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King Saud University Department of Computer Engineering First Semester 1432-1433 CEN 545 Digital Image Processing Lecture Notes 1 and 2 Dr. Naif Alajlan Lecture 1 Introduction Images as Signals: CEN 545 Signal: x(t) Image : f ( x, y ) Intensity at x,y (continuous) f ( x, y ) Intensity at pixel x,y (discrete) Video image: function of space and each space point is function of time. h(m, n) Image processor f (m, n) g (m, n) Linear Image Processing: g (m, n) f (m, n) h(m, n) In frequency domain G (u, v) F (u, v) H (u, v) Some Types of Images: Binary, Gray-scale, Color Systems and Signals Image Processing Time-variant Space-variant 1-Dimensional 2-Dimensional f(x) f(x,y) Deterministic Random Causal Non-causal What is Image Processing? Image processing is the manipulation of an image in order to improve its quality, enhances its ability to convey visual information and make it look better. Image processing addresses three types of image problems: 1. Contrast: Refers to the variability of image intensity across the image. Edges are regions of relatively high local contrast. Global contrast intensity variation across the entire image is referred to as the dynamic range of the image. Usually contrast variation is caused by illumination. 2. Blur: caused by either a resolution or focus problems, or relative motion between the camera and the object during the image capturing. Blur edges correspond to gradual change of intensity locally. 3. Noise: any unwanted intensity variation in the image. Usually modeled as a random process, but it can be highly structured and deterministic in some instances. Major Categories of Image Processing Problems While most image problems are relatively simple to describe, solving them effectively can be quite difficult. There is a broad and continuously expanding spectrum of applications to image processing. The two major branches of image processing are enhancement and restoration. o Enhancement: aimed at improving the subjective quality or the objective utility of the image and includes: point operations, local operations and global operations. Any of these can be linear or non-linear. o Restoration: aims at recovering any image after degradation. Classical linear systems theory forms the foundation for the derivation of restoration methods: Random Noise ns (m, n) Deterministic Deterministic f (m, n) Linear shift invariant hd (m, n) s(m, n) LSI degradation h(m, n) fˆ (m, n) minimum MSE with f (m, n) hd (m, n) Model of the image degradation. ns(m, n) random noise. Optimal methods find a filter h(m, n) whose output is a minimum mean square error estimate of f (m, n) . Drawback: Local statistical models of the image vary with spatial location and undermine the methods which are optimal for stationary processes. Image Processing Techniques 1. Point Operations: map each input pixel to an output pixel intensity according to an intensity transformation. A simple linear point operation which maps the input gray level f (m, n) to an output gray level g (m, n) is given by: g (m, n) af (m, n) b Where a and b are chosen to achieve a desired intensity variation in the image. Note that the output g (m, n) here depends only on the input f (m, n) at m,n. 2. Local Operations: determine the output pixel intensity as some function of a relatively small neighborhood of input pixels in the vicinity of the output location. A general linear operator can be expressed as weighted of picture elements within a local neighborhood N. g (m, n) a k , l N k ,l f (m k , n l ) Simple local smoothing (for noise reduction) and sharpening (for deploring or edge enhancement) operators can be both linear and non-linear. 3. Global Operations: the outputs depend on all input pixels values. If linear, global operators can be expressed using two-dimensional convolution. g (m, n) f (m, n) h(m, n) h( k , l ) f ( m k , n l ) k , lN 4. Adaptive Filters: whose coefficients depend on the input image f (m, n) h ( m, n ) g (m, n) Statistical estimate 5. Non-Linear Filters: Median/order statistics Non-linear local operations Homomorphic filters g (m, n) [ f (m k , n l )]h(k ,l ) k ,lN In addition to enhancement and restoration, image processing generally includes issues of representations, spatial sampling and intensity quantization, compression or coding and segmentation. As part of computer vision, image processing leads to feature extraction and pattern recognition or scene analysis. Point Operations for Contrast Enhancement Contrast refers to the distribution of intensities or grey levels in an image and can be local (textures or edges) or global (the dynamic range across the entire image). Histogram: A convenient representation of contrast of an image. Plots the number of pixels at each intensity value. Low-contrast images have narrow distributions. High-contrast images have broad distributions. Maximum contrast Minimum contrast Black and white image 0 255 0 255 gray image … Optimal contrast 0 255 Simple point operations can achieve dramatic contrast effects. Normalizing the Histogram Let H (k ) the number of pixels whose gray level value is k, 0 k L 1 (for 8-bit quantization: L=256). P(k ) L 1 H (k ) , where N H (k ) is the total number pixels in the image. N k 0 By allowing the gray level value to vary continuously, over [0, 1]: p(r), 0 r 1 is called gray level probability density function (glpdf). How do transformations (i.e., point operations) affect contrast (glpdf) of the image? f (m, n) Point operation g (m, n) Suppose g (m, n) af (m, n) b r g l in the input image Let s g l in the output image s ar b T (r ) s s ar b pg(s) T(r) 2s a 22r Proof: pf(r) 2s E[( s s ) 2 ] E[( ar b a r b) 2 ] E[ a 2 (r r ) 2 ] a 2 2r For a > 1 a<1 r # contrast increasing “stretching” contrast decreasing “compressing” Example: Consider s r 2 g l below 12 are compressed and above s 2r 1 2 r are stretched 12 Enhances contrast for bright images. 1 4 1 2 Similarly, s r enhances contrast for dark images 1 2 More generally, given pf(r) and s = T(r), what is pg(s)? Assume T(r) is monotically increasing and singledvalued pg(s) ds = pf(r) dr p g ( s) p f (r ) ds dr r T 1 ( s ) Example: (a) s ar b Find pg(s) in terms of pf(r) if (b) s r 2 (a) s T (r ) ar b r T 1 ( s) s b a Pg ( s ) Pf s a b ds a dr (b) Pg ( s ) Pf a s 2 s Histogram Equalization Find s = T(r) such that Pg ( s ) 1 0 s 1 r ds Pf (r ) s Pf ( x)dx dr 0 (uniform) k or l Pf ( j ) j 0 which is the cumulative distribution function (cdf) of the input (glpdf). In discrete images with quantized gray level quantization artifacts prevent exact equalization of the histogram. Example: Find T(r) that results in (optimal) contrast enhancement: (a) Pf (r ) 2r , 0 r 1 r T (r ) 2 x dx r 2 ? 0 (b) Pf (r ) 1 cos 2r r sin 2x sin 2r T (r ) (1 cos 2x) dx x r 2 0 2 0 r Example: (Artifacts of quantization) practical issues Pf (k ) 2k 56 0k 7 (a) What T(k) that best enhances the contrast of this image? (b) Plot Hg(l). Solution: 2j 2 k 2 k (k 1) k (k 1) j 56 j 0 56 2 56 j 0 56 k (a) T (k ) (b) k 0 1 2 3 4 5 6 7 Hf(k) 0 2 4 6 8 10 12 14 Pf(k) 0/56 2/56 4/56 6/56 8/56 10/56 12/56 14/56 T(k) 0/56 = 0 2/56 = 0.036 6/56 = 0.107 12/56 = 0.214 20/56 = 0.357 30/56 = 0.536 42/56 = 0.75 56/56 = 1 l 0 0 1 2 3 4 5 7 0 – 0.125 0.125 – 0.250 0.250 – 0.375 0.375 – 0.500 0.500 – 0.625 0.625 – 0.750 0.750 – 0.875 0.875 – 1 Pf (k ) H f (k ) Number of pixels