Lucas Parra - Previous Work

 
MEG for brain computer interface (2000 - 2001) 
We have been interested recently in using magneto-encephalography as a non-invasive modality for the direct communication between the brain and the machine. An important first application of such a brain computer interface (BCI) is as a communication device for severely paralyzed subjects. By linearly combining 122 MEG channels we have identified and localized sources that predicts a left or right thumb button push 30 ms prior to its execution. 
   

 

Left: Mean and variance of discriminating activity over time (in seconds). Activity is linear combination of 122 ME sensor with optimal discrimination between left and righ button push. Signal between green bars is used for discrimination. Right: Single trial discrimination performance shown as receiver operator characteristic.

Projection of discrimnating MEG activity on scalp.

Statistics of natural signals (2000 - current)
As a consequence of much of my work of the last few years  we have realized that the property on non-stationarity of natural signals can explain a number of important higher order statistics. This includes higher order properties that are similar for a variety of natural signals. These properties are well documented, but their universality has not been given much consideration nor explanation. For example we find that speech signals, stock market data, and typical features of natural images all have high kurtosis that can be purely described by second-order non-stationarity.
 

Acoustic source separation (1997 - current) 
The task is to recover acoustic sources by using multiple microphones . The acoustic source can be a user giving spoken commands to a computer, ventilator noise, the music playing in the background, the neighbor's dog barking in the backyard, etc. Mostly we are interested in the user commands, and want to remove the other interfering signals. There are two main difficulty in recovering and "cleaning" these acoustic sources: 1. reverberations, i.e. the reflections of the sounds on walls, floor, ceiling, furniture, and so on. Reverberation requires that we not only combine the microphones with different weighting, i.e. spatial unmixing, but also that we combine signals that arrived with different delays, i.e. temporal deconvolution. 2. The signals are broad band. It turns out that for narrow band signals one can solve the reverberation problem by converting the signal into the frequency domain, and perform spatial unmixing for the specific frequency band. The fact that the signals are broad band requires that we solve the problem for all frequencies at once. The statistical property used here is non-stationarity . For non-stationary signals there can be only one correct set of separating filters that decorrelate the sources at all times. Currently we are working on increasing the robustness of these separation methods by including prior geometric information. 

listen to microphone 1 listen to microphone 2	listen to model source 1 listen to model source 2
listen to microphone 1 listen to microphone 2	listen to model source 1 listen to model source 2

These are two examples of recording an separation performed in real rooms. Signals are recorded at 8 kHz and have been filtered with filters of 1024 taps. Find here more audio demonstration . 

 Spectral unmixing (1999) 
Hyperspectral imagery consists of images taken simultaneously at multiple wavelengths (>200). Typically pixels cover large areas that may contain a combinations of different surface materials (minerals, vegetation, water, etc.). Therefore, the spectra are an overlap of different, and often unknown, reflectance spectra. To identify the material and their corresponding abundance automatically we use prior information based on the physics of the problem. For example we know that the mixture coefficients are positive since there can never be less than 0%. In addition we exploit some statistical properties of typical spectra such as independence of spectra after removing the linear trend (innovation process). We are currently studying the applicability of these spectral analysis methods to medical applications.

Simulation results. Left: Reflectance spectra (endmembers) blindly identified from a mixture of 100 pixels spectra. Center: Target reflectance spectra used in the mixture (unknown to the algorithm). Note almost perfect correspondence with targets. Right: Points represent pixel intensities for two wavelengths. Circles represent reflectance of the endmembers. Note that the points (must) lie all within a simplex spanned by the endmembers. 

Hierarchical image probability model (1998 - current) 
A typical pattern recognition task consists in deriving a classification rule from a set of example images. This is particularly useful for patterns that are not well defined such as tumors and micro-calcification in mammograms, or targets in SAR images (synthetic aperture radar). Rather than building an actual classifier one can also build a probability model for the example images. Classification is then based on the likelihood a images in question has under the different models (likelihood ratio). The advantage of such an approach is that the same models may also be used for compression or sampling, i.e. we can synthesize an images that are likely according to the examples used by the model. This last feature is useful because it allows us to "debug" the models. The synthesized examples have to look like the training examples, otherwise there is something wrong with the model. Motivated by much previous research and some careful math we have designed a probability model which can be considered a Hidden Markov model on a tree for the wavelet coefficients of an image pyramid. There is a few fine points such as: we introduce emission probabilities similar to what is used in speech recognition; the features values at one resolution condition the feature values at a higher resolution; the feature pyramid is sub-sampled such that we obtain a probability model with the same degrees of freedom than the number of pixels. As a result the model is an explicit probability distribution over images. The important point, however, is that we are capturing in an explicit and efficiently computable probability model a number of important statistical properties of natural images, such as non-stationarity, and long range correlations.

Image I₀ is low-pass filtered and sub-sampled to generate a 'Gaussian pyramid'. Features F are extracted at various resolutions. Sub-sampling gives us as many degrees of freedom as in the original image such that we can set Pr(I ₀) = Pr(G₀,G ₁,G₂,...). After some manipulations and assumptions this becomes prod _i Pr(G _i|F _i+1). 

Information that is being ignored by the assumptions above such as long rage dependencies may be captured by introducing hidden variables A_i that can (due to their dependency structure) propagate information over lager distances: Pr(I ₀) = sum_A prod _ii |F_i+1,A_i ) Pr(A_i|A _i+1 ).  See the papers for proper math. 

 PET source reconstruction with list-mode likelihood (1997-1998) 
In nuclear medicine the conventional approaches for source density reconstruction start often by binning the observed events, i.e. event counts are accumulated for events for which measured properties fall within certain bins (small range of locations, time, energy, etc.). The event counts are converted to the source distribution through a linear inverse problem. The binning process can be circumvented if a probability model of the physical imaging process is formulated that assigns to every measured continuous coordinate a corresponding likelihood of occurrence. Maximum likelihood (ML) and in fact the expectation maximization (EM) algorithm can then be used to recover the source distribution. This was demonstrated on time-of-flight PET (positron emission tomography). The same list-mode likelihood formalism and the corresponding EM algorithm can be applied in any other nuclear medicine modality. Once again, this demonstrates that a probabilistic model of the physical world allows to reconstruct the underlying sources.

Time-of-flight PET measures coincident gamma quanta at positions x₁ and x ₂ and the time of arrival difference t. Probabilistic modeling of the measurement process allows reconstruction directly from those measurements.
 

Reconstruction simulations for 200K and 1M events of a head phantom with typical measurement errors.

Compton reconstruction (1995-1999) 
The Compton camera has been proposed as a new modality for nuclear medicine. The reconstruction problem consists in deducing from multiple Compton scatter events the originating 3D source distributions. Not unlike CT or PET here the ambiguity of where the events originated has to be inverted. In CT or PET each event can have originated along a line, while here events can originate from anywhere on a cone surface. The proposed approach is in a sense a classic linear inverse filtering such as in any other tomographic method. The novelty lies in the formulation of the problem in terms of spherical convolution and inverse filtering in the domain of spherical harmonics rather than the Fourier domain of Cartesian coordinates. It shows how careful math can solve seemingly impossible inversion problems.

Compton scatter determines origin * of event only up to a cone surface.  

Key step in the reconstruction is the back-projection of all event cones onto a sphere and inverse filtering of the resulting summation image.

Lucas Parra, May 21, 2002