Research interests

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. Although the main issue that has been addressed by compressed sensing is acquisition, but the application of the tools developed in this field have has gone far beyond that.
As mentioned the "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 able to handle noisy situations properly. 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 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 although faster than the $ell_1$, it 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 they are 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. 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 actually getting involved in other problems that may appear here later:

Manifold Learning and Clustering: Inspired by the manifold learning algorithm ISOMAP we constructed a density weighted distance between any two points. We theoretically and empirically showed that this type of distance works well for clustering and classification problems. We compared the performance of our algorithm with the performance of some of the state of the art algorithms like spectral clustering.
Covariance Regularized Regression: Coming soon.

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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