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Local EnhancementLocal Enhancement
Histogram processing methods are Histogram processing methods are global processing, in the sense that global processing, in the sense that pixels are modified by a pixels are modified by a transformation function based on the transformation function based on the gray-level content of an entire gray-level content of an entire image.image.
Sometimes, we may need to enhance Sometimes, we may need to enhance details over small areas in an image, details over small areas in an image, which is called a local enhancement.which is called a local enhancement.
Local EnhancementLocal Enhancement
define a square or rectangular neighborhood and move the define a square or rectangular neighborhood and move the center of this area from pixel to pixel.center of this area from pixel to pixel.
at each location, the histogram of the points in the at each location, the histogram of the points in the neighborhood is computed and either histogram equalization neighborhood is computed and either histogram equalization or histogram specification transformation function is obtained.or histogram specification transformation function is obtained.
another approach used to reduce computation is to utilize another approach used to reduce computation is to utilize nonoverlapping regions, but it usually produces an nonoverlapping regions, but it usually produces an undesirable checkerboard effect.undesirable checkerboard effect.
a) Original image (slightly blurred to reduce noise)
b) global histogram equalization (enhance noise & slightly increase contrast but the construction is not changed)
c) local histogram equalization using 7x7 neighborhood (reveals the small squares inside larger ones of the original image. (a) (b) (c)
Explain the result in c)Explain the result in c)
Basically, the original image consists of Basically, the original image consists of many small squares inside the larger dark many small squares inside the larger dark ones.ones.
However, the small squares were too close However, the small squares were too close in gray level to the larger ones, and their in gray level to the larger ones, and their sizes were too small to influence global sizes were too small to influence global histogram equalization significantly.histogram equalization significantly.
So, when we use the local enhancement So, when we use the local enhancement technique, it reveals the small areas.technique, it reveals the small areas.
Note also the finer noise texture is resulted Note also the finer noise texture is resulted by the local processing using relatively by the local processing using relatively small neighborhoods.small neighborhoods.
Enhancement using Enhancement using Arithmetic/Logic OperationsArithmetic/Logic Operations
Arithmetic/Logic operations perform Arithmetic/Logic operations perform on pixel by pixel basis between two on pixel by pixel basis between two or more imagesor more images
except NOT operation which perform except NOT operation which perform only on a single imageonly on a single image
Logic OperationsLogic Operations
Logic operation performs on gray-Logic operation performs on gray-level images, the pixel values are level images, the pixel values are processed as binary numbersprocessed as binary numbers
light represents a binary 1, and dark light represents a binary 1, and dark represents a binary 0 represents a binary 0
NOT operation = negative NOT operation = negative transformationtransformation
Example of AND OperationExample of AND Operation
original image AND image mask
result of AND
operation
Example of OR OperationExample of OR Operation
original image OR image mask
result of OR operation
Image SubtractionImage Subtraction
g(x,y) = f(x,y) – h(x,y)g(x,y) = f(x,y) – h(x,y)
enhancement of the differences enhancement of the differences between imagesbetween images
Image SubtractionImage Subtraction
a). original fractal imagea). original fractal image b). result of setting the four b). result of setting the four
lower-order bit planes to zerolower-order bit planes to zero• refer to the bit-plane slicingrefer to the bit-plane slicing• the higher planes contribute the higher planes contribute
significant detailsignificant detail• the lower planes contribute the lower planes contribute
more to fine detailmore to fine detail• image b). is nearly identical image b). is nearly identical
visually to image a), with a visually to image a), with a very slightly drop in overall very slightly drop in overall contrast due to less variability contrast due to less variability of the gray-level values in the of the gray-level values in the image.image.
c). difference between a). and c). difference between a). and b). (nearly black)b). (nearly black)
d). histogram equalization of d). histogram equalization of c). (perform contrast c). (perform contrast stretching transformation)stretching transformation)
aa bb
cc dd
Spatial FilteringSpatial Filtering
use filter (can also be called as use filter (can also be called as mask/kernel/template or window)mask/kernel/template or window)
the values in a filter subimage are the values in a filter subimage are referred to as coefficients, rather than referred to as coefficients, rather than pixel.pixel.
our focus will be on masks of odd our focus will be on masks of odd sizes, e.g. 3x3, 5x5,…sizes, e.g. 3x3, 5x5,…
Spatial Filtering ProcessSpatial Filtering Process
simply move the filter mask from simply move the filter mask from point to point in an image.point to point in an image.
at each point (x,y), the response of at each point (x,y), the response of the filter at that point is calculated the filter at that point is calculated using a predefined relationship.using a predefined relationship.
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Smoothing Spatial FiltersSmoothing Spatial Filters
used for blurring and for noise used for blurring and for noise reductionreduction
blurring is used in preprocessing steps, blurring is used in preprocessing steps, such as such as • removal of small details from an image removal of small details from an image
prior to object extractionprior to object extraction• bridging of small gaps in lines or curvesbridging of small gaps in lines or curves
noise reduction can be accomplished noise reduction can be accomplished by blurring with a linear filter and also by blurring with a linear filter and also by a nonlinear filterby a nonlinear filter
Smoothing Linear FiltersSmoothing Linear Filters
output is simply the average of the output is simply the average of the pixels contained in the neighborhood of pixels contained in the neighborhood of the filter mask.the filter mask.
called averaging filters or lowpass filters.called averaging filters or lowpass filters.
Smoothing Linear FiltersSmoothing Linear Filters
replacing the value of every pixel in an replacing the value of every pixel in an image by the average of the gray levels in image by the average of the gray levels in the neighborhood will reduce the “sharp” the neighborhood will reduce the “sharp” transitions in gray levels.transitions in gray levels.
sharp transitionssharp transitions• random noise in the imagerandom noise in the image• edges of objects in the imageedges of objects in the image
thus, smoothing can reduce noises thus, smoothing can reduce noises (desirable) and blur edges (undesirable)(desirable) and blur edges (undesirable)
3x3 Smoothing Linear Filters3x3 Smoothing Linear Filters
box filter weighted average
the center is the most important and other pixels are inversely weighted as a function of their distance from the center of the mask
Weighted average filterWeighted average filter
the basic strategy behind weighting the basic strategy behind weighting the center point the highest and then the center point the highest and then reducing the value of the coefficients reducing the value of the coefficients as a function of increasing distance as a function of increasing distance from the origin is simply from the origin is simply an attempt an attempt to reduce blurring in the to reduce blurring in the smoothing processsmoothing process..
ExampleExample
a). original image 500x500 a). original image 500x500 pixelpixel
b). - f). results of smoothing b). - f). results of smoothing with square averaging filter with square averaging filter masks of size n = 3, 5, 9, 15 masks of size n = 3, 5, 9, 15 and 35, respectively.and 35, respectively.
Note:Note:• big mask is used to eliminate big mask is used to eliminate
small objects from an image.small objects from an image.• the size of the mask establishes the size of the mask establishes
the relative size of the objects the relative size of the objects that will be blended with the that will be blended with the background.background.
aa bb
cc dd
ee ff
Order-Statistics Filters (Nonlinear Order-Statistics Filters (Nonlinear Filters)Filters)
the response is based on ordering the response is based on ordering (ranking) the pixels contained in the (ranking) the pixels contained in the image area encompassed by the filterimage area encompassed by the filter
exampleexample• median filter : R = median{zmedian filter : R = median{zk k |k = 1,2,|k = 1,2,
…,n x n}…,n x n}• max filter : R = max{zmax filter : R = max{zk k |k = 1,2,…,n x n}|k = 1,2,…,n x n}• min filter : R = min{zmin filter : R = min{zk k |k = 1,2,…,n x n}|k = 1,2,…,n x n}
note: n x n is the size of the masknote: n x n is the size of the mask
Median FiltersMedian Filters
replaces the value of a pixel by the median replaces the value of a pixel by the median of the gray levels in the neighborhood of of the gray levels in the neighborhood of that pixel (the original value of the pixel is that pixel (the original value of the pixel is included in the computation of the median)included in the computation of the median)
quite popular because for certain types of quite popular because for certain types of random noise (random noise (impulse noise impulse noise salt and salt and pepper noisepepper noise) , they ) , they provide excellent provide excellent noise-reduction capabilitiesnoise-reduction capabilities, with , with considering considering less blurring than linear less blurring than linear smoothing filters of similar size.smoothing filters of similar size.
2626
Example : Median FiltersExample : Median Filters
2727
Sharpening Spatial FiltersSharpening Spatial Filters
to highlight fine detail in an image to highlight fine detail in an image or to enhance detail that has been or to enhance detail that has been
blurred, either in error or as a natural blurred, either in error or as a natural effect of a particular method of effect of a particular method of image acquisition.image acquisition.
2828
Blurring vs. SharpeningBlurring vs. Sharpening
as we know that blurring can be done as we know that blurring can be done in spatial domain by pixel averaging in in spatial domain by pixel averaging in a neighbors a neighbors
since averaging is analogous to since averaging is analogous to integrationintegration
thus, we can guess that the sharpening thus, we can guess that the sharpening must be accomplished by must be accomplished by spatial spatial differentiation.differentiation.
2929
Derivative operatorDerivative operator
the strength of the response of a derivative the strength of the response of a derivative operator is proportional to the degree of operator is proportional to the degree of discontinuity of the image at the point at discontinuity of the image at the point at which the operator is applied.which the operator is applied.
thus, image differentiation thus, image differentiation • enhances edges and other discontinuities enhances edges and other discontinuities
(noise)(noise)• deemphasizes area with slowly varying gray-deemphasizes area with slowly varying gray-
level values.level values.
3030
First-order derivativeFirst-order derivative
a basic definition of the first-order a basic definition of the first-order derivative of a one-dimensional derivative of a one-dimensional function f(x) is the differencefunction f(x) is the difference
)()1( xfxfx
f
3131
Second-order derivativeSecond-order derivative
similarly, we define the second-order similarly, we define the second-order derivative of a one-dimensional derivative of a one-dimensional function f(x) is the differencefunction f(x) is the difference
)(2)1()1(2
2
xfxfxfx
f
Response of First and Second Response of First and Second order derivativesorder derivatives
Response of first order derivative is:Response of first order derivative is: zero in flat segments (area of constant grey zero in flat segments (area of constant grey
values)values) Non zero at the onset of a grey level step or rampNon zero at the onset of a grey level step or ramp Non zero along ramps Non zero along ramps
Response of second order derivative is:Response of second order derivative is: Zero in flat areasZero in flat areas Non zero at the onset of a grey level step or rampNon zero at the onset of a grey level step or ramp Zero along ramps of constant slopeZero along ramps of constant slope
First and Second-order derivative First and Second-order derivative of f(x,y)of f(x,y)
when we consider an image function when we consider an image function of two variables, f(x,y), at which time of two variables, f(x,y), at which time we will dealing with partial we will dealing with partial derivatives along the two spatial derivatives along the two spatial axes.axes.
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Laplacian operator
Gradient operator
Discrete Form of LaplacianDiscrete Form of Laplacian
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Result Laplacian maskResult Laplacian mask
Laplacian mask implemented an Laplacian mask implemented an extension of diagonal neighborsextension of diagonal neighbors
Other implementation of Laplacian Other implementation of Laplacian masksmasks
give the same result, but we have to keep in mind that when combining (add / subtract) a Laplacian-filtered image with another image.
Effect of Laplacian OperatorEffect of Laplacian Operator
as it is a derivative operator,as it is a derivative operator,• it highlights gray-level discontinuities in it highlights gray-level discontinuities in
an imagean image• it deemphasizes regions with slowly it deemphasizes regions with slowly
varying gray levelsvarying gray levels tends to produce images that have tends to produce images that have
• grayish edge lines and other grayish edge lines and other discontinuities, all superimposed on a discontinuities, all superimposed on a dark, dark,
• featureless background.featureless background.
Correct the effect of featureless Correct the effect of featureless backgroundbackground
easily by adding the original and easily by adding the original and Laplacian image.Laplacian image.
be careful with the Laplacian filter usedbe careful with the Laplacian filter used
),(),(
),(),(),(
2
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if the center coefficient of the Laplacian mask is negative
if the center coefficient of the Laplacian mask is positive
ExampleExample
a). image of the North a). image of the North pole of the moonpole of the moon
b). Laplacian-filtered b). Laplacian-filtered image with image with
c). Laplacian image c). Laplacian image scaled for display scaled for display purposespurposes
d). image enhanced by d). image enhanced by addition with original addition with original image image
11 11 11
11 -8-8 11
11 11 11
Mask of Laplacian + additionMask of Laplacian + addition
to simply the computation, we can to simply the computation, we can create a mask which do both create a mask which do both operations, Laplacian Filter and operations, Laplacian Filter and Addition the original image.Addition the original image.
Mask of Laplacian + additionMask of Laplacian + addition
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ExampleExample
NoteNote
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Unsharp maskingUnsharp masking
to subtract a blurred version of an image to subtract a blurred version of an image produces sharpening output image.produces sharpening output image.
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sharpened image = original image – blurred imagesharpened image = original image – blurred image
High-boost filteringHigh-boost filtering
generalized form of Unsharp maskinggeneralized form of Unsharp masking A A 1 1
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High-boost filteringHigh-boost filtering
if we use Laplacian filter to create if we use Laplacian filter to create sharpen image fsharpen image fss(x,y) with addition of (x,y) with addition of original imageoriginal image
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High-boost filteringHigh-boost filtering
yieldsyields
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if the center coefficient of the Laplacian mask is negative
if the center coefficient of the Laplacian mask is positive
High-boost MasksHigh-boost Masks
A A 1 1 if A = 1, it becomes “standard” if A = 1, it becomes “standard”
Laplacian sharpeningLaplacian sharpening
ExampleExample
Gradient OperatorGradient Operator
first derivatives are implemented first derivatives are implemented using the using the magnitude of the magnitude of the gradientgradient..
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the magnitude becomes nonlinearyx GGf
commonly approx.
Gradient MaskGradient Mask simplest approximation, 2x2simplest approximation, 2x2
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Gradient MaskGradient Mask
Sobel operators, 3x3Sobel operators, 3x3
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the weight value 2 is to achieve smoothing by giving more important to the center point
NoteNote
the summation of coefficients in all the summation of coefficients in all masks equals 0, indicating that they masks equals 0, indicating that they would give a response of 0 in an area would give a response of 0 in an area of constant gray level.of constant gray level.
ExampleExample
Example of Combining Spatial Example of Combining Spatial Enhancement MethodsEnhancement Methods
want to sharpen want to sharpen the original image the original image and bring out more and bring out more skeletal detail.skeletal detail.
problems: narrow problems: narrow dynamic range of dynamic range of gray level and high gray level and high noise content noise content makes the image makes the image difficult to enhancedifficult to enhance
Example of Combining Spatial Example of Combining Spatial Enhancement MethodsEnhancement Methods
solve : solve :
1.1. Laplacian to highlight fine Laplacian to highlight fine detaildetail
2.2. gradient to enhance gradient to enhance prominent edgesprominent edges
3.3. gray-level transformation to gray-level transformation to increase the dynamic range of increase the dynamic range of gray levelsgray levels
Image Enhancement in the Image Enhancement in the Frequency DomainFrequency Domain
Fourier SeriesFourier Series
Any function that periodically repeats itself can be expressedas the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficients. This sum is called a Fourier series.
Fourier SeriesFourier Series
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Fourier TransformFourier Transform
A function that is not periodic but the area under its curve is finite can be expressed as the integral of sines and/or cosines multiplied by a weighing function. The formulationin this case is Fourier transform.
The One-Dimensional Fourier The One-Dimensional Fourier Transform and its InverseTransform and its Inverse
The fourier transform F(u) :The fourier transform F(u) :
The inverse Fourier transform f(x) :The inverse Fourier transform f(x) :
Two variables Fourier transform F(u, v) : Two variables Fourier transform F(u, v) :
The inverse transform f(x, y) : The inverse transform f(x, y) :
dxexfuF uxj 2)()(
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The One-Dimensional Fourier The One-Dimensional Fourier Transform and its InverseTransform and its Inverse
The discrete Fourier Transform F(u) :The discrete Fourier Transform F(u) :
The inverse DFT :The inverse DFT :
Apply euler’s formula : Apply euler’s formula :
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The One-Dimensional Fourier The One-Dimensional Fourier Transform and its InverseTransform and its Inverse
F(u) in polar coordinates :F(u) in polar coordinates :
spectrum)Fourier theof (square
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The One-Dimensional Fourier The One-Dimensional Fourier Transform and its InverseTransform and its Inverse
The two dimensional DFT and its The two dimensional DFT and its InverseInverse
The discrete fourier transform of a function f(x,y) The discrete fourier transform of a function f(x,y) of size M x N :of size M x N :
The inverse Fourier transform :The inverse Fourier transform :
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The two dimensional DFT and its The two dimensional DFT and its InverseInverse
DFT corresponding (u,v)=(0,0)DFT corresponding (u,v)=(0,0)
If f(x,y) is real, its Fourier transform is conjugate If f(x,y) is real, its Fourier transform is conjugate symmetricsymmetric
The spectrum of the Fourier transform is The spectrum of the Fourier transform is symmetricsymmetric
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The two dimensional DFT and its The two dimensional DFT and its InverseInverse
Basics of filtering in the frequency Basics of filtering in the frequency domaindomain
1.1. Multiply the input image by to Multiply the input image by to center the transformcenter the transform
2.2. Compute F(u,v), by the DFT of the image Compute F(u,v), by the DFT of the image from (1)from (1)
3.3. Multiply F(u,v) by a filter function H(u,v)Multiply F(u,v) by a filter function H(u,v)4.4. Compute the inverse DFT of the result in Compute the inverse DFT of the result in
(3)(3)5.5. Obtain the real part of the result in (4)Obtain the real part of the result in (4)6.6. Multiply the result in (5) by Multiply the result in (5) by
yx )1(
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Basics of filtering in the frequency Basics of filtering in the frequency domaindomain
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Some basic filters and their Some basic filters and their propertiesproperties
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Some basic filters and their Some basic filters and their propertiesproperties
Lowpass filter : less Lowpass filter : less sharpsharp(Smoothing)(Smoothing)
Highpass filter : less Highpass filter : less gray-level variation gray-level variation and emphasized and emphasized edgesedges(Sharpening)(Sharpening)
Correspondence between Filtering Correspondence between Filtering in the Spatial and Frequency in the Spatial and Frequency
DomainsDomains The convolution theorem :The convolution theorem :
The impulse response :The impulse response :
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Correspondence between Filtering Correspondence between Filtering in the Spatial and Frequency in the Spatial and Frequency
DomainsDomains The Fourier transform of a unit impulse functionThe Fourier transform of a unit impulse function
Let Let
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Correspondence between Filtering Correspondence between Filtering in the Spatial and Frequency in the Spatial and Frequency
DomainsDomains Gaussian functions :Gaussian functions :
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Correspondence between Filtering Correspondence between Filtering in the Spatial and Frequency in the Spatial and Frequency
DomainsDomains
Smoothing Frequency-Domain Smoothing Frequency-Domain FiltersFilters
IdealIdeal ButterworthButterworth GaussianGaussian
Ideal lowpass filterIdeal lowpass filter
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Ideal lowpass filterIdeal lowpass filter
Ideal lowpass filterIdeal lowpass filter
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Ideal lowpass filterIdeal lowpass filter
Ideal lowpass filterIdeal lowpass filter
Butterworth Lowpass FiltersButterworth Lowpass Filters
When D(u,v)= , H(u,v)=0.5When D(u,v)= , H(u,v)=0.50D
nDvuDvuH
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Butterworth Lowpass FiltersButterworth Lowpass Filters
Butterworth Lowpass FiltersButterworth Lowpass Filters
Butterworth Lowpass FiltersButterworth Lowpass Filters
Gaussian Lowpass filterGaussian Lowpass filter
letlet
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Gaussian Lowpass filterGaussian Lowpass filter
Gaussian Lowpass filterGaussian Lowpass filter
Sharpening Frequency Domain Sharpening Frequency Domain FiltersFilters
IdealIdeal ButterworthButterworth GaussianGaussian
),(1),( vuHvuH lphp
Sharpening Frequency Domain Sharpening Frequency Domain FiltersFilters
Ideal Highpass FiltersIdeal Highpass Filters
Ideal highpass filterIdeal highpass filter
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Butterworth Highpass FiltersButterworth Highpass Filters
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The Laplacian in the Frequency The Laplacian in the Frequency DomainDomain
The Laplacian in the Frequency The Laplacian in the Frequency DomainDomain
),(),(),( 2 yxfyxfyxg
Unsharp Masking, High-Boost Filtering, and Unsharp Masking, High-Boost Filtering, and High-Frequency Emphasis FilteringHigh-Frequency Emphasis Filtering
Unsharp masking :Unsharp masking : High-Boost filtering :High-Boost filtering :
High-Frequency Emphasis FilteringHigh-Frequency Emphasis Filtering
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Unsharp Masking, High-Boost Filtering, and Unsharp Masking, High-Boost Filtering, and High-Frequency Emphasis FilteringHigh-Frequency Emphasis Filtering