IC Postdoctorial Research Fellowship Program - Physics-based Algorithms

Sponsor: National Intelligence Community

Research Team: Emmett Ientilucci (Post Doc), John Schott

Project Scope: This is a two year post doctoral grant focused on the development of improved physics based algorithms for target detection.

Sponsor: National Intelligence Community Research Team: Emmett Ientilucci (Post Doc), John Schott Project Scope: This is a two year post doctoral grant focused on the development of improved physics based algorithms for target detection. Project Status: In this portion of the post-doc, we are looking at alternative methods for describing background / foreground spaces. One such method is the design of a probabilistic target space used in conjunction with an unstructured pre-clustered background. Together, these background / foreground descriptions can be used as input to the powerful quadratic detector. To date, the approach to such a probabilistic target space has been derived. The approach involves the use of weighted moments. Let us assume we observe n realizations of a p-dimensional vector, which are stored as a p×n matrix X. These realizations are drawn from a normal probability distribution with density N(µ,σ). From this we calculate the probability density (the likelihood-per-unit x) of the parameter, f(x). For multiple parameters (e.g., visibility, water vapor, elevation, etc.) we then compute the product of the marginal normal densities. That is, the joint density (assuming independence) is computed as

where we have N parameters. The case of dependant parameters can also be dealt with. Since we have M combinations we let z_j=(X₁,X₂,...X_N) for j=1, 2,...M. We then turn each likelihood into a weight. Each j weight is computed as a fraction of the "total weight". That is

such that ∑w_j=1 now have a vector of j = 1, 2,...M scalars to weight each combination of visibility, water vapor, elevation, etc. Let w be the n-dimensional vector of those weights. The weighted mean is calculated as x☐_w=Xw and the weighted variance-covariance matrix as

where W is a diagonal n×n matrix with the vector w on the diagonal, and
X_c=X-x☐_w1^T is the matrix of centered data, while 1^T is an n-dimensional row vector of 1's. We can also write X_c≥X(I-w1^T).
Since W and (I-w1^T) are n×n matrices and n is usually large, the following procedure may be more computationally efficient:

1. Calculate X_c=X-x☐_w1^T
2. Calculate A=X_cW by multiplying the i-th column of for all X_c by w_i for all i = 1,...,n
3. Calculate S_w=AW^T_c
The usage of a weighted mean and covariance allows one to describe the target spaces in a probabilistic manor. This provides a mechanism related to "importance" of the most likely target-space vector, which does not exist in the traditional geometric descriptions of the target spaces. The implementation of this scheme on data is forth coming. Ongoing work will explore the incorporation of the probabilistic target spaces into the quadratic detector along with an alternative pre-clustering unstructured approach to background characterization (work that has already begun). This implementation will then be compared to previous, geometric, descriptions of background and foreground subspaces along with potential schemes incorporating feedback for overall semi-autonomous algorithm behavior. It is believed that this approach will be far more forgiving when it comes to sensor calibration errors than structured approaches. The net result will allow us to take advantage of all the previous work on physics based hyperspectral target detection using, what is expected to be, a more robust target detection algorithm.