Abstract
A Fourier transformation changes all the locations of an image to frequencies of gratings, and back again. This allows certain frequencies to be attenuated or amplified, for one purpose or another. The theory is developed here for continuous curves, then for 8-bit grey-scale images and 8-bit sinusoidal gratings. A pair of programs for analysis and synthesis is given, with examples.
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Notes
- 1.
J.B.J. Fourier (1768–1830) Governor of Lower Egypt 1801, Prefect of Isere 1802, created Baron 1808 (by Napoleon, annulled at his fall), Secretary of Academie des Sciences 1823, Academie Francaise 1826.
- 2.
This was doubted by Poisson and others, but proved under increasingly rigorous conditions by Sturm and by Dirichlet.
- 3.
Originated by Whitaker, with further contributions by Nyquist, Kotelniakov, Shannon, and others.
- 4.
Harry T. Nyquist (1889–1976), ATandT Labs 1917–1934, Bell Telephone Labs 1934–1954.
- 5.
Published in 1965 by Cooley and Tukey; it was later found that Gauss had used essentially the same process in 1805 for astronomical calculations. James W. Cooley (born 1926) was a programmer with von Neumann at Princeton 1953–1956, and with IBM 1962–1991. John W. Tukey (1915–2000) was a statistician at Princeton, Professor 1950–1985; coined the term “bit” 1946, and the term “software” 1958.
- 6.
Often called the DC component, following a usage from electrical signal-processing.
References
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Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19:297–301
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ISO Standard 10918-1 (1994) Digital compression and coding of continuous-tone still images. http://www.iso.org/. Accessed 1 Feb 2014
Joint Photographic Experts Group (2014) http://www.jpeg.org/index.html. Accessed 1 Feb 2014
Rao KR, Yip P (1990) Transform algorithms, advantages, applications. Academic Press, Boston
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Parkin, A. (2016). 8-bit Fourier Process. In: Digital Imaging Primer. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85619-1_40
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DOI: https://doi.org/10.1007/978-3-540-85619-1_40
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