Research Interests
My current research interests are in signal representations, quantization and sampling. In general I am open minded and interested in most signal processing problems, especially at the intersection with machine learning. A brief presentation of my recent research follows.
Compressive Sensing, Random Projections and Sparse Representations
Compressive sensing is a new signal acquisition technique for sparse and compressible signals. Rather than uniformly sampling the signal, compressive sensing computes inner products with a few basis functions-often randomly generated-in a way that captures most of the signal information. The signal is then recovered using a convex optimization.
Compressive sensing combines several useful ideas to produce powerful results. Randomness is often used to combat worst-case adversarial conditions, and to ensure robustness to model selection. Sparsity is used as a prior to guide the signal reconstruction. Incoherence is used to guarantee that the acquisition process captures all the necessary signal information. Each of these concepts is important on its own but their combination provides a promising new sampling framework.
Compressive sensing is a broad field with a number of to date open research avenues. My current interests are mostly in issues of quantization and practical data acquisition. Quantization of the samples is an important unavoidable step in any form of digital processing. The interplay of quantization with compressed sensing is not well understood beyond simple additive models. Practical implementation of the sampling operators is also challenging. Techniques from classical sampling theory, such as Sigma-Delta quantization, can be used to implement compressive sensing systems and improve their performance under quantization. Nonetheless, immediately applying such techniques is not possible due to the nature of the compressed sensing sampling operators. Part of my research involves adapting such techniques to compressive sensing systems.
Frames and Robust Representations
Frames are redundant generalizations of bases that have recently received significant attention in the signal processing community. Frame expansions have been proven useful in providing robustness to quantization, erasures, and other types of signal degradation. My doctoral work examines how projections can be used in frame representations to provide robustness to quantization and erasures with very low computational complexity.
An example of a frame expansion is oversampling of bandlimited signals. Oversampling is most often studied in the context of Sigma-Delta quantization noise shaping--a quantization technique that has proven very useful in the design of D/A and A/D converters. Our work views Sigma-Delta noise shaping as a sequence of compensations of the quantization error using projections. This interpretation allows the generalization of Sigma-Delta noise shaping to arbitrary frame expansions. Furthermore, the idea of compensating the error using a projection is extended to compensate for erasure errors. In this case the erasures can be intentional or anticipated or neither.
The ideas presented in this work can also be applied in a variety of contexts, such as sparse representations and sparse sampling. Future work will examine these applications.
Estimation of Wrapped PDFs
A recurring problem in signal processing is unwrapping the phase of the transform of a signal. This problem can arise in system identification and cross-spectral frequency response estimation in broadband microphone arrays. If the learning algorithm uses the frequency domain magnitude and phase data, the wrapped phase may potentially confuse the algorithm, producing incorrect estimates of the parameters.
In order to estimate the parameters, Paris Smaragdis at MERL and I, side-stepped the phase unwrapping issue by explicitly including a wrapped Normal PDF in the model for the phase. The model is estimated using an E-M algorithm, similar to the one used in estimating the parameters of Gaussian Mixture Models. It is also combined with Hidden Markov Models to learn target trajectories instead of just locations.
Basecalling using Hidden Markov Models
One step in automated genome sequencing is the electrophoresis sequencing reaction. With modern equipment, the output of this reaction is a four channel signal, with each channel corresponding to each of the four bases in the DNA molecule, namely A, T, C, and G. Each of the signals contains peaks at locations in which the corresponding base exists in the molecule being sequenced. The goal of the basecalling process is to process these signals and determine the correct sequence of A, T, C, and G for the DNA fragment being sequenced. The quality issues in the signal are usually channel crosstalk, additive noise, and spurious peaks. The quality of the read deteriorates significantly near the end of the sequence. Modern equipment can sequence up to 1000 base-pair long molecules.
The basecalling problem is similar in many ways to the speech recognition problem, thus in this work we propose a Hidden Markov Model framework as a solution. The model uses artificial neural networks to model the state emission densities of the HMM, instead of the mixtures of Gaussians usually used in these cases. Consequently the Baum-Welch training procedure is also modified to incorporate the training of the neural net using the backpropagation algorithm. The classification results on sample data show significant improvement compared to the state of the art algorithm. This work was conducted in the Whitehead Institute for Biomedical Research for my master's research.