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Digital Image Processing I
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Preface ix,1 Basic Principles of Digital Image Processing 1. 1 1 Digital Processing 5,1 2 Digitization 5,2 Review of Sampling 7. 2 0 1 Ideal Sampling of 1 D function 8, 2 1 Aliasing Whittaker Shannon Sampling Theorem 11. 2 2 Realistic Sampling Averaging by the Detector 12. 3 Review of Quantization 21,3 1 Quantization A D Conversion 21. 3 1 1 Tone Transfer Function 21,3 2 Quantization Error Noise 25.
3 2 1 Signal to Noise Ratio 28,3 2 2 Example Variance of a Sinusoid 29. 3 2 3 Example Variance of a Square Wave 30, 3 2 4 Variance of Noise from a Gaussian Distribution 30. 3 2 5 Approximations to SNR 30,3 2 6 SNR of Quantization 32. 3 3 Quantization to Few Bits 35,3 3 1 Improved Gray Scale IGS Quantization 35. 3 3 2 Quantizers with Memory Error Di usion 35,4 Point Operations 39.
4 1 Geometrical Operations 40,4 2 Least Squares Solution for Warping 42. 4 3 Common Geometrical Operations 45,4 4 Pixel Transfers 45. 4 5 Pixel Interpolation 46,4 6 Point Operations on Single Images 46. 4 6 1 Image Histograms 47,4 6 2 Histograms of Typical Images 48. 4 6 3 Cumalative Histogram 48,vi CONTENTS, 4 6 4 Histogram Modi cation for Image Enhancement 49.
4 6 5 Jones Plots 50,4 7 Histogram Equalization Flattening 51. 4 7 1 Example of Histogram Equalization 1 D Image 53. 4 7 2 Nonlinear Nature of Histogram Equalization 53. 4 8 Histogram Speci cation 54, 4 9 Application of Histograms to Tone Transfer Correction 55. 4 10 Application of Histograms to Image Segmentation 56. 5 Local Operations 57,5 1 Window Operators Correlation 57. 5 2 Convolution 59,5 2 1 Convolutions Edges of the Image 63. 5 2 2 Convolutions Computational Intensity 63,5 2 3 Smoothing Kernels Lowpass Filtering 64.
5 2 4 Di erencing Kernels Highpass Filters 66,5 3 Nonlinear Filters 72. 5 3 1 Median Filter 72, 5 3 2 Example of Median Filter of Uniform Distribution 75. 5 4 Median Filter and Gaussian Noise 77, 5 5 Comparison of Histograms after Mean and Median Filter 79. 5 6 E ect of Window Shape 79, 5 6 1 Other Statistical Filters Mode Variance Maximum Minimum 81. 5 6 2 Examples of Nonlinear Filters 81, 5 6 3 Nonlinear Filters on Images with Additive Gaussian Noise 82.
5 6 4 Nonlinear Filters on Noise Free Gray Level Image 82. 5 7 Adaptive Operators 82,5 8 Convolution Revisited Bandpass Filters 82. 5 9 Bandpass Filters for Images 87,5 10 Implementation of Filtering 87. 5 11 Neighborhood Operations on Multiple Images 88. 5 11 1 Image Sequence Processing 88, 5 11 2 Spectral Spatial Neighborhood Operations 88. 5 11 3 Image Division to Enhance Low Contrast Image Structure 89. 6 Shape Based Operations 91,6 1 Pixel Connectivity 92. 6 2 Image Labeling 93,6 2 1 Example 93,6 3 Border Operations 94.
6 4 Cascaded Morphological Operations Opening and Closing 96. 6 5 Applications of Morphological Operations 97,6 5 1 Noise Removal 97. CONTENTS vii,6 5 2 Medial Axis Transform 98,6 6 Binary Morphological Operations 98. 6 6 1 Horizontal Structuring Element 98,6 6 2 Vertical Structuring Element 99. 6 6 3 Complex Structuring Element 99,6 6 4 Binary Morphological Operations 99. 7 Geometric Operations 101,7 1 Least Squares Solution for Warping 103.
7 2 Common Geometrical Operations 106,7 3 Pixel Transfers 106. 7 4 Pixel Interpolation 106,8 Global Operations 109. 8 1 Relationship to Neighborhood Operations 109,8 2 Discrete Fourier Transform DFT 110. 8 3 Fast Fourier Transform FFT 111,8 4 Fourier Transforms of Images 114. 8 5 Image Restoration via Fourier Transforms 115,8 5 1 Examples of Inverse Filters in 1 D 117.
8 5 2 Spectrum and Impulse Response of Inverse Filter 118. 8 5 3 Inverse Filter for SIN C Function Blur 118,8 6 Other Global Operations 119. 8 7 Discrete Cosine Transform DCT 119,8 7 1 Steps in Forward DCT 122. 8 7 2 Steps in Inverse DCT 123,8 8 Walsh Hadamard Transform 123. References, Center for Image Processing in Education lots of links to software and images. http www evisual org homepage html, ImageJ software for image processing and analysis in Java evolution of NIHImage.
http rsb info nih gov ij,Image 2000 from NASA,http www ccpo odu edu SEES ozone oz i2k soft htm. Scion Image Processing Software for PC and MAC OS, http www scioncorp com frames fr scion products htm. Hypercube Image Analysis Software for PC and MAC OS. http www tec army mil Hypercube, GIMP Image Processing Software Gnu IMP for PC MacOS Linux. http www gimp org, Irfanview free image processing viewer with some processing capability. http www irfanview com, Gregory A Baxes Digital Image Processing Principles and Applications.
John Wiley Sons New York 1994, Ronald N Bracewell Two Dimensional Imaging Prentice Hall Englewood. Cli s 1995, Ronald N Bracewell The Fourier Transform and Its Applications Second. Edition Revised McGraw Hill 1986, Ronald N Bracewell The Hartley Transform Oxford University Press New. R N Bracewell The Fourier Transform Scienti c American June 1989. Kenneth R Castleman Digital Image Processing Prentice Hall Englewood. Cli s 1996, E O Brigham The Fast Fourier Transform and its Applications Prentice. Hall Englewood Cli s 1988, Michael P Ekstrom Ed Digital Image Processing Techniques Academic.
Press New York 1984, B R Frieden Probability Statistical Optics and Data Testing Third Edi. tion Springer Verlag Berlin 2002, Jack D Gaskill Linear Systems Fourier Transforms and Optics John. Wiley Sons New York 1978, Rafael C Gonzalez and Richard E Woods Digital Image Processing Second. Edition Prentice Hall Upper Saddle River 2002, Jae S Lim Two Dimensional Signal and Image Processing Prentice Hall. Englewood Cli s 1990, Paul J Nahin An Imaginary Tale Princeton University Press Princeton NJ.
A Nussbaum and R Phillips Contemporary Optics for Scientists and En. gineers Prentice Hall 1976, Wayne Niblack An Introduction to Digital Image Processing Prentice. Hall Englewood Cli s 1986, J Anthony Parker Image Reconstruction in Radiology CRC Press Boca. Raton FL 1990, William K Pratt Digital Image Processing Second Edition John Wiley. Sons New York 1991, Azriel Rosenfeld and Avinash C Kak Digital Picture Processing Second. Edition Academic Press San Diego 1982, Craig Scott Introduction to Optics and Optical Imaging IEEE Press New.
J S Walker Fast Fourier Transforms 2nd Edition CRC Press New York 1996. Basic Principles of Digital Image,Processing, During the last two decades or so inexpensive and powerful digital computers have. become widely available and have been applied to a multitude of tasks By hitching. computers to new imaging detectors and displays very capable systems for creating. analyzing and manipulating imagery have been constructed and are being applied. in many arenas For example they now are used to reconstruct X ray and magnetic. resonance images MRI in medicine to analyze multispectral aerial and satellite. images for environmental and military uses to read Universal Product Codes that. specify products and prices in retail stores to name just a few. Since I rst taught a predecessor of this course in 1987 the capabilities of inex. pensive imaging tools cameras printers computers have exploded no surprise to. you I m sure This has produced a wide and ever expanding range of applications. that we could not even envision in those days To give an idea of the change over the. last two decades consider the set of data shown below which is copied directly from. the rst edition of Digital Image Processing by Gonzalez and Woods. 2CHAPTER 1 BASIC PRINCIPLES OF DIGITAL IMAGE PROCESSING. Coded 64 64 5 bit image 32 gray values, These data represent a 64 64 5 bit image 25 32 gray values This data set was. entered by hand with only 4 mistakes in 1988 by Ranjit Bhaskar an imaging science. graduate student for use by students The image was rendered using the so called. overstrike routine on a line printer where dark pixels were printed using several. overprinted characters and lighter pixels by sparse characters e g and The. subject of the image is shown on the next page, Data in previous photo rendered using overstrike printout with line printer this. is how we used to do it folks, This course will investigate the basic principles of digital imaging systems and. introduce useful applications many simple examples will be provided to illustrate the. concepts First a de nition, IMAGE A reproduction or imitation of form of a person or thing.
The optical counterpart of an object produced by a lens mirror etc. Noah Webster, We normally think of an image in the sense of a picture i e a planar represen. tation of the brightness i e the amount of light re ected or transmitted by an. An image is usually a function of two spatial variables e g f x y which rep. resents the brightness f at the Cartesian location x y Obviously it also may be. graphed in three dimensions with brightness displayed on the z axis. 4CHAPTER 1 BASIC PRINCIPLES OF DIGITAL IMAGE PROCESSING. Image Representation of,Function of Two Spatial Coordinates f x y. It is more and more common to deal with images that have more than two coor. dinate dimensions e g, f x y tn monochrome movie discrete set of images over time. f x y spectral image with continuous domain of wavelengths. f x y n multispectral image discrete set of wavelengths. f x y t time varying monochrome image over continuous time domain. f x y tn time varying monochrome image with discrete time samples cinema. f x y z 3 D monochrome image e g optical hologram, f x y tn m discrete samples in time and wavelength e g color movie. f x y z t reality, It is generally fesible to cut 2 D slices from these multidimensional functions to cre.
ate images but the images need not be pictorial For example consider the 2 D. slices cut from the 3 D function spatial temporal function f x y t the 2 D slice. f x y t t0 is pictorial but f x y y0 t is not That said the units of the axes. have no e ect on the computations it is perfectly feasible for computers to process. and display f x y y0 t as to do the same for f x y t0. After converting image information into an array of integers the image can be. manipulated processed and displayed by computer Computer processing is used. for image enhancement restoration segmentation description recognition coding. reconstruction transformation,1 1 DIGITAL PROCESSING 5. 1 1 Digital Processing, The general digital image processing system may be divided into three components. the input device or digitizer the digital processor and the output device image. 1 The digitizer converts a continuous tone and spatially continuous brightness. distribution f x y to an discrete array the digital image fq n m where n m. and fq are integers, 2 The digital processor operates on the digital image fq n m to generate a new. digital image gq k where k and gq are integers The output image may be. represented in a di erent coordinate system hence the use of di erent indices. 3 The image display converts the digital output image gq k back into a continuous. tone and spatially continuous image g x y for viewing It should be noted that. some systems may not require a display e g in machine vision and arti cial. intelligence applications the output may be a piece of information For ex. ample a digital imaging system that was designed to answer the question Is. there evidence of a cancerous tumor in this x ray image ideally would have. two possible outputs YES or NO i e a single bit of information. Note that the system includes most of the links in what we call the imaging chain. We shall rst consider the mathematical description of image digitizing and display. devices and follow that by a long discussion of useful processing operations. 1 2 Digitization, Digitization is the conversion of a continuous tone and spatially continuous brightness. distribution f x y to an discrete array of integers fq n m by two operations which. will be discussed in turn, 6CHAPTER 1 BASIC PRINCIPLES OF DIGITAL IMAGE PROCESSING.
A SAMPLING a function of continuous coordinates f x y is evaluated on. a discrete matrix of samples indexed by n m You probably saw some discussion of. sampling in the course on linear and Fourier mathematics. B QUANTIZATION the continuously varying brightness f at each sample. is converted to a one of set of integers fq by some nonlinear thresholding process. The digital image is a matrix of picture elements or pixels if your ancestors are. computers Video descendents and imaging science undergraduates often speak of. pels often misspelled pelz Each matrix element is an integer which encodes the. brightness at that pixel The integer value is called the gray value or digital count of. Computers store integers as BInary digiTS or bits 0 1. 2 bits can represent 004 0 014 1 104 2 114 3 a total of 22 4. where the symbol 4 denotes the binary analogue to the decimal point and. thus may be called the binary point which separates the ordered bits with positive. and negative powers of 2,m BITS can represent 2m numbers. 8 BITS 1 BYTE 256 decimal numbers 0 255,12 BITS 4096 decimal numbers 0 4095. 16 BITS 216 65536 decimal numbers 0 65535, Digitized images contain nite numbers of data bits and it probably is apparent. that the process of quantization discards some of the content of the image i e the. quantized image di ers from the unquantized image so errors have been created. Beyond that we can consider the amount of information in the quantized image. which is de ned as the number of bits required to store the image The number. of bits of information usually is smaller than the the number of bits of data. which is merely the product of the number of image pixels and the number of bits. per pixel The subject of information content is very important in imaging and will. be considered in the section on image compression We will discuss digitizing and. reconstruction errors after describing the image display process. Review of Sampling, The process of sampling derives a discrete set of data points at usually uniform. spacing In its simplest form sampling is expressed mathematically as multiplication. of the original image by a function that measures the image brightness at discrete. locations of in nitesimal width area volume in the 1 D 2 D 3 D cases. fs n x f x s x n x,f x brightness distribution of input image.
s x n x sampling function, fs n x sampled input image de ned at coordinates n x. The ideal sampling function for functions of continuous variables is generated from. the so called Dirac delta function x which is de ned by many authors including. Gaskill The ideal sampling function is the sum of uniformly spaced discrete Dirac. delta functions which Gaskill calls the COMB while Bracewell calls it the SHAH. COM B x x n,s x n x x n x COM B,8 CHAPTER 2 REVIEW OF SAMPLING. The COMB function de ned by Gaskill called the SHAH function by Bracewell. For the somewhat less rigorous purpose we have here we may consider the. sampling function to just grab the value of the continuous input function f x y at. the speci c locations separated by the uniform intervals x and y where x y. fs n m f n x m y, In other words we are sweeping some unimportant and possibly confusing mathe. matical details under the rug,2 0 1 Ideal Sampling of 1 D function. Multiplication of the input f x by a COMB function merely evaluates f x on the. uniform grid of points located at n x where n is an integer Because it measures. the value of the input at an in nitesmal point this is a mathematical idealization. that cannot be implemented in practice Even so the discussion of ideal sampling. usefully introduces some essential concepts, Consider ideal sampling of a sinusoidal input function with spatial period X0 that.
is ideally sampled at intervals separated by x,f x 1 cos 0. The amplitude of the function at the sample indexed by n is. fs n x 1 cos 2 0,1 cos 2 n 0, The dimensionless parameter Xx0 in the second expression is the ratio of the sampling. interval to the spatial period wavelength of the sinusoid and is a measurement of. the delity of the sampled image For illustration examples of sampled functions. obtained for several values of Xx0 are,x 1 1 h ni,Case I X0 12 x 0 0 fs n 1 cos. Case II X0 2 x 0 0 fs n 1 cos n 1 1 n,Case III X0 2 x 0 fs n 1 sin n. X0 2 2 2 2,Case IV X0 x 0 0,1 2 n 1 h ni,fs n 1 cos 1 cos 3.

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