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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Chapter 3 Image Enhancement in the Spatial Domain Chapter 3 Image Enhancement in the Spatial Domain

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Image Enhancement in the Spatial Domain 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.

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

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Some Basic Gray Level Transformations

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Image Negatives Let the range of gray level be [0, L-1], then

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Log Transformations where c : constant

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Power-Law Transformation where c, : positive constants

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Power-Law Transformation Example 1: Gamma Correction Power-Law Transformation Example 1: Gamma Correction

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Power-Law Transformation Example 2: Gamma Correction Power-Law Transformation Example 2: Gamma Correction

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Power-Law Transformation Example 3: Gamma Correction Power-Law Transformation Example 3: Gamma Correction

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Piecewise-Linear Transformation Functions Case 1: Contrast Stretching Piecewise-Linear Transformation Functions Case 1: Contrast Stretching

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Piecewise-Linear Transformation Functions Case 3:Bit-plane Slicing 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.

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Piecewise-Linear Transformation Functions Bit-plane Slicing: A Fractal Image Piecewise-Linear Transformation Functions Bit-plane Slicing: A Fractal Image

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Piecewise-Linear Transformation Functions Bit-plane Slicing: A Fractal Image Piecewise-Linear Transformation Functions Bit-plane Slicing: A Fractal Image

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

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

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Histogram Equalization For example:

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Histogram Equalization Histogram equalization is based on a transformation of the probability density function of a random variable. Let p r (r) and p s (s) denote the probability density function of random variable r and s, respectively. If p r (r) and T(r) are known, then the probability density function p s (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.

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Histogram Equalization In discrete version: –The probability of occurrence of gray level r k in an image is n : the total number of pixels in the image n k : the number of pixels that have gray level r k 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 r k in the input image into a corresponding pixel with level s k.

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

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

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods 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 p z (z). We skip the details of implementation.

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods 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(r i ) 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

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Use of Histogram Statistics for Image Enhancement Example: Enhancement based on local statistics Use of Histogram Statistics for Image Enhancement Example: Enhancement based on local statistics

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Use of Histogram Statistics for Image Enhancement Example: Enhancement based on local statistics Use of Histogram Statistics for Image Enhancement Example: Enhancement based on local statistics

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Use of Histogram Statistics for Image Enhancement Example: Enhancement based on local statistics Use of Histogram Statistics for Image Enhancement Example: Enhancement based on local statistics

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Enhancement Using Arithmetic/Logic Operations AND OR

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Image Subtraction Example 1 Image Subtraction Example 1

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

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Noise Reduction by Image Averaging Example: Adding Gaussian Noise Noise Reduction by Image Averaging Example: Adding Gaussian Noise Figure 3.30 (a): An image of Galaxy Pair NGC3314. 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.

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Noise Reduction by Image Averaging Example: Adding Gaussian Noise Noise Reduction by Image Averaging Example: Adding Gaussian Noise Figure 3.31 (a): From top to bottom: Difference images between Fig (a) and the four images in Figs (c) through (f), respectively. Figure 3.31 (b): Corresponding histogram.

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

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

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

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

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Smoothing Spatial Filters Image smoothing with masks of various sizes Smoothing Spatial Filters Image smoothing with masks of various sizes

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Smoothing Spatial Filters Another Example Smoothing Spatial Filters Another Example

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

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Sharpening Spatial Filters An Example Sharpening Spatial Filters An Example

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Use of Second Derivatives for Enhancement The Laplacian 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.

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Use of Second Derivatives for Enhancement The Laplacian Use of Second Derivatives for Enhancement The Laplacian

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Use of Second Derivatives for Enhancement The Laplacian 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

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Use of Second Derivatives for Enhancement The Laplacian: Simplifications Use of Second Derivatives for Enhancement The Laplacian: Simplifications The g(x,y) mask Not only

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Use of First Derivatives for Enhancement The Gradient 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

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

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Use of First Derivatives for Enhancement The Gradient: Using Sobel Operators Use of First Derivatives for Enhancement The Gradient: Using Sobel Operators

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Combining Spatial Enhancement Methods Combining Spatial Enhancement Methods

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Digital Image Processing, 2nd ed. © 2002 R. C. Gonzalez & R. E. Woods Combining Spatial Enhancement Methods Combining Spatial Enhancement Methods

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

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