Despite the fact that my interests since I was an undergrad were defined by the research I do now, I have experience working in other topics and areas where I had the chance to prove myself and to put a personal touch. In particular, I have done research work in compressive spectral imaging, facial image processing and computational electromagnetism. I am not active in any of these research topics, but the papers and manuscripts I wrote about them are still close to my heart.
Sensing Matrix Designs in Compressed Sensing
While doing my Ph.D at University of Delaware, I worked initially in compressed sensing related topics. In particular, I worked in structured matrix design. Initially, I analyzed the structure of the matrices that describe the compressed sensing architecture known as the CASSI system, and as a result I determined theoretically its spectral resolution limits providing also experimental results. The results of this part of my research can be found in these papers:
A. Parada-Mayorga and G.R. Arce, ”Spectral Super-Resolution in Colored Coded Aperture Spectral Imaging,” in IEEE Transactions on Computational Imaging, vol. 2, no. 4, pp. 440-455, Dec. [link]
A. Parada Mayorga, G.R. Arce, ”Spectral Super-Resolution in Colored Coded Aperture Spectral Imaging,” in Imaging and Applied Optics 2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper CTh2E.2. [link]
In the same line of research I worked in the optimal design of compressed sensing matrices in the CASSI system based on the coherence metric. In the process of understanding this problem I had the wonderful experience of studying in depth the spherical code problem, the sphere packing problem and all the beautiful math that can be connected to these problems. Additionally, I ended up studying: polytopes and lattice theory. As a result of this part of my research I published the following paper:
A. Parada-Mayorga and G. R. Arce, ”Colored Coded Aperture Design in Compressive Spectral Imaging via Minimum Coherence,” in IEEE Transactions on Computational Imaging, vol. 3, no. 2,pp. 202-216, June 2017. [link]
A. Parada-Mayorga, A. Cuadros and G.R. Arce, “Coded Aperture Design for Compressive X-ray Tomosynthesis via Coherence Analysis”. IEEE International Symposium on Biomedical Imaging. Melbourne, Australia. April 2017. [link][pdf] — This paper improved the results obtained by a previous journal paper, performing a computation in seconds while the other method spent days/weeks. Interestingly, I get little to no citations in comparison to the other paper —
I am not active in this research topic, but I do not discard in the future to revisit compressed sensing with another philosophy and some new ideas.
Dynamic Analysis of Facial expressions
This part of my research was indeed a mixture of several branches of signal processing and machine learning: digital image processing, nonlinear dimensionality reduction, and statistical shape analysis. All of the them are beautiful topics and in all of them I had the chance to learn a lot. My goal was to develop a systematic way to track dynamically features of facial expressions.
Here I met the first time with manifold theory in data analysis, and in the processes I met several works of David Donoho. This in particular was one of the best parts of my research. I did not publish my work in a prestigious journal like I did in my Ph.D, instead I published the two papers in a local event at a university. You can find these publications here and here (terrible english!, I did not have a good english at the time) and my whole master thesis is here (in spanish!, bad english at the time again!). I feel proud of this work for the countless problems I solved, and also because of the mathematics I involved in the different analysis that I performed. Specially smooth manifolds and Riemmanian geometry in data analysis.
As an undergrad I devoted my research efforts to computational electromagnetism. In particular, I analyzed the problem of heating in ferromagnetic materials as a consequence of induced currents by magnetic fields. My research goals at the time were to solve numerically the Maxwell equations describing the Eddy currents problem by means of finite element analysis and to solve at the same time the heat equation describing the heating of cubic and rectangular geometries of ferromagnetic materials.
While doing my research I involved my self with the application of calculus of variations, functional analysis, partial differential equations and formal finite element analysis. Although I did not published a paper, I wrote a complete thesis that you can find here (in spanish!, my english was not good at that time). I feel very proud of this work not only because of the countless problems I solved in the way to achieve the goals of the project, but also because I learned with hands on experience the role that branches of mathematics far from the basic curriculum can have in a given problem.