Friday, June 4, 2010

The Gibbs Phenomenon

In chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Fourier analysis, a fascinating but very mathematical subject. One of the most surprising results of Fourier analysis is the Gibbs phenomenon, which we describe at the end of Sec. 11.5 (Fourier Series for a Periodic Function).
“Table 11.4 shows the first few coefficients for the Fourier series representing the square wave, obtained from Eq. 11.34. […] Figure 11.16 shows the fits for n=3 and n=39. As the number of terms in the fit is increased, the value of Q [measuring the least squares fit between the function at its Fourier series] decreases. However, spikes of constant height (about 18% of the amplitude of the square wave or 9% of the discontinuity in y) remain. These are seen in Fig. 11.16. These spikes appear whenever there is a discontinuity in y and are called the Gibbs phenomenon.”
You have to be amazed by the Gibbs phenomenon. Think about it: as you add terms in the sum, the fit between the function and its Fourier series gets better and better, but the overshoot in amplitude does not get any smaller. Instead, the region containing ringing near the discontinuity gets narrower and narrower. If you want to see a figure like our Fig. 11.16 presented as a neat animation, take a look at http://www.sosmath.com/fourier/fourier3/gibbs.html. Also, check out http://ocw.mit.edu/ans7870/18/18.06/javademo/Gibbs/ for an interactive demo that will let you include up to 200 terms in the Fourier series.

The Gibbs phenomenon is important in medical imaging. The entry for the Gibbs phenomenon from the Encyclopedia of Medical Imaging is reproduced below.
"Gibbs phenomenon, (J. Willard Gibbs, 1839-1903, American physicist), phenomenon occurring whenever a ‘curve’ with sharp edges is subject to Fourier analysis. The Gibbs phenomenon is relevant in MR imaging, where it is responsible for so-called Gibbs artefacts. Consider a signal intensity profile across the skull, where at the edge of the brain the signal intensity changes from virtually zero to a finite value. In MR imaging the measurement process is a breakdown of such intensity profiles into their Fourier harmonics. i.e. sine and cosine functions. Representation of the profiles measured with a limited number of Fourier harmonics is imperfect, resulting in high frequency oscillations at the edges, and the image can therefore exhibit some noticeable signal intensity variations at intensity boundaries: the Gibbs phenomenon, overshoot artefacts, or ‘ringing.’ The artefacts can be suppressed by filtering the images. However, filtering can in turn reduce spatial resolution.”
Figures 12.24 and 12.25 of our book show a CT scan with ringing inside the skull and its removal by filtering, an example of the Gibbs phenomenon.

Josiah Willard Gibbs was a leading American physicist from the 19th century. He is particularly well known for his contributions to thermodynamics. Gibbs appears at several places in Intermediate Physics for Medicine and Biology. Section 3.17 discusses the Gibbs free energy, a quantity that provides a simple way to keep track of the changes in total entropy when a system is in contact with a reservoir at constant temperature and pressure. A footnote on page 68 addresses the Gibbs paradox (which deserves an entire blog entry of its own), and Problem 47 in Chapter 3 introduces the Gibbs factor (similar to the Boltzmann factor but including the chemical potential).

The preface to Gibbs' book on statistical mechanics is reproduced in Selected Papers of Great American Physicists: The Bicentennial Commemorative Volume of the American Physical Society 1976, edited by Spencer Weart. I recall being quite impressed by this book when in graduate school at Vanderbilt University. Below is a quote from Weart's biographical notes about Gibbs.
“Gibbs, son of a Yale professor of sacred literature, descended from a long line of New England college graduates. He studied at Yale, received his Ph.D. there in 1863—one of the first doctorates granted in the United States—tutored Latin and natural philosophy there, and then left for three decisive years in Europe. Up to that time, Gibbs had shown interest in both mathematics and engineering, which he combined in his dissertation ‘On the Form of the Teeth in Wheels in Spur Gearing.’ The lectures he attended in Paris, Berlin and Heidelberg, given by some of the greatest men of the day, changed him once and for all. In 1871, two years after his return from Europe, he became Yale’s first Professor of Mathematical Physics. He had not yet published any papers on this subject. For nine years he held the position without pay, living on the comfortable inheritance his father had left; only when Johns Hopkins University offered Gibbs a post did Yale give him a small salary.

Gibbs never married. He lived out a calm and uneventful life in the house where he grew up, which he shared with his sisters. He was a gentle and considerate man, well-liked by those who knew him, but he tended to avoid society and was little known even in New Haven. Nor was he known to more than a few of the world’s scientists—partly because his writings were extremely compact, abstract and difficult. As one of Gibb’s European colleagues wrote, ‘Having once condensed a truth into a concise and very general formula, he would not think of churning out the endless succession of specific cases that were implied by the general proposition; his intelligence, like his character, was of a retiring disposition.’ The Europeans paid for their failure to read Gibbs: A large part of the work they did in thermodynamics before the turn of the century could have been found already in his published work."

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