Target Detection in Hyperspectral Imaging

Summary:

The goal in a target detection algorithm is to detect every pixel in the image which contains a significant portion of spectra of a known type, called the target spectra. There are two classes of standard target detection algorithms. The structured, or subspace, algorithms attempt to find a linear subspace of the k-dimensional space containing all the background pixels of the image. Target pixels are expected to be outside of this space in the direction of the target spectra. So for each pixel in the image, the angle between the vector from the line orthogonal to the linear subspace to the test pixel and the vector from the linear subspace to the target pixel is computed. Small angles indicate pixels whose spectral is similar to the target. Statistical, or unstructured, algorithms work in a similar manner using the assumption that the background is Gaussian, not linear. Each test pixel in the image is matched to a target spectra using a normalized Mahalanobis distance between the test and target spectra. This is equivalent to projecting the data onto the principle component vectors, called whitening the data, and then computing the difference between the test spectra and target spectra in whitened space. This works very well when the data has a Gaussian distribution, a condition that is sometimes met by simple hyperspectral scenes.

The standard algorithms for target detection can be extremely effective. For example, it is sometimes possible to find a single pixel in a image of a forest with a given paint signature even if the object is beneath the trees and only a small percentage of photons reaching the sensor come from the given paint. Again, it is possible to identify a few pixels out of millions that contain trace amounts of a hazardous chemical. However, performance is not consistent, and can be particularly poor in heterogeneous scenes which have multiple nonlinear distributions of spectra. The matched filters are optimized to detect signal from white noise, but the background in HSI comes from the phenomenology of the imaging, not noise.

Topology can be used to greatly improve the performance of target detection algorithms. The primary idea is to use topology to identify anomalies in an image that decrease the Gaussian distribution of the data and compute the covariance matrix using only the non-anomalous pixels. In short, we use topology to remove the nonlinear portions of the image to get a better estimation of the background.