Chapter 20

1. Chapter 20 Matched Filter Comparison

2. Targets in a Structured Background The geometric interpretations of an dimensional data set suggests that the image space can be spanned by a set of p bases vectors according to (1)

3. When we use the subspace approach we can express the hypothesis as follows: (2) (3)

4. For the example used in this paper, Manolakis et al. 2001 used the first Q significant eigen vectors of the correlation matrix computed over the background scene (in this case, was equal to 10) to form U. Comment: note it is also possible to form Uby running a background end member selection approach such as PPI, MDM or N Finder. The end members form the columns of U. To simplify calculations, we could also run a SVD on the end member spectra to form an orthogonal set of vectors spanning the background subspace and use these to make up U.

5. adaptive subspace detector (ASD) given by: (4) where Pb and Ps are the projection operators nullify the background and the image space respectively i.e. (5) (6) where is the pseudo inverse of U and (7)

6. The target abundance value a (8)

7. The OSP detector developed by Harsanyi and Chang 1994 is the numerator of this abundance estimator. Manolakis et al. 2000 suggests using the normalized value since it provides the abundance estimate (9) If we don’t know anything about the background, we can use the spectral angle mapper defined as (10) the SAM algorithm is related to the matched filter for signals in white noise (MFWN) expressed as (11)

8. Detection of Targets in an Unstructured (i.e. Stochastic) Backgrounds In this case, we assume the clutter is composed of stochastic variation in the background plus system noise captured in the random variable v. The hypothesis can then be expressed as (12) or (13)

9. The analysis requires the following assumptions: 1) v is multivariate normal with zero mean (i.e. we need to demean the data) and has covariance matrix (un-normalized) given by (14) 2) we have a training region with a set of pixels which are independent, identically distributed (i i d) and target free 3) the test and training pixels are statistically independent The generalized likelihood ratio applied to this representation yields (15)

10. Two variants of the GLRT are the adaptive matched filter AMF (16) and the adaptive coherence estimator (17)

11. Note if we use the normalized or maximum likelihood estimate of the covariance matrix i.e. (18) then the detectors take on the following forms (19)

12. then the detectors take on the following forms con’t (20) (21)

13. Finally, we introduce the CEM which, provides the maximum likelihood estimates of target abundance given by: (22)

14. Test Methodology Manolakis et al. 2001 first point out the limitations (assumptions) of these approaches for: A: the structured background A1 the spectra can be modeled by linear mixing A2 the modeling error has uncorrelated components with the same variance A3 the modeling error is multivariate normal A4 the background subspace is perfectly known A5 the target vector is known Next page B: the stochastic background

15. B: the stochastic background B1 the background is homogeneous and normally distributed B2 the background for the target is the same as the training set B3 test and training pixels are independent B4 the target and background spectra are additive (additive verses replacement model) B5 the target vector is known

16. Manolakis Figure 1

17. Manolakis Figure 2

18. Manolakis Figure 5