1051-718 Digital Image Mathematics Image Processing Notes Roger L. Easton, Jr. 25 January 2012 Contents Preface ix 1 Basic Principles of Digital Image Processing 1 1.1 DigitalProcessing............................. 4 1.2 Digitization ................................ 4 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 ..................... 31 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 . 36 3.3.3 Halftoning............................. 38 3.4 Collecting and Quantizing Color Images . 39 4 Image Processing Operations 41 4.1 Geometrical Operations . 42 4.1.1 Common Geometrical Operations . 43 4.1.2 Power Series for Coordinates . 44 4.1.3 A¢ ne Transformation . 45 4.1.4 Bilinear Transformation –Pseudoinverse Calculation . 46 4.2 Least-Squares Solution for A¢ ne Transformation . 47 4.3 PixelTransfers .............................. 53 v vi CONTENTS 4.4 Pixel Interpolation . 54 5 Point Operations 59 5.1 Image Histograms . 59 5.1.1 Histograms of Typical Images . 62 5.1.2 Other Examples of Histograms . 67 5.1.3 Histogram Modi…cation for Image Enhancement . 68 5.1.4 JonesPlots ............................ 70 5.2 Cumulative Histogram . 75 5.2.1 Nonlinear Nature of Histogram Equalization . 83 5.2.2 Histogram Speci…cation or “Matching” . 86 5.3 Examples of Point Operations . 87 5.4 Application of Histograms to Tone-Transfer Correction . 88 5.5 Application of Histograms to Image Segmentation . 88 5.5.1 Thresholding Based on a priori Information . 91 6 Point Operations on Multiple Images 95 6.1 Color and “Spectral”Images . 96 6.2 Multidimensional Histograms for Segmentation . 98 6.2.1 Spectral Images . 100 6.2.2 Principal Component Analysis –PCA . 104 6.3 ColorSpaces................................ 114 6.3.1 Red, Green, Blue . 114 6.3.2 Hue, Saturation, Lightness (or “Brightness,”or “Value”): . 115 6.3.3 Conversion from RGB to HSL . 120 6.3.4 Example: Rotation of Hue Angle . 123 6.3.5 Hue Angle Rotation of PC Bands of Livingstone Diary Page . 125 6.4 Time-Sequence Images: Video . 127 6.5 Color-Space Transformations for Video Compression . 128 6.6 Segmentation by Logical Operations on Multiple Images . 131 6.7 Arithmetic Operations on Multiple Images . 132 6.7.1 Multiple-Frame Averaging . 132 6.7.2 Required Number of Bits for image Sums, Averages, and Dif- ferences .............................. 136 6.7.3 Image Subtraction . 137 6.7.4 Di¤erence Images as Features . 137 6.7.5 “Mask”or “Template”Multiplication: Image Mattes . 140 6.7.6 Image Division . 140 7 Local Operations 145 7.1 Window Operators –Correlation . 147 7.2 Convolution ................................ 149 7.2.1 Convolutions –Edges of the Image . 153 7.2.2 Convolutions –Computational Intensity . 154 7.2.3 Smoothing Kernels –Lowpass Filtering . 154 CONTENTS vii 7.2.4 Di¤erencing Kernels –Highpass Filters . 157 7.3 E¢ ects of Averaging and Di¤erencing on Noisy Images . 163 7.3.1 Application of the Laplacian to Texture Segmentation . 164 7.4 Application to Image Sharpening . 165 7.4.1 Unsharp Masking . 165 7.4.2 Other Image Sharpeners . 168 7.4.3 Generalized Laplacian . 171 7.5 Directional Derivatives: Gradient . 173 7.5.1 Roberts’Gradient . 175 7.5.2 “Laplacian of Gaussian” . 176 7.6 Nonlinear Filters . 180 7.6.1 Median Filter . 180 7.6.2 Example of Median Filter of Uniform Distribution . 182 7.6.3 Median Filter and Gaussian Noise . 185 7.6.4 Comparison of Histograms after Mean and Median Filter . 187 7.6.5 Bilateral Filters . 193 7.7 Adaptive Operators . 199 7.8 Convolution Revisited –Bandpass Filters . 199 7.8.1 Bandpass Filters for Images . 203 7.9 Pattern Matching . 203 7.9.1 Other Matching Kernels . 208 7.9.2 Normalization of Contrast of Detected Features . 208 7.10 Implementation of Filtering . 209 7.10.1 Nonlinear and Shift-Variant Filtering . 209 7.11 Neighborhood Operations on Multiple Images . 211 7.11.1 Image Sequence Processing . 211 7.11.2 Spectral + Spatial Neighborhood Operations . 211 7.11.3 “Pseudounsharp Masking” . 212 8 Global Operations 215 8.1 Relationship to Neighborhood Operations . 215 8.2 Discrete Fourier Transform (DFT) . 216 8.3 Fast Fourier Transform (FFT) . 217 8.4 Fourier Transforms of Images . 220 8.5 Image Restoration via Fourier Transforms . 230 8.5.1 Examples of Inverse Filters in 1-D . 231 8.5.2 Spectrum and Impulse Response of Inverse Filter . 232 8.5.3 Inverse Filter for SINC-Function Blur . 233 8.6 Other Global Operations . 233 8.7 Discrete Cosine Transform (DCT) . 234 8.7.1 Steps in Forward DCT . 237 8.7.2 Steps in Inverse DCT . 237 8.8 Walsh-Hadamard Transform . 238 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 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 York, 1986. R.N. Bracewell, “The Fourier Transform”, Scienti…c American, June 1989, pp.86-95. 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, 1998. ix x Preface 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 York, 1998. J.S.Walker, Fast Fourier Transforms 2nd Edition, CRC Press, New York, 1996. 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 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’msure). 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: 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: 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.