DIGITAL IMAGE PROCESSING

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DIGITAL IMAGE PROCESSING

Instructors: Dr J. [email protected]@gmail.com

( J.Shanbehzadeh M.Gholizadeh )DIGITAL IMAGE PROCESSING

Instructors: Dr J. [email protected] [email protected]

Chapter 3 - Intensity Transformations and Spatial Filtering

( J.Shanbehzadeh M.Gholizadeh )Road map of chapter 3

3.13.33.43.53.13.1- Background3.2- Some Basic Intensity Transformation Functions3,3- Histogram Processing3.4- Fundamentals of Spatial Filtering3.5 - Smoothing Spatial Filters3.6- Sharpening Spatial Filters3.7- Combining Spatial Enhancement Tools3.8- Using Fuzzy Techniques for Intensity Transformations and Spatial Filtering

Background3.23.2Some Basic Intensity Transformation FunctionsHistogram Processing3.33.4Fundamentals of Spatial Filtering3.5Smoothing Spatial Filters3.63.6Sharpening Spatial Filters3.73.73.83.8Combining Spatial Enhancement ToolsUsing Fuzzy Techniques for Intensity Transformations and Spatial Filtering

( J.Shanbehzadeh M.Gholizadeh )Image EnhancementInput Image I(r,c)Frequency Sequency DomainSpatial DomainApplication Specific FeedbackOutput Image E(r,c)

( J.Shanbehzadeh M.Gholizadeh )Principle Objective of EnhancementProcess an image so that the result will be more suitable than the original image for a specific application.

The suitableness is up to each application.

A method which is quite useful for enhancing an image may not necessarily be the best approach for enhancing another image

( J.Shanbehzadeh M.Gholizadeh )Image Enhancement MethodsSpatial Domain : (image plane)Techniques are based on direct manipulation of pixels in an image

Frequency Domain : Techniques are based on modifying the Fourier transform of an image

There are some enhancement techniques based on various combinations of methods from these two categories.

( J.Shanbehzadeh M.Gholizadeh )3.1Background

( J.Shanbehzadeh M.Gholizadeh )Elements of Visual PerceptionThe Basics of Intensity Transformations and Spatial FilteringAbout the Examples in This Chapter3.1- Background

3.2- Some Basic Intensity Transformation Functions

3,3- Histogram Processing

3.4- Fundamentals of Spatial Filtering

3.5 - Smoothing Spatial Filters

3.6- Sharpening Spatial Filters

3.7- Combining Spatial Enhancement Tools

3.8- Using Fuzzy Techniques for Intensity Transformations and Spatial Filtering

The Basics of Intensity Transformations and Spatial Filtering

( J.Shanbehzadeh M.Gholizadeh )The Basics of Intensity Transformations and Spatial Filtering3.1- Background








Neighborhood of a point (x,y) can be defined by using a square/rectangular (common used) or circular sub image area centered at (x,y)

The center of the sub image is moved from pixel to pixel starting at the top of the corner









Spatial domain process

G(x,y)=T[f(x,y)]

Where f(x,y) is the input image, g(x,y) is the processed image, and T is an operator on f, defined over some neighborhood of (x,y)









Neighborhood of a point (x,y) can be defined by using a square/rectangular (common used) or circular sub image area centered at (x,y)

The center of the sub image is moved from pixel to pixel starting at the top of the corner









Gray-level transformation function

S=T(r)

where r is the gray level of f(x,y) and s is the gray level of g(x,y) at any point (x,y)

( J.Shanbehzadeh M.Gholizadeh )Elements of Visual PerceptionThe Basics of Intensity Transformations and Spatial FilteringAbout the Examples in This Chapter3.1- Background








About the Examples in This Chapter( J.Shanbehzadeh M.Gholizadeh )3.2 Some Basic Inventory Transformation Functions ( J.Shanbehzadeh M.Gholizadeh )Some Basic Intensity Transformation FunctionsImage NegativesLog Transformations3.1- Background








Power-Law(Gamma) TransformationsPiecewise-Linear Transformation FunctionsImage Negatives( J.Shanbehzadeh M.Gholizadeh )Image NegativesAn image with gray level in the range [0, L-1] where L = 2n ; n = 1, 2Negative transformation :s = L 1 r Reversing the intensity levels of an image.Suitable for enhancing white or gray detail embedded in dark regions of an image, especially when the black area dominant in size.

Input gray level, rOutput gray level, sNegativeLognth rootIdentitynth powerInverse Log3.1- Background








( J.Shanbehzadeh M.Gholizadeh )Image Negatives3.1- Background








Image negatives:Enhance white or gray details

s=L-1-r

( J.Shanbehzadeh M.Gholizadeh )Image Negatives3.1- Background








Lx0Lyy=L-x( J.Shanbehzadeh M.Gholizadeh )Some Basic Intensity Transformation FunctionsImage NegativesLog Transformations3.1- Background








Power-Law(Gamma) TransformationsPiecewise-Linear Transformation FunctionsLog Transformations( J.Shanbehzadeh M.Gholizadeh )Log Transformations3.1- Background








c is a constant and r 0

Log curve maps a narrow range of low gray-level values in the input image into a wider range of output levels.

Used to expand the values of dark pixels in an image while compressing the higher-level values.

( J.Shanbehzadeh M.Gholizadeh )Log Transformations3.1- Background








Compress the dynamic range of images with large variations in pixel valuesStretch dark region, suppress bright region

From the range 0 1.5106 to the range 0 to 6.2We cant see the significant degree of detail as it will be lost in the display.

( J.Shanbehzadeh M.Gholizadeh )Range Compression3.1- Background








Lx0

c=100

Y=clog10ddy( J.Shanbehzadeh M.Gholizadeh )Some Basic Intensity Transformation FunctionsImage NegativesLog Transformations3.1- Background








Power-Law(Gamma) TransformationsPiecewise-Linear Transformation FunctionsPower-Law(Gamma) Transformations( J.Shanbehzadeh M.Gholizadeh )Power-Law(Gamma) Transformations3.1- Background








Power-law transformations: s=cr or s=c(r+) 1 maps a narrow range of bright input values into a wider range of output values

: gamma, gamma correction

( J.Shanbehzadeh M.Gholizadeh )Power-Law(Gamma) Transformations3.1- Background








( J.Shanbehzadeh M.Gholizadeh )Power-Law(Gamma) Transformations3.1- Background








(a) a magnetic resonance image(MRI) of an upper thoracic human spine with a fracture dislocation and spinal cord impingementThe picture is predominately darkAn expansion of gray levels are desirable needs < 1

(b) result after power-law transformation with = 0.6, c=1

(c) transformation with = 0.4 (best result)

(d) transformation with = 0.3 (under acceptable level)(a)(b)(c)(d)( J.Shanbehzadeh M.Gholizadeh )26Power-Law(Gamma) Transformations(Effect of decreasing gamma)3.1- Background








When the is reduced too much, the image begins to reduce contrast to the point where the image started to have very slight wash-out look, specially in the background

( J.Shanbehzadeh M.Gholizadeh )Power-Law(Gamma) Transformations(Effect of decreasing gamma)3.1- Background








(a) image has a washed-out appearance, it needs a compression of gray levels needs > 1

(b) result after power-law transformation with = 3.0 (suitable)

(c) transformation with = 4.0 (suitable)

(d) transformation with = 5.0 (high contrast, the image has areas that are too dark, some detail is lost)Original imageabcd( J.Shanbehzadeh M.Gholizadeh )Some Basic Intensity Transformation FunctionsImage NegativesLog Transformations3.1- Background








Power-Law(Gamma) TransformationsPiecewise-Linear Transformation FunctionsPiecewise-Linear Transformation Functions( J.Shanbehzadeh M.Gholizadeh )Piecewise-Linear Transformation FunctionsAdvantage:The form of piecewise functions can be arbitrarily complex.A Practical Implementation of some important transformations can be formulated only as piecewise functions.

Disadvantage: Their specification requires considerably more user input.3.1- Background








( J.Shanbehzadeh M.Gholizadeh )30Gray-Scale Modification3.1- Background








( J.Shanbehzadeh M.Gholizadeh )31Contrast Stretching

Lx0abyayb

y3.1- Background








( J.Shanbehzadeh M.Gholizadeh )32Contrast Stretching3.1- Background








( J.Shanbehzadeh M.Gholizadeh )33Gray-level Slicing

Highlighting a specific range of gray levels in an imageDisplay a high value of all gray levels in the range of interest and a low value for all other gray levels

(a) transformation highlights range [A,B] of gray level and reduces all others to a constant level

(b) transformation highlights range [A,B] but preserves all other levels Brighten in the range3.1- Background








( J.Shanbehzadeh M.Gholizadeh )34Gray-level Slicing3.1- Background









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( J.Shanbehzadeh M.Gholizadeh )37Digital Image Processing Piecewise-Linear Transformation Functions Gray-level Slicing

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( J.Shanbehzadeh M.Gholizadeh )38Bit-plane SlicingHighlighting the contribution made to total image appearance by specific bitsSuppose each pixel is represented by 8 bitsHigher-order bits contain the majority of the visually significant dataUseful for analyzing the relative importance played by each bit of the image

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( J.Shanbehzadeh M.Gholizadeh )39Bit-plane Slicing - Fractal ImageThe (binary) image for bit-plane 7 can be obtained by processing the input image with a thresholding gray-level transformation.Map all levels between 0 and 127 to 0Map all levels between 129 and 255 to 255

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( J.Shanbehzadeh M.Gholizadeh )40Bit-plane Slicing - Fractal Image

Bit-plane 7Bit-plane 6Bit-plane 5Bit-plane 4Bit-plane 3Bit-plane 2Bit-plane 1Bit-plane 03.1- Background








( J.Shanbehzadeh M.Gholizadeh )41Bit-plane Slicing

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( J.Shanbehzadeh M.Gholizadeh )42Bit-plane Slicing

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( J.Shanbehzadeh M.Gholizadeh )43Clipping

Lx0ab

y3.1- Background








( J.Shanbehzadeh M.Gholizadeh )44Summary of Point OperationSo far, we have discussed various forms of f(x) that leads to different enhancement results

The natural question is: How to select an appropriate f(x) for an arbitrary image?

One systematic solution is based on the histogram information of an image.3.1- Background








( J.Shanbehzadeh M.Gholizadeh )453.3 Histogram Processing( J.Shanbehzadeh M.Gholizadeh )Histogram ProcessingHistogram ModificationHistogram Equalization3.1- Background








Adaptive Contrast Enhancement (ACE)Histogram Matching (Specification)Local Histogram ProcessingUsing Histogram Statistics for Image EnhancementHistogram Modification( J.Shanbehzadeh M.Gholizadeh )47Histogram Processing(What is Histogram?)Histogram: h(rk)=nk

Where rk is the kth gray level and nk is the number of pixels in the image having gray level rk

Normalized histogram: P(rk)=nk/n

Histogram of an image represents the relative frequency of occurrence of various gray levels in the image3.1- Background








( J.Shanbehzadeh M.Gholizadeh )48Histogram Examples

Dark imageBright imageComponents of histogram are concentrated on the low side of the gray scale.Components of histogram are concentrated on the high side of the gray scale.3.1- Background








( J.Shanbehzadeh M.Gholizadeh )49Histogram Examples

Low-contrast imageHigh-contrast imageHistogram is narrow and centered toward the middle of the gray scaleHistogram covers broad range of the gray scale and the distribution of pixels is not too far from uniform, with very few vertical lines being much higher than the others3.1- Background








( J.Shanbehzadeh M.Gholizadeh )50Why Histogram?Histogram information reveals that image is under-exposed

It is a baby in the cradle!3.1- Background








( J.Shanbehzadeh M.Gholizadeh )51Why Histogram?3.1- Background








Over-exposed image( J.Shanbehzadeh M.Gholizadeh )52Histogram Modification Histogram Stretching Histogram Shrink Histogram Sliding3.1- Background








Histogram Modification

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( J.Shanbehzadeh M.Gholizadeh )54Histogram Stretching

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( J.Shanbehzadeh M.Gholizadeh )57Histogram Shrinking

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( J.Shanbehzadeh M.Gholizadeh )58Histogram Shrinking

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( J.Shanbehzadeh M.Gholizadeh )59Histogram Sliding

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( J.Shanbehzadeh M.Gholizadeh )60Histogram ProcessingHistogram ModificationHistogram Equalization3.1- Background








Adaptive Contrast Enhancement (ACE)Histogram Matching (Specification)Local Histogram ProcessingUsing Histogram Statistics for Image EnhancementHistogram Equalization( J.Shanbehzadeh M.Gholizadeh )61Histogram EqualizationAs the low-contrast images histogram is narrow and centered toward the middle of the gray scale, if we distribute the histogram to a wider range the quality of the image will be improved.

We can do it by adjusting the probability density function of the original histogram of the image so that the probability spread equally

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( J.Shanbehzadeh M.Gholizadeh )62Histogram EqualizationHistogram equalization:

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( J.Shanbehzadeh M.Gholizadeh )63Histogram Equalization

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( J.Shanbehzadeh M.Gholizadeh )64

Histogram EqualizationProbability density functions (PDF):

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BeforeAfterThe quality is not improved much because the original image already has a broaden gray-level scale 3.1- Background









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HistogramCumulativeNormalized Histogram

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Histogram Equalizationtransformed intensitygk = (L-1) * T(k).To encompass the whole dynamic range.

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Adaptive Contrast Enhancement (ACE)Histogram Matching (Specification)Local Histogram ProcessingUsing Histogram Statistics for Image EnhancementAdaptive Contrast Enhancement (ACE)( J.Shanbehzadeh M.Gholizadeh )77Adaptive Contrast Enhancement (ACE)

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Adaptive Contrast Enhancement (ACE)3.1- Background


































Adaptive Contrast Enhancement (ACE)Histogram Matching (Specification)Local Histogram ProcessingUsing Histogram Statistics for Image EnhancementHistogram Matching (Specification)( J.Shanbehzadeh M.Gholizadeh )82Histogram Matching (Specification) Histogram equalization has a disadvantage which is that it can generate only one type of output image.

With Histogram Specification, we can specify the shape of the histogram that we wish the output image to have.

It doesnt have to be a uniform histogram

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( J.Shanbehzadeh M.Gholizadeh )83Given a desired histogram, the goal is to transform the image intensity such that the transformed image has a histogram matches the desired histogram.

Let the initial image histogram be pr, the desired image histogram be pz.

Let T be the function that equalizes the original image and G be the function that equalizes the desired image. r: The initial image intensity. s: The image intensity after equalization z: The image intensity of the desired image. v: The image intensity after equalization of the desired image.

Histogram Matching (Specification) 3.1- Background








( J.Shanbehzadeh M.Gholizadeh )84Histogram matching (specification)

is the desired PDFHistogram Matching (Specification) 3.1- Background

















( J.Shanbehzadeh M.Gholizadeh )86Histogram matching:Obtain the histogram of the given image, T(r)Precompute a mapped level Sk for each level rk Obtain the transformation function G from the given pz(z)Precompute Zk for each value of rkMap rk to its corresponding level Sk; then map level Sk into the final level ZkHistogram Matching (Specification) 3.1- Background













































Image is dominated by large, dark areas, resulting in a histogram characterized by a large concentration of pixels in pixels in the dark end of the gray scaleHistogram Matching (Specification) 3.1- Background









Notice that the output histograms low end has shifted right toward the lighter region of the gray scale as desired.Histogram Matching (Specification) 3.1- Background

























Adaptive Contrast Enhancement (ACE)Histogram Matching (Specification)Local Histogram ProcessingUsing Histogram Statistics for Image EnhancementLocal Histogram Processing( J.Shanbehzadeh M.Gholizadeh )95Local Histogram ProcessingHistogram specification is a trial-and-error process

There are no rules for specifying histograms, and one must resort to analysis on a case-by-case basis for any given enhancement task.

Histogram processing methods are global processing, in the sense that pixels are modified by a transformation function based on the gray-level content of an entire image.

Sometimes, we may need to enhance details over small areas in an image, which is called a local enhancement.

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( J.Shanbehzadeh M.Gholizadeh )96Local enhancement:Histogram using a local neighborhood, for example 7*7 neighborhood

Original image (slightly blurred to reduce noise)global histogram equalization (enhance noise & slightly increase contrast but the construction is not changed)local histogram equalization using 7x7 neighborhood (reveals the small squares inside larger ones of the original image. define a square or rectangular neighborhood and move the center of this area from pixel to pixel.at each location, the histogram of the points in the neighborhood is computed and either histogram equalization or histogram specification transformation function is obtained.another approach used to reduce computation is to utilize no overlapping regions, but it usually produces an undesirable checkerboard effect.Local Histogram Processing3.1- Background









Basically, the original image consists of many small squares inside the larger dark ones.

However, the small squares were too close in gray level to the larger ones, and their sizes were too small to influence global histogram equalization significantly.

So, when we use the local enhancement technique, it reveals the small areas.

Note also the finer noise texture is resulted by the local processing using relatively small neighborhoods.

Local Histogram Processing3.1- Background








( J.Shanbehzadeh M.Gholizadeh )98Histogram using a local 3*3 neighborhood

Local Histogram Processing3.1- Background
















Adaptive Contrast Enhancement (ACE)Histogram Matching (Specification)Local Histogram ProcessingUsing Histogram Statistics for Image EnhancementUsing Histogram Statistics for Image Enhancement( J.Shanbehzadeh M.Gholizadeh )100Use of histogram statistics for image enhancement:r denotes a discrete random variableP(ri) denotes the normalized histogram component corresponding to the ith value of r

Mean:

The nth moment:

The second moment:

Using Histogram Statistics 3.1- Background








( J.Shanbehzadeh M.Gholizadeh )101Global enhancement: The global mean and variance are measured over an entire image

Local enhancement: The local mean and variance are used as the basis for making changes3.1- Background








Using Histogram Statistics ( J.Shanbehzadeh M.Gholizadeh )102 rs,t is the gray level at coordinates (s,t) in the neighborhood

P(rs,t) is the neighborhood normalized histogram component

mean:

local variance:

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Using Histogram Statistics ( J.Shanbehzadeh M.Gholizadeh )103 E,K0,K1,K2 are specified parametersMG is the global meanDG is the global standard deviation

Mapping:

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Using Histogram Statistics ( J.Shanbehzadeh M.Gholizadeh )104

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Using Histogram Statistics ( J.Shanbehzadeh M.Gholizadeh )1073.4 Fundamentals of Spatial Filtering( J.Shanbehzadeh M.Gholizadeh )The Mechanics of Spatial FilteringSpatial Correlation and ConvolutionVector Representation of Linear FilteringGenerating Spatial Filter MasksThe Mechanics of Spatial Filtering3.1- Background








Fundamentals of Spatial Filtering( J.Shanbehzadeh M.Gholizadeh )109The Mechanics of Spatial Filtering:

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Fundamentals of Spatial Filtering( J.Shanbehzadeh M.Gholizadeh )110Image size: MNMask size: mn

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

x= 0,1,2,,M-1 and y= 0,1,2,,N-1

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Fundamentals of Spatial Filtering( J.Shanbehzadeh M.Gholizadeh )111

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Fundamentals of Spatial Filtering( J.Shanbehzadeh M.Gholizadeh )113The Mechanics of Spatial FilteringSpatial Correlation and ConvolutionVector Representation of Linear FilteringGenerating Spatial Filter MasksSpatial Correlation and Convolution3.1- Background









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Spatial Correlation and Convolution( J.Shanbehzadeh M.Gholizadeh )115

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Spatial Correlation and Convolution( J.Shanbehzadeh M.Gholizadeh )116The Mechanics of Spatial FilteringSpatial Correlation and ConvolutionVector Representation of Linear FilteringGenerating Spatial Filter MasksVector Representation of Linear Filtering3.1- Background








Fundamentals of Spatial Filtering( J.Shanbehzadeh M.Gholizadeh )117Vector Representation of Linear Filtering:

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Vector Representation of Linear Filtering( J.Shanbehzadeh M.Gholizadeh )1183.5 Smoothing Spatial Filters( J.Shanbehzadeh M.Gholizadeh )Smoothing Linear FiltersOrder-Static (Nonlinear) FiltersSmoothing Linear Filters3.1- Background








Smoothing Spatial Filters( J.Shanbehzadeh M.Gholizadeh )120Smoothing Linear Filters :Noise reductionSmoothing of false contoursReduction of irrelevant detail

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Smoothing Spatial Filters( J.Shanbehzadeh M.Gholizadeh )121Image size: MNMask size: mn

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

x= 0,1,2,,M-1 and y= 0,1,2,,N-1

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Smoothing Spatial Filters( J.Shanbehzadeh M.Gholizadeh )122

Image smoothing with masks of various sizes.3.1- Background








Smoothing Spatial Filters( J.Shanbehzadeh M.Gholizadeh )123

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Smoothing Spatial Filters( J.Shanbehzadeh M.Gholizadeh )124Smoothing Linear FiltersOrder-Static (Nonlinear) FiltersOrder-Static (Nonlinear) Filters3.1- Background








Smoothing Spatial Filters( J.Shanbehzadeh M.Gholizadeh )125Order-statistic filters:Median filter: Replace the value of a pixel by the median of the gray levels in the neighborhood of that pixel

Noise-reduction3.1- Background








Order-Static (Nonlinear) Filters( J.Shanbehzadeh M.Gholizadeh )126

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Order-Static (Nonlinear) Filters( J.Shanbehzadeh M.Gholizadeh )1273.6 Sharpening Spatial Filters( J.Shanbehzadeh M.Gholizadeh )FoundationUsing the Second Derivative for Image Sharpening - The LaplacianFoundation3.1- Background








Sharpening Spatial FiltersUnsharp Masking and Highboost FilteringUsing First-Order Derivative for (Nonlinear) Image Sharpening - The Gradient( J.Shanbehzadeh M.Gholizadeh )129The first-order derivative:

The second-order derivative

Foundation3.1- Background


























( J.Shanbehzadeh M.Gholizadeh )132FoundationUsing the Second Derivative for Image Sharpening - The LaplacianUsing the Second Derivative for Image Sharpening - The LaplacianSharpening Spatial FiltersUnsharp Masking and Highboost FilteringUsing First-Order Derivative for (Nonlinear) Image Sharpening - The Gradient3.1- Background








( J.Shanbehzadeh M.Gholizadeh )133Use of second derivatives for enhancement-The Laplacian:Development of the method

Laplacian3.1- Background



































( J.Shanbehzadeh M.Gholizadeh )137Simplifications:`


















( J.Shanbehzadeh M.Gholizadeh )139FoundationUsing the Second Derivative for Image Sharpening - The LaplacianSharpening Spatial FiltersUnsharp Masking and Highboost FilteringUsing First-Order Derivative for (Nonlinear) Image Sharpening - The GradientUnsharp Masking and Highboost Filtering3.1- Background








( J.Shanbehzadeh M.Gholizadeh )140Unsharp maskingSubtract a blurred version of an image from the image itself

f(x,y) : The image, f(x,y): The blurred image

High boost Filtering:

Unsharp Masking and Highboost Filtering3.1- Background


























( J.Shanbehzadeh M.Gholizadeh )143FoundationUsing the Second Derivative for Image Sharpening - The LaplacianUnsharp Masking and Highboost FilteringUsing First-Order Derivative for (Nonlinear) Image Sharpening - The GradientUsing First-Order Derivative for (Nonlinear) Image Sharpening - The GradientUsing First-Order Derivative for (Nonlinear) Image Sharpening - The Gradient3.1- Background








( J.Shanbehzadeh M.Gholizadeh )144Using first-order derivatives for (nonlinear) image sharpening, The gradient:The gradient:

The magnitude is rotation invariant (isotropic)

Using First-Order Derivative for (Nonlinear) Image Sharpening - The Gradient3.1- Background








( J.Shanbehzadeh M.Gholizadeh )145Computing using cross differences, Roberts cross-gradient operators

and

Using First-Order Derivative for (Nonlinear) Image Sharpening - The Gradient3.1- Background








( J.Shanbehzadeh M.Gholizadeh )146Sobel operators:A weight value of 2 is to achieve some smoothing by giving more importance to the center point

Using the Gradient for Image Sharpening3.1- Background


























( J.Shanbehzadeh M.Gholizadeh )1493.7 Combining Spatial Enhancement Tools( J.Shanbehzadeh M.Gholizadeh )3.1- Background








Combining Spatial Enhancement ToolsAn example:

Laplacian to highlight fine detailGradient to enhance prominent edgesSmoothed version of the gradient image used to mask the Laplacian imageIncrease the dynamic range of the gray levels by using a gray-level transformation


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Combining Spatial Enhancement Tools( J.Shanbehzadeh M.Gholizadeh )152

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Combining Spatial Enhancement Tools( J.Shanbehzadeh M.Gholizadeh )1533.8 Using Fuzzy Techniques for Intensity Transformations and Spatial filtering( J.Shanbehzadeh M.Gholizadeh )3.1- Background








Using Fuzzy Techniques for Intensity Transformations and Spatial filteringIntroductionPrinciples of Fuzzy Set TheoryIntroductionUsing Fuzzy SetsUsing Fuzzy Sets for Intensity TransformationsUsing Fuzzy Sets for spatial Filtering( J.Shanbehzadeh M.Gholizadeh )155IntroductionFuzzy set theory is the extension of conventional (crisp) set theory It handles the concept of partial truth (truth values between 1 (completely true) and 0 (completely false)) It was introduced by Prof. Lotfi A. Zadeh of UC/Berkeley in 1965 as a mean to model the vagueness and ambiguity in complex systems The idea of fuzzy sets is simple and natural

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Using Fuzzy Techniques for Intensity Transformations and Spatial filtering( J.Shanbehzadeh M.Gholizadeh )1573.1- Background








Using Fuzzy Techniques for Intensity Transformations and Spatial filteringIntroductionPrinciples of Fuzzy Set TheoryPrinciples of Fuzzy Set TheoryUsing Fuzzy SetsUsing Fuzzy Sets for Intensity TransformationsUsing Fuzzy Sets for spatial Filtering( J.Shanbehzadeh M.Gholizadeh )158A fuzzy set A in Z (the universe of discourse) is characterized by a membership function for all :Empty setA(z) = 0EqualityA = B, if and only if A(z) = B(z) Complement(z) = 1 - A(z) Subset , if and only if A(z) B(z)UnionU(z) = max[A(z) , B(z)] IntersectionI(z) = min[A(z) , B(z)]

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Principles of Fuzzy Set Theory( J.Shanbehzadeh M.Gholizadeh )159

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Principles of Fuzzy Set Theory( J.Shanbehzadeh M.Gholizadeh )160Some Common Membership FunctionsTriangular:

Trapezoidal:

Sigma

S-shape

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( J.Shanbehzadeh M.Gholizadeh )161Bell-shape

Truncated Gaussian

Some Common Membership Functions3.1- Background









Some Common Membership Functions3.1- Background








( J.Shanbehzadeh M.Gholizadeh )1633.1- Background








Using Fuzzy Techniques for Intensity Transformations and Spatial filteringIntroductionPrinciples of Fuzzy Set TheoryUsing Fuzzy SetsUsing Fuzzy Sets for Intensity TransformationsUsing Fuzzy Sets for spatial FilteringUsing Fuzzy Sets( J.Shanbehzadeh M.Gholizadeh )164R1: IF the color is green, THEN the fruit is verdant

R2: IF the color is yellow, THEN the fruit is half-mature

R3: IF the color is red, THEN the fruit is mature

Using Fuzzy Sets3.1- Background



























3(z,v) = min{red(z), mat(v)} Using Fuzzy Sets3.1- Background









Q3(v) = min{red(z0), 3(z0,v)} Q2(v) = min{yellow(z0), 2(z0,v)}Q1(v) = min{green(z0), 1(z0,v)}Using Fuzzy Sets3.1- Background








( J.Shanbehzadeh M.Gholizadeh )169Q = Q1 OR Q2 OR Q3

Q(v) = maxr{mins{s(z0), r(z0,v)}}


















( J.Shanbehzadeh M.Gholizadeh )171A Fuzzy Rule-Based SystemFuzzify the inputs

Perform any required fuzzy logical operation

Apply an implication method

Apply an aggregation method

Defuzzify the final output

3.1- Background









A Fuzzy Rule-Based System3.1- Background








( J.Shanbehzadeh M.Gholizadeh )173Dealing withM IF-THEN rulesN input variables : z1,,zNOne output variable : vAij : the fuzzy set associated with the ith rule and jth input variableBi : the fuzzy set associated with the ith rule

IF (z1,A11) AND (z2,A12) AND AND (zN,A1N) THEN (v,B1)IF (z1,A21) AND (z2,A22) AND AND (zN,A2N) THEN (v,B2)..IF (z1,AM1) AND (z2,AM2) AND AND (zN,AMN) THEN (v,BM) ELSE (v,BE)i = min{Aij(zj); j = 1,2,,M}E = min{1- i ; i = 1,,M}

A Fuzzy Rule-Based System3.1- Background








( J.Shanbehzadeh M.Gholizadeh )174What does Fuzzy Image Processing Mean? For instance, we want to define a set of gray levels that share the property darkIn classical set theory, we have to determine a threshold, say the gray level 100All gray levels between 0 and 100 are element of this set, the others do not belong to the setBut the darkness is a matter of degreeSo, a fuzzy set can model this property much betterTo define this set, we also need two thresholds, say gray levels 50 and 150

Fuzzy image processing is not a unique theory. It is a collection of different fuzzy approaches to image processing, Nevertheless, the following definition can be regarded as an attempt to determine

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DIGITAL IMAGE PROCESSING