In the last five years I have been working in the following reseach areas:
 Compressed Sensing is a new acquisition scheme that aims to perform the sampling and the compression simultaneously. Instead of sampling the signals (like images) at the Nyquist rate and compressing them afterwards, compressed sensing tries to sample the signal at the 'information rate'. This lets the sensing devices sample faster and\or with a fewer number of sensors. While the main issue that has been addressed by compressed sensing is acquisition, the application of the tools developed in this field have  gone far beyond that.
As mentioned before, compressed sensing is faster than the traditional sensing, but this comes at a price. Reconstruction of the signal from the measurements is more complicated than the interpolation schemes used in the traditional sampling devices. The first proposed recovery method $\ell_1$ is very nice from the theoretical point of view and has the best 'performance' at least among the rigorous algorithms that are also robust to noise. But due to its high compuational cost it is not useful for even a medium size problem with tens of thousands of variables and thousands of measurements. 
   My main research interest in compressed sensing has been focused on developing faster recovery methods. In a joint work with David Donoho and Andrea Montanari, we have proposed a new fast recovery algorithm, which we call AMP. The AMP algorithm while faster than the $ell_1$, exhibits the same phase transition as the phase transiton of $ell_1$. From the theoretical point of view, I believe this algorithm is even nicer than $ell_1$ and can be analyzed more conveniently. You may refer to the publication list for more information.
 Although an old area of research, image processing is still an exciting field with lots of open practical and theoretical problems. My research in this area has mainly been focused on image representation and compression. It is well-known that the wavelet transform is better than the Fourier transform in representing the geometries of images (this is the main reason for the adoption of wavelet for JPEG2000), but they are for sure, not the "optimal" transform.  Intuitively speaking the 2D separable wavelet transform is very efficient in representing the point singularities but it is blind to the connectivity of the singularities and to the smoothness of edges. This makes the 2D wavelet inefficient in representing two dimensional geometries.
    A few other dictionaries like ridgelet, curvelet, shearlet, contourlet have been proposed to overcome this problem. The main drawback of all these systems is their overcompleteness that makes them inappropriate choices for compression. In a recent paper we proposed a system that is able to get close to the rate-distortion behavior of certain image geometries and this compression schme outperforms wavelet on these geometries. I believe by extending this approach to other geometies such as texture it is possible to outperform wavelet with a large margin. Check the publications list for more information.
Again this is a very broad and active area of research. Being very interested in this field, I have been involved in some researches in this area that are mentioned below and I am getting involved in other problems that may appear here later:
I believe that reproducible research is more than a slogan, and I tried to stick to this concept during my PhD. You may check the website of Wavelab, or 'optimal tuning' paper. They are both mentioned in the publications list. You may also check our paper on the cons and pros of the reproducible research which appeared in "Computing in Science and Engineering". 



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