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Computer Vision Metrics: Chapter Three (Part E)

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For Part D of Chapter Three, please click here.

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Basis Space Metrics

Features can be described in a basis space, which involves transforming pixels into an alternative basis and describing features in the chosen basis, such as the frequency domain. What is a basis space and what is a transform? Consider the decimal system, which is base 10, and the binary system which is base 2. We can change numbers between the two number systems by using a transform. A Fourier transform uses sine and cosine as basis functions in frequency space, so that the Fourier transform can move pixels between the time-domain pixel space and the frequency space. Basis space moments describe the projection of a function onto a basis space [518]—for example, the Fourier transform projects a function onto a basis of harmonic functions.

Basis spaces and transforms are useful for a wide range of applications, including image coding and reconstruction, image processing, feature description, and feature matching. As shown in Figure 3-18, image representation and image coding are closely related to feature description. Images can be described using coding methods or feature descriptors, and images also can be reconstructed from the encodings or from the feature descriptors. Many methods exist to reconstruct images from alternative basis space encodings, ranging from lossless RLE methods to lossy JPEG methods; in Chapter 4, we provide illustrations of images that have been reconstructed from only local feature descriptors (see Figures 4-16 and 4-17).

Figure 3-18. An oversimplified spectrum of basis space options, showing feature set size and complexity of description and reconstruction

As illustrated in Figure 3-18, a spectrum of basis spaces can be imagined, ranging from a continuous real function or live scene with infinite complexity, to a complete raster image, a JPEG compressed image, a frequency domain, or other basis representations, down to local feature descriptor sets. Note that the more detail that is provided and used from the basis space representation, the better the real scene can be recognized or reconstructed. So the tradeoff is to find the best representation or description, in the optimal basis...