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CHAPTER 1

VECTOR SPACES, SIGNALS, AND IMAGES

1.1 OVERVIEW

In this chapter we introduce the mathematical framework of vector spaces, matrices, and inner products. We motivate the mathematics by using it to model signals and images, both outside the computer (the analog signal as it exists in the “real world”) and inside the computer (the digitized signal, suitable for computer storage and processing). In either case the signal or image may be viewed as an element of a vector space, so we define and develop some essential concepts concerning these spaces. In particular, to analyze signals and images, we will decompose them into linear combinations of basic sinusoidal or complex exponential waveforms. This is the essence of the discrete Fourier and cosine transforms.

The process of sampling the analog signal and converting it to digital form causes an essential loss of information, called “aliasing” and “quantization error.” We examine these errors rather closely. Analysis of these errors motivates the development of methods to quantify the distortion introduced by an image compression technique, which leads naturally to the concept of the “energy” of a signal and a deeper analysis of inner products and orthogonality.

1.2 SOME COMMON IMAGE PROCESSING PROBLEMS

To open this chapter, we discuss a few common challenges in image processing, to try to give the reader some perspective on the subject and why mathematics is such an essential tool. In particular, we take a very short look ...

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