3 image enhancement in the spatial domain - …140.115.154.40/vclab/teacher/2014dip/3 image...

3 Image Enhancement in

the Spatial Domain

Chih-Wei Tang (唐唐唐唐之之之之瑋瑋瑋瑋)

Department of Communication Engineering

National Central University

JhongLi, Taiwan

C.E., NCU, Taiwan 2

Outline

� Gray level transformations

� Histogram processing

� Enhancement using arithmetic operations

� Spatial filtering

Background

)],([),( yxfTyxg =

� Spatial domain methods

� operate directly on the pixels

� 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/windows

� e.g., image sharpening

� the center of the window

moves from pixel to pixel

� the simplest form:

gray-level transformation

� s=T(r)

� T can operate on a set of input images

� e.g., sum of input images for

noise reduction3C.E., NCU, Taiwan

C.E., NCU, Taiwan 4

Point Processing

� Examples

� 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 –

Image Negatives

� 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

Original image Negative image

6C.E., NCU, Taiwan


Log 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

� An example: Fourier spectrum of wide range ( )

� a significant degree of detail would lost in the display

Fourier spectrum( )

After log transformation(c=1� s:0~6.2)

610~0

6105.1~0 ×

Display in a 8-bit display

7C.E., NCU, Taiwan 7


Power-Law Transformations

� , c & γ: positive constants

� fractional γ : map a narrow range of dark input values into a wider range of output values

� a family of possible transformation curves: varying γ

� A variety of devices used for image capture, printing & display respond according to a power law

� device-dependent value of gamma

γ= crs

8C.E., NCU, Taiwan

Example #1 of Power-Law

Transformations – Gamma Correction� Correct the power-law response phenomena

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

� darker than intended

5.2=γ

4.05.2/1 ==γ

9C.E., NCU, Taiwan


Transformations – Contrast Manipulation

Magnetic resonance (MR) image of a fractured human spine

Original image c=1, γ=0.6

�c=1, γ=0.4 c=1, γ=0.3

10


Transformations – Contrast Manipulation

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

Original image c=1, γ=3.0

�c=1, γ=4.0 c=1, γ=5.0

11

Piecewise-Linear Transformation

Functions

� Advantage: arbitrarily complex

� Disadvantage: the specification requires considerably more user inputs

� Example: contrast-stretching transformation

� 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 of gray levels is often desired

� Various ways to accomplish this

� highlight some range and reduce all others to a constant level

� highlight some range but preserve all other levels

Original image Transformed image by (a)

13C.E., NCU, Taiwan

Bit-plane Slicing

� Instead of highlighting gray-level ranges, highlighting the contribution made to total image appearance by specific bits

� Higher-order bits: the majority of the visually significant data

� Lower-order bits: subtle details

� Obtain bit-plane images

� thresholding gray-level transformation function

More significant

More significant

14C.E., NCU, Taiwan

C.E., NCU, Taiwan 15

Outline






Histogram Processing

� Histogram: A plot vs.

� : kth gray level

� : number of pixels in the image having gray level

� k=0,1,..,L-1

� Normalized histogram

� Purposes� image enhancement

� image compression

� segmentation

� Simple to calculate� economic hardware implementation

� proper for real-time image processing

kr kk nrH =)(

kr

knkr

nnrp kk /)( =

Histogram Equalization/

Histogram Linearization (I)

� Having gray-level values automatically cover the entire gray scale

� controlling the probability density function (pdf) of its gray levels via the transformation function T(r), where

� Continuous transformation S=T(r): assume T(r) satisfies

� (a.1) singled-valued: guarantee that the inverse transformation will exist

� (a.2) monotonically increasing: preserves the increasing order from black to white in the output image

� not monotonically increasing: at least a section of the intensity range being inverted

� (b) : the output gray levels will be in the same range as the input levels

∑ ∑= =

−====k

j

k

j

j

jrkk Lkn

nrprTs

0 0

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

10for 10 ≤≤≤≤ rTr

17C.E., NCU, Taiwan


Histogram Equalization (II)

� According to an elementary probability theory, if and T(r) are known and satisfies condition (a), then the probability density function of the transformed variable s can be obtained by

� 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 function

ds

drrpsp rs )()( =

)(sps

)(rpr

)(1

sT−


Histogram Equalization (III)

� Consider the transformation function

∫==r

r dwwprTs0

)()(

)(])([)(

)(

0rpdwwp

dr

d

rd

rdT

dr

dsr

r

r === ∫

1s0 ,1]1[)(

1)()(

)()(

1

1

≤≤==

= −

−=

=sTr

sTrr

rsrp

rpsp

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

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

)(rpr

)(sps

)(rpr

Results of

Histogram

Equalization (1/2)

Transformation functions

20C.E., NCU, Taiwan

Results of Histogram Equalization (2/2)

21C.E., NCU, Taiwan

Histogram Matching

(Histogram Specification) (1/2)� Motivation

� sometimes, it is useful to specify the shape of the histogram of the processed image instead of uniform histogram

� the original image & histogram specification are given

∫==r

r dwwprTs0

)()(

sdttpzGz

z == ∫0 )()(, define

)]([)(11

rTGsGz−− ==

)()( zTzG =Obtain T(r)

Obtain G(z))(zpzGive

Give input image r,

Continuous gray level r & z

Discrete gray level

1,...,2,1,0),(1 −== −

LksGz kk

Histogram Equalization

22C.E., NCU, Taiwan


Inverse mapping from sto the corresponding z

Histogram Matching (2/2)

Local Enhancement� Motivation

� to enhance details over small areas in an image

� 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

Enhancement

� Provide simple while powerful image enhancement

� Histogram statistics

� global mean

� a measure of average gray level in an image

� global variance

� a measure of average contrast

� local mean

� local variance

∑−

=

=1

0

)(L

i

ii rprm

∑−

=

−=µ1

0

22 )()()(

L

i

ii rpmrr

∑∈

=xy

xySts

tstss rprrm),(

,, )()(

∑∈

−=σxy

xyxySts

tsStsS rpmr),(

,2

,2

)(][

∑−

=

−=µ1

0

)()()(L

i

in

in rpmrr

nth moment of r


Outline





Enhancement Using Arithmetic

Operations – Image Subtraction

�

� The enhancement of the differences between images

� to bring out more details, contrast stretching can be performed

� histogram equalization

� power-law transformation

Original imageOriginal image

Zero the 4 lower-order bit plane

Zero the 4 lower-order bit plane

),(),(),( yxhyxfyxg −=

27C.E., NCU, Taiwan

Image Subtraction –

Mask 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 contrast medium into the bloodstream

� The bright arterial paths carrying the medium are unmistakably 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 28


The Range of the Image Values

After Image Subtraction

� For 8-bit image: -255~255

� Two principle ways to scale a different image

� 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]

� more complex


Outline





Basics of Spatial Filtering (1/2)

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

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

+++++++

+−−+−−−−=

yxfwyxfwyxfw

yxfwyxfwR

L )1,1()1,1(),1()0,1(),()0,0(

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

+++++++

+−−+−−−−=

yxfwyxfwyxfw

yxfwyxfwR

L

� The concept of filtering comes from the use of the Fourier transform for signal processing in the frequency domain.

� At each point (x,y), the response of the filter at that point is calculated using a predefined relationship.

� Linear spatial filtering: the response is the sum of products

3131


Basics of Spatial Filtering (2/2)

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

� filtering mask / convolution mask / convolution kernel

� 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 median gray-level value in the neighborhood

When the Center of the Filter Approaches the

Border of the Image ?

� 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

� 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/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 increases



Smoothing Linear Filters (1/2)

� 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 edges

� Reduce irrelevant detail in an image and get a gross representation of objects of interest

� irrelevant: pixel regions that are small with respect to the size of the filter mask

� Applications

� noise reduction

� smooth false contouring

Smoothing Linear

Filters (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 equal

� 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 other

� 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


Examples of Smoothing

Linear Filters (1/2)

� n=3: general slight blurring through the entire image, but details are of approximately the same size as the 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 image

� n=35: black order due to zero padding

36

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 mask

� Smoothing linear filtering + thresholding to eliminate objects based on their intensity

37


Order-Statistics Filters

� 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 pixel

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

� for 3x3 neighborhood

(10,20,20,20,15,20,20,25,100) -> median is 20

� excellent noise-reduction capabilities, with considerably less blurring than linear smoothing filters of similar size

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


An Example of Applying Median Filters

� 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

� 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

� nonzero at the onset and end of a gray-level step or ramp

� zero along ramps of constant slope


Foundations of Sharpening Spatial

Filters (2/2)

� The first-order derivative of a 1-D function f(x)

� The second-order derivative of f(x)

� The two definitions satisfy the conditions stated previously

)()1( xfxfx

f−+=

∂

∂

)(2)1()1(2

2

xfxfxfx

f−−++=

∂

∂

An Example of Applying Sharpening

Spatial Filters (1/3)

42

An Example of Applying Sharpening

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-order derivatives, much finer ones

� Isolated noise point: the response at and around the point is 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 derivatives

� 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. 43


Foundation of

Sharpening Spatial Filters (3/3)

� In most applications, the second derivative is better suited than the first derivative for image enhancement.

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

Second Derivatives for

Enhancement – The Laplacian

� Isotropic filters

� rotation invariant: rotating the image and applying the filter gives the same result as applying the filter to the image first and then rotating the result

� The simplest isotropic derivative operator: Laplacian

� Laplacian is a linear operator : because derivatives of any order are linear operator

y

f

x

ff

2

2

2

22

∂

∂+

∂

∂=∇

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

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

2

2

2

2

yxfyxfyxfx

f

yxfyxfyxfx

f

−−++=∂

∂

−−++=∂

∂

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

yxfyxfyxfyxfyxff −−+++−++=∇45

Filter Masks – The Laplacian (1/2)

Isotropic for increments of 45 deg.

Isotropic for increments of 90 deg.

4646


Filter Masks – The Laplacian (2/2)

� The Laplacian is a derivative operator

� highlights gray-level discontinuities in an image & deemphasizes regions with slowly varying gray levels

� Being used frequently for sharpening digital images

� The Lapalcian for image enhancement

∇+

∇−=

),(),(

),(),(),(

2

2

yxfyxf

yxfyxfyxg

If the center coefficient of theLaplacian mask is negativeIf the center coefficient of the Lapalcian mask is positive

Examples of the Laplacian

Enhanced Image (1/2)

� Notice that the Laplacian image may contain both positive and negative values

� scaling is desired

� Adding the original image to the Laplacian image

� small details were enhanced

� the background tonality is reasonably preserved


� The results obtainable with the mask containing the diagonal terms usually are a little sharper than those obtained with the more basic mask

Examples of the Laplacian

Enhanced Image (2/2)

49C.E., NCU, Taiwan


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

� the magnitude of this vector (gradient)

� nonlinear operator

� isotropic

∂

∂∂

∂

=

=∇

y

fx

f

G

G

y

xf

2/122

2/122][)f(

∂

∂+

∂

∂=+=∇=∇

y

f

x

fGGmagf yx

First Derivatives for Enhancement –

The Gradient (2/2)

� 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 only for a limited number of rotational increments (90 deg.)

� Digital approximation of the gradient

yx GG +≈∇f

5658 , zzGzzG yx −=−=

6859 , zzGzzG yx −=−=

1

2

51C.E., NCU, Taiwan

� Roberts cross-gradient operators

6859 zzzzf −+−≈∇

)2()2(

)2()2(

741963

321987

zzzzzz

zzzzzzf

++−++

+++−++≈∇

(From )2

Since even size are awkward toimplement �� 3x3 Sobel Operator

A 3x3 Region of an Image & Masks

for Computing the Gradient

derivative in xderivative in y52C.E., NCU, Taiwan


An Example of Using Sobel Gradient

� An optical image of a contact lens illuminated by a lighting arrangement to highlight imperfections

� Sobel gradient: the edge defects are quite visible, and the slowly varying shades of gray are eliminated.

� simplify the computational task required for automated inspection.

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