## Previous Research

Despite the fact that my interests since I was an undergrad were defined by the research I do now, I had the experience of working in other topics where I had the chance to prove myself and to put a personal touch in areas such as computational 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 (not active)

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 additionally experimental results. The results of this part of my research can be found in this paper:

**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]

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** and the **sphere packing problem** and all the beautiful math that can be connected to this problem, among others: **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]

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 (not active)

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 part 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** **manifold**s and **Riemmanian geometry** in data analysis. **I am currently not active in this topic**, **but I will be definitely in touch with smooth manifolds and Riemmanian geometry.**

### Computational Electromagnetism (not active)

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 of electrical engineering, can have in a given problem. **I am currently not active in this topic.**