Fundamentals of Digital Image Processing Roger L. Easton, Jr. 22 November 2010 Contents Preface ix 1 Basic Principles of Digital Image Processing 1 1.1 Digital Processing............................. 4 1.2 Digitization ................................ 4 2ReviewofSampling 7 2.0.1 IdealSamplingof1-Dfunction................. 8 2.1Aliasing—Whittaker-ShannonSamplingTheorem........... 11 2.2RealisticSampling—AveragingbytheDetector............ 12 3 Review of Quantization 21 3.1Quantization:A/DConversion..................... 21 3.1.1 ToneTransferFunction...................... 21 3.2QuantizationError(“Noise”)...................... 25 3.2.1 Signal-to-NoiseRatio....................... 28 3.2.2 Example:VarianceofaSinusoid................. 29 3.2.3 Example:VarianceofaSquareWave:.............. 30 3.2.4 Varianceof“Noise”fromaGaussianDistribution....... 30 3.2.5 Approximations to SNR ..................... 31 3.2.6 SNR ofQuantization....................... 32 3.3QuantizationtoFewBits......................... 35 3.3.1 ImprovedGrayScale(IGS)Quantization............ 35 3.3.2 Quantizers with Memory — Error Diffusion........... 36 3.3.3 Halftoning ............................. 38 4 Image Processing Operations 41 4.1GeometricalOperations......................... 42 4.1.1 CommonGeometricalOperations................ 43 4.1.2 PowerSeriesforCoordinates................... 44 4.1.3 AffineTransformation...................... 45 4.1.4 Bilinear Transformation — Pseudoinverse Calculation ..... 46 4.2 Least-Squares Solution for AffineTransformation........... 47 4.3 Pixel Transfers .............................. 53 4.4 Pixel Interpolation ............................ 54 v vi CONTENTS 5 Point Operations 59 5.1 Image Histograms ............................. 59 5.1.1 HistogramsofTypicalImages.................. 61 5.1.2 OtherExamplesofHistograms................. 65 5.1.3 Histogram ModificationforImageEnhancement........ 66 5.1.4 Jones Plots . ........................... 67 5.2CumulativeHistogram.......................... 73 5.2.1 NonlinearNatureofHistogramEqualization.......... 82 5.2.2 Histogram Specificationor“Matching”............. 85 5.3ExamplesofPointOperations...................... 86 5.4ApplicationofHistogramstoTone-TransferCorrection........ 87 5.5ApplicationofHistogramstoImageSegmentation........... 87 6 Point Operations on Multiple Images 91 6.1Colorand“Spectral”Images....................... 92 6.2MultispectralHistogramsforSegmentation............... 94 6.2.1 MultidimensionalHistogramsofColorImages......... 95 6.2.2 SpectralImages.......................... 96 6.2.3 PrincipalComponentAnalysis—PCA............. 99 6.3ColorSpaces................................ 104 6.3.1 Red,Green,Blue......................... 104 6.3.2 Hue,Saturation,Lightness(orBrightness,orValue):..... 105 6.3.3 ConversionfromRGBtoHSL.................. 110 6.3.4 Example: Wax “Coat of Arms” in a French Epic Poem . 113 6.4Time-SequenceImages:Video...................... 115 6.5 Color-Space Transformations for Video Compression .......... 116 6.6SegmentationbyLogicalOperationsonMultipleImages....... 120 6.7ArithmeticOperationsonMultipleImages............... 121 6.7.1 Multiple-FrameAveraging.................... 121 6.7.2 Required Number of Bits for image Sums, Averages, and Dif- ferences.............................. 125 6.7.3 ImageSubtraction........................ 126 6.7.4 DifferenceImagesasFeatures.................. 126 6.7.5 “Mask”or“Template”Multiplication:ImageMattes..... 129 6.7.6 ImageDivision.......................... 129 7 Local Operations 133 7.1WindowOperators—Correlation.................... 133 7.2Convolution................................ 135 7.2.1 Convolutions—EdgesoftheImage............... 139 7.2.2 Convolutions—ComputationalIntensity............ 140 7.2.3 SmoothingKernels—LowpassFiltering............. 141 7.2.4 DifferencingKernels—HighpassFilters............. 143 7.3 Effiects of Averaging and DifferencingonNoisyImages........ 150 7.3.1 ApplicationoftheLaplaciantoTextureSegmentation..... 151 CONTENTS vii 7.4 ....................................... 152 7.5 Applications of Differencing—ImageSharpening............ 152 7.5.1 UnsharpMasking......................... 152 7.5.2 OtherImageSharpeners..................... 153 7.5.3 GeneralizedLaplacian...................... 156 7.6DirectionalDerivatives:Gradient.................... 158 7.6.1 Roberts’Gradient......................... 160 7.6.2 “LaplacianofGaussian”..................... 161 7.7 Nonlinear Filters ............................. 165 7.7.1 Median Filter ........................... 165 7.7.2 ExampleofMedianFilterofUniformDistribution....... 167 7.8MedianFilterandGaussianNoise.................... 170 7.9ComparisonofHistogramsafterMeanandMedianFilter....... 172 7.9.1 EffectofWindow“Shape”onMedianFilter.......... 172 7.9.2 Other Statistical Filters (Mode, Variance, Maximum, Minimum) 174 7.9.3 ExamplesofNonlinearFilters.................. 175 7.9.4 Nonlinear Filters on Images with Additive Gaussian Noise . 177 7.9.5 NonlinearFiltersonNoise-FreeGray-LevelImage....... 177 7.10 Adaptive Operators............................ 178 7.11ConvolutionRevisited—BandpassFilters................ 179 7.11.1BandpassFiltersforImages................... 183 7.12 Pattern Matching............................. 183 7.12.1OtherMatchingKernels..................... 186 7.12.2 Normalization of Contrast of Detected Features . ...... 188 7.13ImplementationofFiltering....................... 189 7.13.1NonlinearandShift-VariantFiltering.............. 189 7.14NeighborhoodOperationsonMultipleImages............. 190 7.14.1 Image Sequence Processing .................... 190 7.14.2Spectral+SpatialNeighborhoodOperations.......... 191 7.14.3“PseudounsharpMasking”.................... 191 8 Global Operations 195 8.1RelationshiptoNeighborhoodOperations............... 195 8.2DiscreteFourierTransform(DFT).................... 196 8.3FastFourierTransform(FFT)......................197 8.4FourierTransformsofImages...................... 200 8.5 Image Restoration via Fourier Transforms ............... 210 8.5.1 ExamplesofInverseFiltersin1-D................ 211 8.5.2 SpectrumandImpulseResponseofInverseFilter....... 212 8.5.3 Inverse Filter for SINC-FunctionBlur............. 213 8.6OtherGlobalOperations......................... 213 8.7DiscreteCosineTransform(DCT).................... 214 8.7.1 StepsinForwardDCT...................... 217 8.7.2 StepsinInverseDCT....................... 217 viii CONTENTS 8.8Walsh-HadamardTransform....................... 218 Preface 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) (free 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 Cliffs, 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 York, 1986. R.N. Bracewell, “The Fourier Transform”, ScientificAmerican, June 1989, pp.86-95. Kenneth R. Castleman, Digital Image Processing, Prentice Hall, Englewood Cliffs, 1996. E.O.Brigham, The Fast Fourier Transform and its Applications,Prentice Hall, Englewood Cliffs, 1988. MichaelP.Ekstrom,(Ed.),Digital Image Processing Techniques,Academic Press, New York, 1984. B.R. Frieden, Probability, Statistical Optics, and Data Testing,ThirdEdi- 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. 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Chapter 1 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