Week 4

Validation of 1D Edge Enhanced Fourier Reconstruction Schemes for accuracy, computational cost and robustness.

(Continuing) Research Paper Reviews

Complete reading 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
Complete reading 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.

Pen and Paper Exercises

Assume that you are the exact jump locations and jump heights of a piecewise-smooth function. We want to bound (analytically) the error in an edge-augmented Fourier reconstruction (which you computed last week).
- (you should consult Folland for this) Start by writing down an analytic expression for the error in a partial Fourier sum approximation; i.e., write down an upper bound for the error

Extend your result to the edge-augmented Fourier sum you computed last week; i.e., write down an upper bound for the error

Verify these bounds with numerical simulations in Matlab.

Note: Use the usual jump-based estimate for the Fourier coefficients,

Matlab Exercises

Empirical Evaluation of Edge Detection/Estimation Schemes

Concentration Kernel Based Edge Detection

Implement the concentration kernel based edge detection method.
- Evaluate and plot the concentration/conjugate Fourier sum .
- Use Matlab's findpeaks() function to pinpoint the jump locations and jump heights.
How accurate are these jump estimates? How does the performance change with N (the number of Fourier modes used)?
Compute edge-augmented Fourier reconstructions using the estimated jump information to improve the accuracy of your Fourier reconstruction. How does the accuracy change with N (the number of Fourier modes used)?

Prony-Based Edge Detection

Implement Prony's method for detecting jump locations and jump heights.
How accurate are these jump estimates? How does the performance change with N (the number of Fourier modes used) and other Prony parameters?
Compute edge-augmented Fourier reconstructions using the estimated jump information to improve the accuracy of your Fourier reconstruction. How does the accuracy change with N (the number of Fourier modes used)?
What happens when the number of jumps in the function is not exactly known?

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., 6/16) 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.
(due end of day Thurs., 6/16) Continue preparing your 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. By now, you should have 1D theoretical and numerical results.