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Computer Vision Metrics: Chapter Two (Part D)

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For Part C of Chapter Two, please click here.

Bibliography references are set off with brackets, i.e. "[XXX]". For the corresponding bibliography entries, please click here.


Transform Filtering, Fourier, and Others

This section deals with basis spaces and image transforms in the context of image filtering, the most common and widely used being the Fourier transform. A more comprehensive treatment of basis spaces and transforms in the context of feature description is provided in Chapter 3. A good reference for transform filtering in the context of image processing is provided by Pratt [9].

Why use transforms to switch domains? To make image pre-processing easier or more effective, or to perform feature description and matching more efficiently. In some cases, there is no better way to enhance an image or describe a feature than by transforming it to another domain—for example, for removing noise and other structural artifacts as outlier frequency components of a Fourier spectrum, or to compact describe and encode image features using HAAR basis features.

Fourier Transform Family

The Fourier transform is very well known and covered in the standard reference by Bracewell [227], and it forms the basis for a family of related transforms. Several methods for performing fast Fourier transform (FFT) are common in image and signal processing libraries. Fourier analysis has touched nearly every area of world affairs, through science, finance, medicine, and industry, and has been hailed as “the most important numerical algorithm of our lifetime” [290]. Here, we discuss the fundamentals of Fourier analysis, and a few branches of the Fourier transform family with image pre-processing applications.

The Fourier transform can be computed using optics, at the speed of light [516]. However, we are interested in methods applicable to digital computers.

Fundamentals

The basic idea of Fourier analysis [227,4,9] is concerned with decomposing periodic functions into a series of sine and cosine waves (Figure 2-14). The Fourier transform is bi-directional, between a periodic wave and a corresponding series of harmonic basis functions in the frequency domain, where each basis function is a sine or cosine function, spaced at whole harmonic multiples from the base frequency. The result of the forward FFT is a complex number composed of magnitude and phase data for each sine and cosine component in the series, also referred to as real data and imaginary data.

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