that are small with respect to p g p

the size of the filter mask Applications

i d ti noise reduction smooth false contouring


Smoothing Linear gFilters (2/2)

Computationally efficient: instead of being 1/9 (1/16), the coefficients of the filter are all 1’s

Box filter: a spatial averaging filter in which all coefficients are equalequal

Weighted average: give more importance (weight) to some pixels reduce blurring in the smoothing process

the pixel at the center of the mask is multiplied by a higher value than any otherthe other pixels are inversely weighted as a function of their the other pixels are inversely weighted as a function of their distance from the center of the mask

The attractive feature of “16”: integer power of 2 for computer implementation

38C.E., NCU, Taiwan Angela Chih-Wei Tang, 2012 38

Examples of Smoothing Li Filt (1/2)Linear Filters (1/2)

n=3: general slight blurring through the n=3: general slight blurring through the entire image, but details are of approximately the same size as the filter mask are affected considerably filter mask are affected considerably more

n=9: the significant further smoothing of the noisy rectangles

n=15 & 35: generally eliminate small objects from an imageobjects from an image

n=35: black order due to zero padding

39

Examples of Smoothing Linear Filters (2/2) Goal: get a gross representation of objects of interest, such

that the intensity of smaller objects blends with the background and larger objects become bloblike the size of the maskSmoothing linear filtering + thresholding to eliminate Smoothing linear filtering + thresholding to eliminate objects based on their intensity

40

Order-Statistics Filters

N li ti l filt h i b d Nonlinear spatial filters whose response is based on ordering (ranking) the pixels in the area to be filtered

Examples: max filter, min filter & median filter Median filter

replace the value of a pixel by the median of the gray levels in the neighborhood of that pixelg p

force points with distinct gray levels to be more like their neighbors

for 3x3 neighborhood for 3x3 neighborhood(10,20,20,20,15,20,20,25,100) -> median is 20

excellent noise-reduction capabilities, with considerably l bl i th li thi filt f i il iless blurring than linear smoothing filters of similar size

particular effective for impulse noise (salt-and-pepper noise


An Example of Applying Median Filtersp pp y g

The image processed with the averaging filter has less The image processed with the averaging filter has less visible noise, but the price paid is significant blurring.


Foundations of Sharpening Spatial Filters (1/2)

Highlight fine detail in an image or to enhance detail that has been blurred

Averaging is analogous to integration, and sharpening can be accomplished by spatial differentiation

For our use what we require for a first derivative are For our use, what we require for a first derivative are zero in flat areas nonzero at the onset of a gray-level step or ramp nonzero along ramps

For our use, what we require for a second derivative zero in flat areas zero in flat areas nonzero at the onset and end of a gray-level step or ramp zero along ramps of constant slope


Foundations of Sharpening Spatial p g pFilters (2/2)

Th fi d d i i f 1 D f i f( ) The first-order derivative of a 1-D function f(x)

)()1( xfxfxf

The second-order derivative of f(x)x

)(2)1()1(2

ffff

The two definitions satisfy the conditions stated previously

)(2)1()1(2 xfxfxfxf


An Example of Applying Sharpening Spatial Filters (1/3)Spatial Filters (1/3)

45

An Example of Applying Sharpening S ti l Filt (2/3)Spatial Filters (2/3)

Ramp: the first-order derivative is nonzero along the entire ramp, while the second-order derivative is nonzero only at the onset and end of the ramp. first-order derivatives produce thick edges and second- first order derivatives produce thick edges and second

order derivatives, much finer ones Isolated noise point: the response at and around the point is

h t f th d th f th fi t d much stronger for the second- than for the first-order derivative. second-order derivative enhances much more than a first-

order derivative does. Thin line: essentially the same difference between the two

derivativesderivatives Gray-level step: the response of the two derivatives is the

same double-edge effect: The second derivative has a transition

from positive back to negative. 46

Foundation of Sharpening Spatial Filters (3/3)

l h d d b In most applications, the second derivative is better suited than the first derivative for image enhancement.

The principle use of first derivatives in image The principle use of first derivatives in image processing: edge extraction.


Second Derivatives for E h t Th L l iEnhancement – The Laplacian

Isotropic filters Isotropic filters rotation invariant: rotating the image and applying the

filter gives the same result as applying the filter to the f d h h limage first and then rotating the result

The simplest isotropic derivative operator: Laplacianff 22

2

Laplacian is a linear operator : because derivatives of any order yf

xff 22

2

are linear operator

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

2

yxfyxfyxfxf

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

2

yxfyxfyxfxfx

),(4)]1,()1,(),1(),1([2 yxfyxfyxfyxfyxff 48

Filter Masks – The Laplacian (1/2)Isotropic for

increments of 45 deg.Isotropic for

increments of 90 deg.

4949

Filter Masks – The Laplacian (2/2)p ( )

Th L l i i d i i The Laplacian is a derivative operator highlights gray-level discontinuities in an image &

deemphasizes regions with slowly varying gray levelsdee p as es eg o s s o y a y g g ay e e s Being used frequently for sharpening digital images

The Lapalcian for image enhancement

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

2

ffyxfyxfyxg

If the center coefficient of theLaplacian mask is negativeIf the center coefficient of the ),(),( 2 yxfyxf If the center coefficient of the Lapalcian mask is positive


Examples of the Laplacian Enhanced Image (1/2)Enhanced Image (1/2)

Notice that the Laplacian o a ap a aimage may contain both positive and negative valuesvalues scaling is desired

Adding the original image to the Laplacian image small details were small details were

enhanced the background

t lit i bl tonality is reasonably preserved

51C.E., NCU, Taiwan 51

Examples of the Laplacian Enhanced Image (2/2)Enhanced Image (2/2)

The results obtainable with the

k i i mask containing the diagonal terms usually are a little sharper than those obtained with the more basic mask


First Derivatives for Enhancement –The Gradient (1/2) First derivatives in image processing are implemented using

the magnitude of the gradient gradient is defined as the 2-D column vector gradient is defined as the 2-D column vector

xf

Gxf

the magnitude of this vector (gradient)

yfx

Gyf

the magnitude of this vector (gradient)2/122

2/122 ][)f(

yf

xfGGmagf yx

nonlinear operator isotropic

yx


isotropic

First Derivatives for Enhancement –The Gradient (2/2) Problem: the computational burden of implementing the Problem: the computational burden of implementing the

gradient The approximation of the magnitude of the gradient

the isotropic properties of the digital gradient are preserved

yx GG f

the isotropic properties of the digital gradient are preserved only for a limited number of rotational increments (90 deg.)

Digital approximation of the gradient

5658 , zzGzzG yx

6859 zzGzzG 1

2 6859 , zzGzzG yx2


A 3x3 Region of an Image & Masks for Comp ting the Gradient

R b di

for Computing the Gradient Roberts cross-gradient

operators

6859 zzzzf (From )26859 zzzzf (From )2

Since even size are awkward toimplement 3x3 Sobel Operator

)2()2(

)2()2( 321987

zzzzzz

zzzzzzf

implement 3x3 Sobel Operator

)2()2( 741963 zzzzzz

derivative in x derivative in y55C.E., NCU, Taiwan Angela Chih-Wei Tang, 2012

An Example of Using Sobel Gradientp g An optical image of a contact lens illuminated by a lighting

t t hi hli ht i f tiarrangement to highlight imperfections Sobel gradient: the edge defects are quite visible, and the

slowly varying shades of gray are eliminated.y y g g y simplify the computational task required for automated

inspection.


3 image enhancement in the spatial domain.ppt [相容模 …140.115.154.40/vclab/teacher/2012dip/3...

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