Portraits in C with Wegert_Wickerhauser_Weiss

November 2012


“The mind is like a richly woven tapestry in which the colors are distilled from the experiences of the senses, and the design drawn from the convolutions of the intellect.” -Carson McCullers, American writer, 1917-1967

"Somehow, at the end of it, it goes through the filtering process and out comes the Radiohead thing." -Ed O'Brien, British musician, b.1968

Elias Wegert of the Technische Universität Bergakademie Freiberg announced last week the completion of his new book, Visual Complex Functions: An Introduction with Phase Portraits, published by Birkhäuser; see [1].  Wegert describes the work as “a systematic introduction to analytic functions using special color representations which visualize functions as images on their domains.” The colorful volume summarizes one mathematician’s effort to graph hard to visualize 4-dimensional data and is brimming with the intricate images Wegert expertly creates to reveal essential information on mathematical functions in a beautiful way.

This phase portrait shows the sum of the first ten terms of the Lambert (1728-1777) series

(1)

which can be rewritten as a power series

with the remarkable property that the coefficient d(n) of zn coincides with the number of [positive] divisors of n. Therefore f, as stated in (1), is called the generating function of the divisor function d(n). See Wikipedia for  a synopsis of the Lambert series and [2] to download a Complex Beauties calendar containing phase portraits.

"Since phase plots are painted with the restricted palette of saturated colors from the color circle, Leonardo’s Mona Lisa will certainly never appear," Elias Wegert wrote in an article submitted to the Notices of the AMS in 2009 that describes his work and methods; see [3] (pub. June 2011). And while Wegert tends to be practical, I have always appreciated that Wegert’s phase plots afford all viewers a glimpse of the beautiful intricacy and symmetry of certain structures that fuel both analytic and practical mathematical musings as much as they provide mathematicians with information about complex-valued functions or some geometry of C.

With the phase plots not only showing the location of zeros and poles (the points where colors converge in the portraits) but also revealing their multiplicity, one field of application of phase plots is the visual analysis of transfer functions in filter design. “Filter design is a classical problem in electrical engineering,” explains WUSTL Professor of Mathematics Victor Wickerhauser, an expert in audio and image filters.  An affable educator, Wickerhauser summarizes the mathematical process to obtain a filtered audio signal using a Discrete Fourier Transform, a Transfer Function, and the inverse DFT thus:

.

 

The Transfer Function (or frequency response of the generalized filter) is traditionally discussed as a quotient of two polynomials

  where   is the wavenumber,

 

which poles (roots of the denominator) and zeros (roots of the numerator) can today be specified by graphically manipulating points in a phase plot with the help of software running Padé (1863-1953) approximations.

“Filter design is the optimization of a device at a fixed cost,” says Wickerhauser. An 8-tap filter corresponds to a polynomial of degree 8. The higher the tap, the higher the cost of the filter. Wickerhauser uses Wavelets, transfer functions represented as an infinite product:

wavelet.

“A transfer function with this characteristic is easier to apply,” Wickerhauser explains, “and what T to use is the object of much research.”

Here the name Ingrid Daubechies —currently at Duke University and a close friend of another Wavelet expert at WUSTL, Professor of Mathematics Guido Weiss— must be mentioned, as should the lovely survey paper on Wavelets written by Guido Weiss, Ed Wilson, and Demetrio Labate that is due to appear in The Notices of the AMS this January 2013; see [4] for a look at the paper's introduction page. The Daubechies 4-tap wavelet is the most famous pair of father wavelet (scaling function in the time domain) and mother wavelet (wavelet function in the time domain) ever devised. “Yes! She walks on water,” a mathematician once chided a Dean of Faculty.  In the end, Mathematics did not win the repartee at that University. In the words of Churchill: “I saw as one might see the transit of Venus, a quantity passing through infinity and changing its sign from plus to minus... [5] 

Phase plots such as those created by Elias Wegert “absolutely, can be used to find T,” Victor Wickerhauser concluded.

Elias Wegert plans to write a follow-up volume to his book and discuss the many applications of phase plots. In the meantime, “I am fascinated by the Riemann Zeta-Function and its colorful phase plot,” he confesses. “So I printed the function along the critical strip, about 20 units in imaginary direction in one rectangle, and arranged these rectangles side by side." Readers can view the ζ plots here» “One discovers a kind of diagonal structure. A little thought then leads to a conjecture for the `period' 2 π / log 2.” When Wegert showed the images to Jörn Steuding (Universität Würzburg), Steuding had an idea on how this observation could be converted into a mathematical theorem. This led to a joint paper which appeared this September in Experimental Mathematics, see [6].

Whether you are taking a Complex Analysis class or are busy designing filters this semester, we hope that phase plots might suggest interesting topics for your research.

—Math news, stories, videos and interviews by Marie Taris, http://www.math.wustl.edu/marietaris/math.html»
(pub. 10.24, 2012)

Enjoyed this story? See also October 2012 News story  Composing Microbial Bebop with Peter Larsen»

[1] Visual Complex Functions: An Introduction with Phase Portraits; E. Wegert; Birkhäuser Basel (2012)
Link to the book at http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0179-9. The eBook can be accessed via its electronic identifier at http://dx.doi.org/10.1007/978-3-0348-0180-5.
[2] Complex Beauties 2012 calendar; E. Wegert, G. Semmler; TU Bergakademie Freiberg.  Download a calendar at www.mathcalendar.net. The 2013 calendar is at  www.mathe-kalender.de, a version in English will soon be available.
[3] Phase Plots of Complex Functions: A Journey in Illustration; Elias Wegert and Gunter Semmler; Notices of the AMS (June/July 2011).  Link to  www.ams.org/notices/201106/rtx110600768p.pdf»
[4] Wavelets, D.Labate, G.Weiss, E. Wilson, to appear in the Notices of the AMS (January 2013). Read the introduction here»
[5] "I saw, as one might see the transit of Venus, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened... but it was after dinner and I let it go." -Winston Churchill (My early life, 1930)
[6] The Riemann Zeta Function on Arithmetic Progressions; J. Steuding, E. Wegert; Experimental Mathematics (September 2012) 21:3, 235-240.  Link to http://www.tandfonline.com/doi/abs/10.1080/10586458.2012.651410»