Week 3

The Big Picture

Edge Detection from Fourier Data and Edge-Augmented Fourier Reconstructions.

(Note: Some of these assignments and exercises may spill over to Week 4; however, we will try and make as much headway as possible!)

Reading Assignments

• Complete reading Chapter 2 of Folland.
• Read Chapter 9 from Spectral Methods for Time-Dependent Problems by Hesthaven et al.

Research Paper Reviews

• Read Sections 4, 5 and 7 from Filters, mollifiers and the computation of the Gibbs phenomenon, Eitan Tadmor, Acta Numerica, Vol. 16 (2007), pp. 305-378, DOI: 10.1017/S0962492906320016
• Read the paper Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data, Dmitry Batenkov, Math. Comp., 84 (2015), pp. 2329-2350, DOI:10.1090/S0025-5718-2015-02948-2. We will be generalizing, extending and improving the results from this paper.

Pen and Paper Exercises

Last week, you started working on incorporating (apriori) edge information into the Fourier reconstruction. Specifically, you implemented Fourier sums of the form

where the estimated Fourier coefficients were computed using the jump "heights" and locations as

You will now work out what happens when M goes to infinity. Start by writing down a closed-form expression for the infinite sum

Can you incorporate this into the first equation? You should also evaluate the resulting modified/augmented partial sum in Matlab and observe the reconstruction quality.

Chalk Talk

Present a section from one of the above papers (for example Sections 4-5 from the Filters, Mollifiers and Computation of the Gibbs Phenomenon paper; Sections 1-3 from the Complete Algebraic Reconstruction... paper).

The ability to read and summarize papers is important when conducting research. Plan to speak for 15-30 minutes. You should introduce the problem, write down notation, list the main theoretical results and summarize any numerical simulations. You don't need to go through all steps in verbose proofs - just provide an outline.

Matlab Coding Exercises

• Recreate Figures 5.1 and 7.1/7.2 from the Filters, Mollifiers and Computation of the Gibbs Phenomenon paper.
• Implement the method introduced in the Complete Algebraic Reconstruction... paper.

Note: Try and write Matlab code (that you can eventually make public on the web) that is readable, well-documented and can recreate your figures/tables/other technical results.

Documents to Prepare

• (due end of day Thurs., 6/9) Prepare a 15 minute Beamer/LaTeX presentation summarizing your research findings from the week. You will be presenting (as a group) to your fellow REU students/groups during the Friday meeting/discussion at the Holmes Hall seminar room.

As you read chapters from the text and the papers, note down/TeX up important definitions, results and theorems. You can include these in your technical reports and presentations to make them more mathematically rigorous.

(For example, when discussing Fourier series, you could state Theorem 2.1 - about convergence of Fourier series - from Folland; if anyone asks about piecewise-smooth functions, you should recall the definition from page 32 of Folland; similarly, pp. 156-160 in Chapter 9 from Spectral Methods for Time-Dependent Problems by Hesthaven et al. discusses non-uniform convergence of the Fourier series for piecewise-smooth functions - the Gibbs ringing artifacts; Defn. 9.6 in this chapter also defines a filter function).

You should also start preparing a technical report in LaTeX detailing your research findings to date. Structure this report as if it were a technical paper - it should contain (i) an abstract (ii) introduction (iii) background material (summarizing the main results/theorems) on Fourier series and the Gibbs phenomenon (iv) theoretical development of edge detection and edge-augmented Fourier reconstructions (v) numerical results. You may find it easier to divide up the topics among yourselves and work collaboratively.