Inverse Tomography

A few years ago I had the pleasure to attend Pedro Goldman's presentation (Argentinian scientist) about his research in inverse tomography and applications to the calculation of the optimal dosage of radiation for radiotherapy.

Fast Optimization or the Radiation Therapy of Tumors -the Impossible Possible


Sitting down on my chair I would not expect that this was going to be one of the most shocking and interesting presentations I ever attended.

This research field is mathematically interesting and extremely important for medical physics.

Computational Law

I did not know about this this research field until my last trip to Beijing China where I met Set Chandler, who develops his programs in Mathematica

http://www.wolfram.com/products/mathematica/portraits/sethchandler/

The general idea is to develop models as response of the implementation of the laws. The idea is to minimize the subjectivity in today's approach.

A much more ambitious project is to analyse the self consistency of the laws and their level of complexity. The complexity does not only depend on the document's length but also in how intricate is the relation with itself and other laws.

Videos created in Mathematica

I just learned about a video channel for videos created in Mathematica.

http://vimeo.com/channels/mathematica

Medical Imaging

Medical Imaging is today a very active multidisciplinary research field where differential geometry, high performance computing  and diverse mathematics meet together.

In my trip to China I had the opportunity to see some of the latests developments at professor Bart M. ter Haar Romeny's group at the University of Eindhoven in Holland.

Some of the lastest thesis in his group that can be donwloaded are

• E.M. Franken, Enhancement of Crossing Elongated Structures in Images, PhD. Thesis, 2008, Technische Universiteit Eindhoven. Advisors: B.M. ter Haar Romenij, R. Duits

• M.A.van Almsick, Context Models of Lines and Contours, PhD. Thesis, 2007, Technische Universiteit Eindhoven. Advisors: B.M. ter Haar Romeny, L.M.J. Florack

• E. Balmashnova, Scale-Euclidean Invariant Object Retrieval, PhD. Thesis, 2007, Technische Universiteit Eindhoven.Advisors: B.M. ter Haar Romenij,co-advisor: L.M.J. Florack

• F.M.W. Kanters, Towards Object Based Image Editing, PhD. Thesis, 2007, TU/e. Advisors: B.M. ter Haar Romenij, L.M.J. Florack

• B. Platel, Exploring the Deep Structure of Images, PhD. Thesis, 2007, Technische Universiteit Eindhoven. Advisors: B.M. ter Haar Romenij, Co-advisor: L.M.J. Florack

Additionally, professor Romeny has a book entitled


Front-End Vision and Multi-Scale Image Analysis: Multi-scale Computer Vision Theory and Applications, written in Mathematica

Online version

The complete package can be found at
http://extras.springer.com/2003/978-1-4020-1507-6

If you are not afraid of modestly advanced mathematics and want something really applied, try this.

The canonical coset decomposition of unitary matrices through Householder transformations

My paper describing the connection between the Householder decomposition of and the canonical coset parametrization of unitary operators was published today in the Journal of Mathematical Physics

J. Math. Phys. 51, 082101 (2010)

http://arxiv.org/abs/1008.2477

This parametrization is very useful for calculating the parameters that uniquely characterises a given unitary operator U(N). I show that instead of solving a complicated systems of non-linear equations one can perform very stable and efficient Householder transformations.  Right now I am working on some practical applications based on that result.

Independent Component Analysis

I was already familiar with the Principal component analysis, which is a method for finding a linear combination of variables such that the possible correlation between them disappears i.e. elimination of the diagonal elements of the covariance matrix. Statistical independence implies the elimination of correlation but the opposite is not necessarily true because there is possible association between the variables on higher orders.  The Independent Component Analysis seems to be the generalization needed to go beyond the simple correlation.

• Bartlett, M.S. and Movellan, J.R. and Sejnowski, T.J., Face recognition by independent component analysis, IEEE Transactions on neural networks,13, pp 1450--1464, 2002

• Hyvarinen, A. and Oja, E. Independent component analysis: a tutorial, Neural Networks, 13, pp 411--430 2000

10th International Mathematica Symposium

I just returned from the 10th International Mathematica Symposium held in Beijing China, where I gave a talk. It was a great opportunity to see the work of other people around the world and the new features of Mathematica 8.

Some of the topics included medical imaging, stochastic simulations, video editing, real-time control, automatic C code generation, CUDA and more.