Digital Image Processing
Third Edition
Rafael C. Gonzalez
University of Tennessee
Richard E. Woods
3.1The Basics of Intensity Transformations and Spatial Filtering
All the image processing techniques discussed in this section are implemented in the spatial domain, which we know from the discussion in Section 2.4.2 is simply the plane containing the pixels of an image. As noted in Section 2.6.7,spatial domain techniques operate directly on the pixels of an image as op-posed, for example, to the frequency domain (the topic of Chapter 4) in which operations are performed on the Fourier transform of an image, rather than on the image itself. As you will learn in progressing through the book, some image processing tasks are easier or more meaningful to implement in the spatial domain while others are best suited for other approaches. Generally, spatial do-main techniques are more efficient computationally and require less processing resources to implement.
The spatial domain processes we discuss in this chapter can be denoted by the expression
g(x, y) = T[f(x, y)] (3.1-1)
where is the input image, is the output image, and T is an operator on f defined over a neighborhood of point (x, y).The operator can apply to a single image (our principal focus in this chapter) or to a set of images, such as performing the pixel-by-pixel sum of a sequence of images for noise reduction, as discussed in Section 2.6.3. Figure 3.1 shows the basic implementation of Eq.(3.1-1) on a single image. The point (x, y) shown is an arbitrary location in the image, and the small region shown containing the point is a neighborhood of (x, y),as explained in Section 2.6.5.Typically,the neighborhood is rectangular, centered on (x, y),and much smaller in size than the image.
3.2 Some Basic Intensity Transformation Functions
Intensity transformations are among the simplest of all image processing techniques. The values of pixels, before and after processing, will be denoted by r and s, respectively. As indicated in the previous section, these values are related by an expression of the form where T is a transformation that maps a pixel value r into a pixel value s. Because we are dealing with digital quantities, values of a transformation function typically are stored in a one-dimensional array and the mappings from r to s are implemented via table lookups. For an 8-bit environment, a lookup table containing the values of Twill have 256 entries.
As an introduction to intensity transformations, consider Fig. 3.3, which shows three basic types of functions used frequently for image enhancement: linear (negative and identity transformations), logarithmic (log and inverse-log transformations), and power-law (nth power and nth root transformations).The identity function is the trivial case in which output intensities are identical to input intensities. It is included in the graph only for completeness.
3.2.1 Image Negatives
The negative of an image with intensity levels in the range [0, L - 1] is obtained by using the negative transformation shown in Fig.3.3, which is given by the expression
s = L - 1 - r (3.2-1)
Reversing the intensity levels of an image in this manner produces the equivalent of a photographic negative. This type of processing is particularly suited for enhancing white or gray detail embedded in dark regions of an image, especially when the black areas shows an example. The original image is a digital mammogram showing a small lesion. In spite of the fact that the visual content is the same in both images, note how much easier it is to analyze the breast tissue in the negative image in this particular case.
3.2.2 Log Transformations
The general form of the log transformation in Fig.3.3 is
s = clog (1 r) (3.2-2)
where c is a constant, and it is assumed that r.The shape of the log curve in Fig.3.3 shows that this transformation maps a narrow range of low intensity values in the input into a wider range of output levels. The opposite is true of higher values of input levels. We use a transformation of this type to expand the values of dark pixels in an image while compressing the higher-level values. The opposite is true of the inverse log transformation.
Any curve having the general shape of the log functions shown in Fig. 3.3 would accomplish this spreading/compressing of intensity levels in an image, but the power-law transformations discussed in the next section are much more versatile for this purpose. The log function has the important characteristic that it compresses the dynamic range of images with large variations in pixel values. A classic illustration of an application in which pixel values have a large dynamic range is the Fourier spectrum, which will be discussed in Chapter 4.At the moment, we are concerned only with the image characteristics of spectra. It is not unusual to encounter spectrum values that range from 0 to or higher.While processing numbers such as these presents no problems for a computer, image display systems generally will not be able to reproduce faithfully such a wide range of intensity values. The net effect is that a significant degree of intensity detail can be lost in the display of a typical Fourier spectrum.
As an illustration of log transformations, Fig. 3.5(a) shows a Fourier spectrum with values in the range 0 to When these values are scaled linearly for display in an 8-bit system, the brightest pixels will dominate the display, at the expense of lower (and just as important) values of the spectrum. The effect of this dominance is illustrated vividly by the relatively small area of the image in Fig.3.5(a) that is not perceived as black. If, instead of displaying the values in this manner, we first apply Eq.(3.2-2) (wi
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Digital Image Processing
Third Edition
Rafael C. Gonzalez
University of Tennessee
Richard E. Woods
3.1The Basics of Intensity Transformations and Spatial Filtering
All the image processing techniques discussed in this section are implemented in the spatial domain, which we know from the discussion in Section 2.4.2 is simply the plane containing the pixels of an image. As noted in Section 2.6.7,spatial domain techniques operate directly on the pixels of an image as op-posed, for example, to the frequency domain (the topic of Chapter 4) in which operations are performed on the Fourier transform of an image, rather than on the image itself. As you will learn in progressing through the book, some image processing tasks are easier or more meaningful to implement in the spatial domain while others are best suited for other approaches. Generally, spatial do-main techniques are more efficient computationally and require less processing resources to implement.
The spatial domain processes we discuss in this chapter can be denoted by the expression
g(x, y) = T[f(x, y)] (3.1-1)
where is the input image, is the output image, and T is an operator on f defined over a neighborhood of point (x, y).The operator can apply to a single image (our principal focus in this chapter) or to a set of images, such as performing the pixel-by-pixel sum of a sequence of images for noise reduction, as discussed in Section 2.6.3. Figure 3.1 shows the basic implementation of Eq.(3.1-1) on a single image. The point (x, y) shown is an arbitrary location in the image, and the small region shown containing the point is a neighborhood of (x, y),as explained in Section 2.6.5.Typically,the neighborhood is rectangular, centered on (x, y),and much smaller in size than the image.
3.2 Some Basic Intensity Transformation Functions
Intensity transformations are among the simplest of all image processing techniques. The values of pixels, before and after processing, will be denoted by r and s, respectively. As indicated in the previous section, these values are related by an expression of the form where T is a transformation that maps a pixel value r into a pixel value s. Because we are dealing with digital quantities, values of a transformation function typically are stored in a one-dimensional array and the mappings from r to s are implemented via table lookups. For an 8-bit environment, a lookup table containing the values of Twill have 256 entries.
As an introduction to intensity transformations, consider Fig. 3.3, which shows three basic types of functions used frequently for image enhancement: linear (negative and identity transformations), logarithmic (log and inverse-log transformations), and power-law (nth power and nth root transformations).The identity function is the trivial case in which output intensities are identical to input intensities. It is included in the graph only for completeness.
3.2.1 Image Negatives
The negative of an image with intensity levels in the range [0, L - 1] is obtained by using the negative transformation shown in Fig.3.3, which is given by the expression
s = L - 1 - r (3.2-1)
Reversing the intensity levels of an image in this manner produces the equivalent of a photographic negative. This type of processing is particularly suited for enhancing white or gray detail embedded in dark regions of an image, especially when the black areas shows an example. The original image is a digital mammogram showing a small lesion. In spite of the fact that the visual content is the same in both images, note how much easier it is to analyze the breast tissue in the negative image in this particular case.
3.2.2 Log Transformations
The general form of the log transformation in Fig.3.3 is
s = clog (1 r) (3.2-2)
where c is a constant, and it is assumed that r.The shape of the log curve in Fig.3.3 shows that this transformation maps a narrow range of low intensity values in the input into a wider range of output levels. The opposite is true of higher values of input levels. We use a transformation of this type to expand the values of dark pixels in an image while compressing the higher-level values. The opposite is true of the inverse log transformation.
Any curve having the general shape of the log functions shown in Fig. 3.3 would accomplish this spreading/compressing of intensity levels in an image, but the power-law transformations discussed in the next section are much more versatile for this purpose. The log function has the important characteristic that it compresses the dynamic range of images with large variations in pixel values. A classic illustration of an application in which pixel values have a large dynamic range is the Fourier spectrum, which will be discussed in Chapter 4.At the moment, we are concerned only with the image characteristics of spectra. It is not unusual to encounter spectrum values that range from 0 to or higher.While processing numbers such as these presents no problems for a computer, image display systems generally will not be able to reproduce faithfully such a wide range of intensity values. The net effect is that a significant degree of intensity detail can be lost in the display of a typical Fourier spectrum.
As an illustration of log transformations, Fig. 3.5(a) shows a Fourier spectrum with values in the range 0 to When these values are scaled linearly for display in an 8-bit system, the brightest pixels will dominate the display, at the expense of lower (and just as important) values of the spectrum. The effect of this dominance is illustrated vividly by the relatively small area of the image in Fig.3.5(a) that is not perceived as black. If, instead of displaying the values in this manner, we first apply Eq.(3.2-2) (wi
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