Presentation: Curvelets and Approximation Theory

Find below a set of slides that I used for my talk at the IMA in the Thematic Year on Mathematical Imaging. On them, there is a detailed construction of my generalized curvelets, some results by Donoho and Candès explaining their main properties, and a bunch of applications to Imaging. Click on the slide below to retrieve the pdf file with the presentation.

Poster: Curvelets vs. Wavelets (Mathematical Models of Natural Images)

Together with Professor Bradley J. Lucier, we presented a poster in the Workshop on Natural Images during the thematic year on Mathematical Imaging at the IMA. We experimented with wavelet and curvelet decompositions of 24 high quality photos from a CD that Kodak® distributed in the late 90s. All the experiment details and results can be read in the file Curvelets/talk.pdf.

The computations concerning curvelet coefficients were carried out in Matlab, with the Curvelab 2.0.1 toolbox developed by Candès, Demanet, Donoho and Ying. The computations concerning wavelet coefficients were performed by Professor Lucier’s own codes.

Wavelet Coefficients

To aid in my understanding of wavelets, during the first months I started studying this subject I wrote a couple of scripts to both compute wavelet coefficients of a given pgm gray-scale image (the decoding script), and recover an approximation to the original image from a subset of those coefficients (the coding script). I used OCaml, a multi-paradigm language: imperative, functional and object-oriented.

The decoding script uses the easiest wavelets possible: the Haar functions. As it was suggested in the article “Fast wavelet techniques for near-optimal image processing“, by R. DeVore and B.J. Lucier, rather than computing the actual raw wavelet coefficients, one computes instead a related integer value (a code). The coding script interprets those integer values and modifies them appropriately to obtain the actual coefficients. The storage of the integers is performed using Huffman trees, but I used a very simple one, not designed for speed or optimization in any way.

Following a paper by A.Chambolle, R.DeVore, N.Y.Lee and B.Lucier, “Non-linear wavelet image processing: Variational problems, compression and noise removal through wavelet shrinkage“, these scripts were used in two experiments later on: computation of the smoothness of an image, and removal of Gaussian white noise by the wavelet shrinkage method proposed by Donoho and Johnstone in the early 90’s.

Progressive reconstruction of a grey-scale image of size with the largest (in absolute value) coefficients,

Modeling the Impact of Ebola and Bushmeat Hunting on Western Lowland Gorillas

In May 2003, together with fellow Mathematician Stephanie Gruver, Statistician Young-Ju Kim, and Forestry Engineer Carol Rizkalla, we worked on this little project to apply ideas from Dynamical Systems to an epidemiology model of the Ebola hemorrhagic fever in the Republic of Congo. The manuscript ebola/root.pdf is a first draft, and contains most of the mathematics behind the study. Carol worked in a less-math-more-biology version: ebola/Ecohealth.pdf “Modeling the Impact of Ebola and Bushmeat Hunting on Western Lowland Gorillas,” and presented it to EcoHealth, where it has been published (June 2007). She also prepared a poster for the Sigma-Xi competition: Click on the image below to retrieve a PowerPoint version of it.

Triangulations

As part of a project developed by Professor Bradley J. Lucier, to code a PDE solver written in scheme, I worked in some algorithms to perform “good triangulations” of polygons with holes (“good triangulations” meaning here, those where all the triangles have their three angles as close to 60º as possible). I obtained the necessary theoretical background and coding strategies from the following references:

• Mark de Berg et al., “Computational Geometry by Example.“
• Francis Chin and Cao An Wang, “Finding the Constrained Delaunay Triangulation and Constrained Voronoi Diagram of a simple Polygon in Linear Time.“
• Joseph O’Rourke, “Computational Geometry in C.“
• Jim Ruppert, “A Delaunay Refinement Algorithm for Quality 2-dimensional Mesh Generation.“

Mechanical Geometry Theorem Proving

In 1977, Professor Wen-Tsun Wu succeeded in developing a method of mechanical geometry theorem proving. This method has been applied to prove or even discover hundreds of non-trivial difficult theorems in elementary and differential geometries on a computer in an almost trivial manner. Usign Ritt’s differential algebra, Wu established a method for solving algebraic and differential equations by transforming an equation system in the general form to equation systems in triangular form. This is the Ritt-Wu decomposition algorithm, that later on was shown to be equivalent to perform a series of operations on ideals, very easily carried out by means of Gröbner basis manipulation.

I wrote a script in MAPLE to perform evaluations of the validity of some simple theorems in Euclidean Geometry, and wrote a small paper (in Spanish) on one of my findings, that was published in Bol. Asoc. Prof. Puig Adams, in October’99: “Sobre demostración automática de un problema geométrico“.

The example I cover in that short article can be seen below.

Given: Circles , that intersect each other in points and , and given points , in circle , consider line through and , and line through and . The intersections of line with circle are and . The intersections of line with circle are and . Consider the segments (connecting with ) and (connecting with ).

To prove: Segments and are parallel.