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# CH24 - CHAPTER 24 Linear Image Processing Linear image...

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397 CHAPTER 24 Linear Image Processing Linear image processing is based on the same two techniques as conventional DSP: convolution and Fourier analysis . Convolution is the more important of these two, since images have their information encoded in the spatial domain rather than the frequency domain. Linear filtering can improve images in many ways: sharpening the edges of objects, reducing random noise, correcting for unequal illumination, deconvolution to correct for blur and motion, etc. These procedures are carried out by convolving the original image with an appropriate filter kernel, producing the filtered image. A serious problem with image convolution is the enormous number of calculations that need to be performed, often resulting in unacceptably long execution times. This chapter presents strategies for designing filter kernels for various image processing tasks. Two important techniques for reducing the execution time are also described: convolution by separability and FFT convolution . Convolution Image convolution works in the same way as one-dimensional convolution. For instance, images can be viewed as a summation of impulses , i.e., scaled and shifted delta functions. Likewise, linear systems are characterized by how they respond to impulses; that is, by their impulse responses . As you should expect, the output image from a system is equal to the input image convolved with the system's impulse response. The two-dimensional delta function is an image composed of all zeros, except for a single pixel at: row = 0, column = 0, which has a value of one . For now, assume that the row and column indexes can have both positive and negative values, such that the one is centered in a vast sea of zeros. When the delta function is passed through a linear system, the single nonzero point will be changed into some other two-dimensional pattern. Since the only thing that can happen to a point is that it spreads out , the impulse response is often called the point spread function (PSF) in image processing jargon.

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The Scientist and Engineer's Guide to Digital Signal Processing 398 a. Image at first layer b. Image at third layer FIGURE 24-1 The PSF of the eye. The middle layer of the retina changes an impulse, shown in (a), into an impulse surrounded by a dark area, shown in (b). This point spread function enhances the edges of objects. The human eye provides an excellent example of these concepts. As described in the last chapter, the first layer of the retina transforms an image represented as a pattern of light into an image represented as a pattern of nerve impulses. The second layer of the retina processes this neural image and passes it to the third layer, the fibers forming the optic nerve. Imagine that the image being projected onto the retina is a very small spot of light in the center of a dark background. That is, an impulse is fed into the eye. Assuming that the system is linear, the image processing taking place in the retina can be determined by inspecting the image appearing at the optic nerve.
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