Digital Image Processing 2 nd ed www imageprocessingbook

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Image Enhancement in the Spatial Domain • The spatial domain: – The image plane – For a digital image is a Cartesian coordinate system of discrete rows and columns. At the intersection of each row and column is a pixel. Each pixel has a value, which we will call intensity. • The frequency domain : – A (2 -dimensional) discrete Fourier transform of the spatial domain – We will discuss it in chapter 4. • Enhancement : – To “improve” the usefulness of an image by using some transformation on the image. – Often the improvement is to help make the image “better” looking, such as increasing the intensity or contrast. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Background • A mathematical representation of spatial domain enhancement: where f(x, y): the input image g(x, y): the processed image T: an operator on f, defined over some neighborhood of (x, y) © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. Gray-level Transformation © 2002 R. C. Gonzalez &

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Some Basic Gray Level Transformations

Digital Image Processing, 2 nd ed. Image Negatives • Let the range of gray

Digital Image Processing, 2 nd ed. Log Transformations where c : constant © 2002

Digital Image Processing, 2 nd ed. Power-Law Transformation where c, © 2002 R. C.

Digital Image Processing, 2 nd ed. Power-Law Transformation Example 1: Gamma Correction © 2002

Digital Image Processing, 2 nd ed. Power-Law Transformation Example 2: Gamma Correction © 2002

Digital Image Processing, 2 nd ed. Power-Law Transformation Example 3: Gamma Correction © 2002

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Piecewise-Linear Transformation Functions Case 1:

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Piecewise-Linear Transformation Functions Case 2: Gray-level Slicing An image © 2002 R. C. Gonzalez & R. E. Woods Result of using the transformation in (a)

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Piecewise-Linear Transformation Functions Case 3: Bit-plane Slicing • Bit-plane slicing: – It can highlight the contribution made to total image appearance by specific bits. – Each pixel in an image represented by 8 bits. – Image is composed of eight 1 -bit planes, ranging from bit-plane 0 for the least significant bit to bit plane 7 for the most significant bit. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Piecewise-Linear Transformation Functions Bit-plane Slicing:

Digital Image Processing, 2 nd ed. Histogram Processing © 2002 R. C. Gonzalez &

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Histogram Equalization • Histogram equalization: – To improve the contrast of an image – To transform an image in such a way that the transformed image has a nearly uniform distribution of pixel values • Transformation: – Assume r has been normalized to the interval [0, 1], with r = 0 representing black and r = 1 representing white – The transformation function satisfies the following conditions: • T(r) is single-valued and monotonically increasing in the interval • © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. Histogram Equalization • For example: © 2002 R.

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Histogram Equalization • Histogram equalization is based on a transformation of the probability density function of a random variable. • Let pr(r) and ps(s) denote the probability density function of random variable r and s, respectively. • If pr(r) and T(r) are known, then the probability density function ps(s) of the transformed variable s can be obtained • Define a transformation function where w is a dummy variable of integration and the right side of this equation is the cumulative distribution function of random variable r. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Histogram Equalization • Given transformation function T(r), ps(s) now is a uniform probability density function. • T(r) depends on pr(r), but the resulting ps(s) always is uniform. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Histogram Equalization • In discrete version: – The probability of occurrence of gray level rk in an image is n : the total number of pixels in the image nk : the number of pixels that have gray level rk L : the total number of possible gray levels in the image – The transformation function is – Thus, an output image is obtained by mapping each pixel with level rk in the input image into a corresponding pixel with level sk. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. Histogram Equalization © 2002 R. C. Gonzalez &

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Histogram Equalization • Transformation functions (1) through (4) were obtained form the histograms of the images in Fig 3. 17(1), using Eq. (3. 3 -8). © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Histogram Matching • Histogram matching is similar to histogram equalization, except that instead of trying to make the output image have a flat histogram, we would like it to have a histogram of a specified shape, say pz(z). • We skip the details of implementation. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Local Enhancement • The histogram processing methods discussed above are global, in the sense that pixels are modified by a transformation function based on the gray-level content of an entire image. • However, there are cases in which it is necessary to enhance details over small areas in an image. original © 2002 R. C. Gonzalez & R. E. Woods global local

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Use of Histogram Statistics for Image Enhancement • Moments can be determined directly from a histogram much faster than they can from the pixels directly. • Let r denote a discrete random variable representing discrete gray-levels in the range [0, L-1], and p(ri) denote the normalized histogram component corresponding to the ith value of r, then the nth moment of r about its mean is defined as where m is the mean value of r • For example, the second moment (also the variance of r) is © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Use of Histogram Statistics for Image Enhancement • Two uses of the mean and variance for enhancement purposes: – The global mean and variance (global means for the entire image) are useful for adjusting overall contrast and intensity. – The mean and standard deviation for a local region are useful for correcting for large-scale changes in intensity and contrast. ( See equations 3. 3 -21 and 3. 3 -22. ) © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Enhancement Using Arithmetic/Logic Operations • Two images of the same size can be combined using operations of addition, subtraction, multiplication, division, logical AND, OR, XOR and NOT. Such operations are done on pairs of their corresponding pixels. • Often only one of the images is a real picture while the other is a machine generated mask. The mask often is a binary image consisting only of pixel values 0 and 1. • Example: Figure 3. 27 © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Enhancement Using Arithmetic/Logic Operations AND

Digital Image Processing, 2 nd ed. Image Subtraction Example 1 © 2002 R. C.

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Image Subtraction Example 2 • When subtracting two images, negative pixel values can result. So, if you want to display the result it may be necessary to readjust the dynamic range by scaling. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Image Averaging • When taking pictures in reduced lighting (i. e. , low illumination), image noise becomes apparent. • A noisy image g(x, y) can be defined by where f (x, y): an original image : the addition of noise • One simple way to reduce this granular noise is to take several identical pictures and average them, thus smoothing out the randomness. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Noise Reduction by Image Averaging Example: Adding Gaussian Noise • Figure 3. 30 (a): An image of Galaxy Pair NGC 3314. • Figure 3. 30 (b): Image corrupted by additive Gaussian noise with zero mean and a standard deviation of 64 gray levels. • Figure 3. 30 (c)-(f): Results of averaging K=8, 16, 64, and 128 noisy images. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Noise Reduction by Image Averaging Example: Adding Gaussian Noise • Figure 3. 31 (a): From top to bottom: Difference images between Fig. 3. 30 (a) and the four images in Figs. 3. 30 (c) through (f), respectively. • Figure 3. 31 (b): Corresponding histogram. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Basics of Spatial Filtering • In spatial filtering (vs. frequency domain filtering), the output image is computed directly by simple calculations on the pixels of the input image. • Spatial filtering can be either linear or non-linear. • For each output pixel, some neighborhood of input pixels is used in the computation. • In general, linear filtering of an image f of size MXN with a filter mask of size mxn is given by where a=(m-1)/2 and b=(n-1)/2 • This concept called convolution. Filter masks are sometimes called convolution masks or convolution kernels. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. Basics of Spatial Filtering © 2002 R. C.

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Basics of Spatial Filtering • Nonlinear spatial filtering usually uses a neighborhood too, but some other mathematical operations are use. These can include conditional operations (if …, then…), statistical (sorting pixel values in the neighborhood), etc. • Because the neighborhood includes pixels on all sides of the center pixel, some special procedure must be used along the top, bottom, left and right sides of the image so that the processing does not try to use pixels that do not exist. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Smoothing Spatial Filters • Smoothing linear filters – Averaging filters (Lowpass filters in Chapter 4)) • Box filter • Weighted average filter Box filter © 2002 R. C. Gonzalez & R. E. Woods Weighted average

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Smoothing Spatial Filters • The general implementation for filtering an MXN image with a weighted averaging filter of size mxn is given by where a=(m-1)/2 and b=(n-1)/2 © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Smoothing Spatial Filters Image smoothing

Digital Image Processing, 2 nd ed. Smoothing Spatial Filters Another Example © 2002 R.

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Order-Statistic Filters • Order-statistic filters – Median filter: to reduce impulse noise (salt-andpepper noise) © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Sharpening Spatial Filters • Sharpening filters are based on computing spatial derivatives of an image. • The first-order derivative of a one-dimensional function f(x) is • The second-order derivative of a one-dimensional function f(x) is © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. Sharpening Spatial Filters An Example © 2002 R.

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Use of Second Derivatives for Enhancement The Laplacian • Development of the Laplacian method – The two dimensional Laplacian operator for continuous functions: – The Laplacian is a linear operator. © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Use of Second Derivatives for Enhancement The Laplacian • To sharpen an image, the Laplacian of the image is subtracted from the original image. • Example: Figure 3. 40 © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. © 2002 R. C. Gonzalez & R. E.

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Use of First Derivatives for Enhancement The Gradient • Development of the Gradient method – The gradient of function f at coordinates (x, y) is defined as the two-dimensional column vector: – The magnitude of this vector is given by © 2002 R. C. Gonzalez & R. E. Woods

Digital Image Processing, 2 nd ed. www. imageprocessingbook. com Use of First Derivatives for Enhancement The Gradient Roberts cross-gradient operators Sobel operators © 2002 R. C. Gonzalez & R. E. Woods