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Chapter 5:
Quality Time with the Fractious Fourier Family
Abstract
Fourier analysis is a common tool in image science, with applications in image processing, image understanding, and image-quality assessment. The term Fourier analysis, however, encompasses a wide variety of mathematical transforms, and all too often it is not clear just what we mean when we say we “take a Fourier transform.” The Fourier integral transform is often used in theoretical developments, but with the implicit understanding that a discrete Fourier transform (DFT) will be substituted when it comes time to write a program. Similarly, an imaging system will often be described by a transfer function, which maps a function of a continuous frequency variable to another such function, even when the actual image data are a discrete set of numbers. Moreover, in order to make use of any Fourier method at all, it is often assumed that a system or image has symmetries that it could not possibly have in reality - continuous imaging systems are assumed to be completely invariant to shifts, or digital images are assumed to have nonphysical wraparound (circulant) properties. A premise of this chapter is that Fourier analysis, properly understood, is useful even when one is careful to distinguish between continuous and discrete quantities, or finite-dimensional and infinite-dimensional vector spaces, and even when one is not willing to assume nonphysical symmetries. To demonstrate this point, we first need to look closely at the meaning of the word “useful” as it is used above. A scientific or medical imaging system is useful only when it conveys meaningful information about an object to some observer. In the view of the author, image quality must be defined in terms of the average performance of the observer on some task of interest. By the same token, a mathematical technique is useful in imaging only if it improves the observer’s ability to extract that task-specific information, or if it improves the ability of the designer of an imaging system to optimize that transfer of information. Fourier analysis, in a broader sense than usual, comes into play for both of these aspects. Our plan in this chapter is to first present the essentials of task-based or objective assessment of image quality, and then to survey the Fourier family with emphasis on some lesser-known members. Then we shall bring these two threads together and show just where and how Fourier methods enter into the assessment and optimization of image quality.
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CHAPTER 5
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