Week 6

Draft report writeup - 1D problem; 2D Fourier partial sums

Pen and Paper Exercises

Compute (analytically) the Fourier coefficients of the following functions:

Matlab Exercises

• Compute the 2D Fourier partial sum reconstructions of the box and circle functions. Can you observe Gibbs artifacts in your reconstruction? How do these artifacts compare with those from your 1D reconstructions? (You can implement the 2D sum in Matlab using two for loops; you can then replace the for loops using matrix-matrix multiplication for improved efficiency.)
• Compute the 2D Fourier partial sum reconstruction as a series of 1D Fourier partial sums.
  • Start by using replacing one of the for loops (or matrix multiplications) from your previous simulation with the (1D) ComputeFourierReconstruction Matlab script. You can do this by reconstructing the function along either rows or columns. Verify that both variants yield the same reconstruction.
  • Next, use exact jump information and the EdgeEnhancedReconstruction Matlab script to improve the quality of your reconstruction. Describe qualitatively the reconstruction properties. Are the Gibbs artifacts completely eliminated?
  • Replace exact jump information with estimated jump information computed using the FindJumps Matlab script.

In each case, generate a plot of the error (in 2-norm and infinity-norm) as a function of N - the number of Fourier coefficients used.

Documents to Prepare

• (due end of day Thurs., 6/30) 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. Now that you have explored all facets of the 1D problem, it would be a good time to start generating and refining a complete set of slides - which you would be comfortable presenting at a research seminar to a new audience.
• (due end of day Wed., 6/29) 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 complete 1D theoretical and numerical results.