USE OF KERNELS FOR HYPERSPECTRAL TRAGET DETECTION Nasser M. Nasrabadi Senior Research Scientist U.S. Army Research Laboratory, Attn: AMSRL-SE-SE 2800 Powder Mill Road, Adelphi, MD 20783, USA

2 Outline Develop nonlinear detection algorithms. Exploiting higher order correlations. Why Kernels Kernel Trick Conventional matched filters Kernel matched filters Detection results

Nonlinear Mapping of Data Exploitation of Nonlinear Correlations Nonlinear mapping Statistical learning (VC): Mapping into a higher dimensional space increases data separability However, because of the infinite dimensionality implementing conventional detectors in the feature space is not feasible using conventional methods Convert the detector expression into dot product forms Kernel-based nonlinear version of the conventional detector Input space High dimensional feature space Input space Kernel trick :

4 Kernel Trick Example of the kernel trick Consider 2-D input patterns, where If a 2nd order monomial is used as the nonlinear mapping This property generalizes for and

5 Examples of Kernels 1. Gaussian RBF kernel: 4. Polynomial kernel: 3. Spectral angle-based kernel: 2. Inverse multiquadric kernel: Possible realization of

Linear Spectral Matched Filter Spectral signal model target present target spectral signature, background clutter noise Linear matched filter design (signal-to-clutter ratio)  - Average output power of the filter for - Constrained energy minimization: no target, target present

To stabilize the inverse of the covariance matrix, usually regularization is used, equivalent to minimizing: The regularized matched filter is given by: Linear Spectral Matched Filter & Regularized Spectral Matched Filter Linear matched filter is given as:

8 Nonlinear Spectral Matched Filter In the feature space, the equivalent signal model The equivalent matched filter in the feature space Output of the matched filter in the feature space

Kernelization of Spectral Matched Filter in Feature space Using the following properties of PCA and Kernel PCA Each eigenvector can be represented in terms of the input data Inverse Covariance matrix is now Kernel matrix spectral decomposition (kernel PCA) The kernelized version of matched filter

10 Conventional MF vs. Kernel MF Conventional spectral matched filter Nonlinear matched filter Kernel matched filter

11 Matched Subspace Detection (MSD) Consider a linear mixed model: where and represent orthogonal matrices whose column vectors span the target and background subspaces and are unknown vectors of coefficients, is a Gaussian random noise distributed as The log Generalized likelihood ratio test (GLRT) is given by where

12 Kernel Matched Subspace Detection Define the matched subspace detector in the feature space To kernelize we use the kernel PCA, and kernel function properties as shown below

13 MSD vs. Kernel MSD GLRT for the MSD: Kernelized GLRT for the kernel MSD: Nonlinear GLRT for the MSD in feature space:

14 The model in the nonlinear feature space is The MLE for   in feature space is given as The kernel version of   is given as Orthogonal Subspace Projector vs. Kernel OSP

15 Consider a linear mixed model: where U represent an orthogonal matrix whose column vectors span the target subspace and C is the background covariance. is unknown vector of coefficients, is a Gaussian random noise distributed as The log Generalized likelihood ratio test (GLRT) is given by Adaptive Subspace Detection (ASD)

16 The model in the nonlinear feature space is The GLRT ASD in feature space is given as Where  is a nonlinear function. Substituting the following identities into the above Eq. Nonlinear ASD

17 ASD vs. Kernel ASD GLRT for the ASD: Kernelized GLRT for the kernel ASD: Nonlinear GLRT for the ASD in feature space:

18 A 2-D Gaussian Toy Example Red dots belong to class H 1, blue dots belong to H 0 (a)MSD (h) KSMF(b) KMSD (c) ASD (d) KASD (e) OSP (f) KOSP (g) SMF

19 Red dots belong to class H 1, blue dots belong to H 0 A 2-D Toy Example (a)MSD (h) KSMF(b) KMSD (c) ASD (d) KASD (e) OSP (f) KOSP (g) SMF

20 Test Images Forest Radiance I Desert Radiance II

21 Results for DR-II Image (a)MSD (h) KSMF(b) KMSD (c) ASD (d) KASD (e) OSP (f) KOSP (g) SMF

22 ROC Curves for DR-II Image

23 (a)MSD (h) KSMF(b) KMSD (c) ASD (d) KASD (e) OSP (f) KOSP (g) SMF Results for FR-II Image

24 ROC Curves for FR-II Image

25 SMF & KSMF Results for Mine Image Mine Hyperspectral Image KSMF for mine imageSMF for mine image

26 ROC Curves for Mine Image

27 Conclusions Nonlinear target detection techniques are valuable. Use of kernels and regularization in filter design. Choice of kernels? Nonlinear sensor fusion using kernels.