Efficient adaptive subspace tracking algorithm for automatic target recognition P. Ragothaman, T. Yang, W.B. Mikhael, R.R. Muise and A. Mahalanobis Automatic target recognition using quadratic correlation filters has been reported recently. It requires the eigenvalue decomposition (EVD) of a large matrix computed using the autocorrelation matrices of target and clutter training images. In practice, situations arise where new images need to be incorporated, which perturbs the EVD. Proposed is a novel computationally efficient method to obtain the new EVD adaptively. Sample results using an infrared dataset illustrate the effectiveness of the technique. Introduction: In [1] a novel method is proposed for designing optimal quadratic correlation filters (QCFs) for shift-invariant target detection in infrared (IR) images. The task of detecting targets in background clutter is cast as a two-class pattern recognition problem. The class- separation metric is formulated as a Rayleigh quotient that is maxi- mised by the QCF solution. As a result, the means of the two classes are well separated while simultaneously ensuring that the variance of each class is small. In the training mode, sets of target and clutter sub- images, referred to as chips, are extracted from IR imagery. Auto- correlation matrices for target and clutter are computed. These matrices are appropriately combined to obtain a non-symmetric matrix A, as shown later. The eigenvalue decomposition (EVD) of A is performed. The dominant eigenvalues of A are l ti , corresponding to target, and l ci , corresponding to clutter, i ¼ 1 to k. The QCF coefficients are the set of eigenvectors w ti and w ci , corresponding to l ti and l ci . In the running mode, the QCF is correlated with a given IR image. It is able to recognise those targets that were used in training based on the response. Frequently, the system is required to incorpo- rate few new targets relative to the typically large number of data points used in training. Thus w ti and w ci become wˆ ti and wˆ ci , which are in the vicinity of w ti and w ci . In such cases, the target and=or clutter set has to be updated and the QCF has to be retrained. This involves the inversion and EVD of large matrices. Since this requires a large amount of computation it is desirable to have an adaptive subspace-tracking algorithm that reduces the amount of computation. There are several contributions in the literature that address the problem of adaptive EVD. Some of the more recent contributions are given in [2, 3]. These are typically formulated for the adaptive EVD of symmetric matrices. In this Letter, an efficient adaptive technique, in terms of speed and computational complexity without sacrificing accuracy, is proposed that utilises the old EVD to search for the new EVD. The technique, called the optimal adaptive eigenvalue decomposition (OAEVD) technique, avoids matrix inversion and direct EVD, thus providing substantial computational savings. In addition, the OAEVD adaptively updates any particular set of eigenvalues and corresponding eigenvectors of interest. In our application, these are the dominant eigenpairs. Proposed OAEVD technique: Since the OAEVD is based on the RQQCF technique [1], for the sake of completeness a brief summary of the RQQCF technique is given first. In the RQQCF technique, M target and M clutter training chips are obtained from IR imagery. Each chip, having dimensions p n  p n, is converted into a 1-D vector of dimensions n  1 by concatenating its columns. Target and clutter training sets of size n  M each, are obtained by placing the respective vectors in matrices. The n  n autocorrelation matrices of the target and clutter sets R x and R y are computed and used to obtain a matrix A given by, A ¼ðR x þ R y Þ 1 ðR x  R y Þ ð1Þ As a result of (1), the eigenvalues of A vary from –1 to þ1. The dominant eigenvalues for clutter, l ci , are close to or equal to 1 and those for targets, l ti , are close to or equal to þ1. The RQQCF coefficients, w ci and w ti , are mapped to the corresponding eigenvalues. To identify a data point as target or clutter, the sum of the absolute value of the k inner products of the data point with w ci and w ti , p t and p c , are calculated. If p t > p c , the data point is identified as a target. Otherwise, it is identified as clutter. When new data points have to be incorporated, R x and=or R y change, and A has to be recomputed. In addition, the EVD of A has to be performed again. Hence an efficient adaptive approach is proposed. In the following formulation, quantities in lower case with an underscore are vectors, and quantities in upper case are matrices. The first step in the adaptive formulation is to identify a ‘Cost Function’ to be minimised when new data is incorporated. According to (1), the following is chosen as the ‘Error Signal’ at the jth iteration: eðjÞ¼½R x  R y  l i ðjÞðR x þ R y Þ w i ðjÞ ð2Þ where e( j) is n  1, l i and w i are the ith eigenvalue and the correspond- ing eigenvector, respectively. The energy in the error signal, e T ( j)e( j), is the cost function to be minimised. The error signal contains both l i and w i . It can be easily seen that the error surface is multimodal. In addition, in practice, the new l i ’s and w i ’s, l ˆ i ’s and wˆ i ’s, are in the vicinity of the old ones. Thus, in the adaptive algorithm the l i ’s and w i ’s are updated in an alternating manner. In each iteration, l i ( w i ) is adjusted while w i (l i ) is kept constant. To obtain l i ( j þ 1) ð w i ð j þ 1ÞÞ , the most recent updated w i ’s (l i ’s) and l i ’s (w i ’s) are used. Again, since the l ˆ i ’s and wˆ i ’s are close to the old ones, only the first-order terms are retained in the Taylor series expansion for eð j þ 1Þ in terms of e( j). This yields e l ðj þ 1Þffi e l ðjÞþ P M k¼1 @e l ð jÞ @w i;k ð jÞ Dw i;k ðjÞþ @e l ðjÞ @l i ðjÞ Dl i ðjÞ ð3Þ where l ¼ 1, ... , n. Writing (3) for l ¼ 1, ... , n: eðj þ 1Þffi eðjÞþðS 1  l i ðjÞS 2 Þ Dw i ðjÞ S 2 w i ðjÞDl i ðjÞ ð4Þ where, S 1 ¼ R x  R y ; S 2 ¼ R x þ R y . Now, to obtain l i ( j þ 1), the most recent w i is used. Therefore (4) becomes eðj þ 1Þffi eðjÞ S 2 w i ðjÞDl i ðjÞ ð5Þ To obtain w i ðj þ 1Þ , the most recent l i is employed. Thus, (4) becomes eðj þ 1Þffi eðjÞþðS 1  l i ðjÞS 2 Þ Dw i ðjÞ ð6Þ In the steepest descent adaptation, D w i ( j) and Dl i ( j) are proportional to the negative gradient of the cost function in the jth iteration, e T ( j)e( j), with respect to w i ( j) and l i ( j), respectively. Therefore, Dl i ðjÞ¼k l i @ð e T ðjÞeðjÞÞ @l i ðjÞ ¼2k l i w T i ðjÞS 2 ðl i ðjÞS 2  S 1 Þ w i ðjÞ ð7Þ Dw i ðjÞ ¼ ½MU  j @ð e T ðjÞeðjÞÞ @ w i ðjÞ ¼ 2 M ½MU  j S eðjÞ ð8Þ where k l i is a real scalar, and [MU] is a diagonal matrix with real, positive entries. From (5), (6), (7) and (8) it can be shown that the update equations are: Dl i ðjÞ¼ V T ðjÞeðjÞ V T ðjÞV ðjÞ w T i ðjÞS 2 ðl i ðjÞS 2  S 1 Þ w i ðjÞ Dw i ðjÞ ¼ ½R 1 j qðjÞ where V T ( j) ¼ S 2 w i ( j)(l i ( j)w i T ( j)S 2  w i T ( j)S 1 )S 2 w i ( j), q( j) ¼ S e( j) and [R] j ¼ [S] 2 . The above algorithm avoids the EVD required for the updated A. As a further computational reduction in the proposed adaptive solution, [R] 1 is approximated by a diagonal matrix B containing the reciprocals of R’s diagonal elements, r ii (i ¼ 1 to n). Additionally, a convergence factor m is introduced in the update equations to ensure reliable convergence. Thus Dl i ðjÞ¼ m V T ðjÞeðjÞ V T ðjÞV ðjÞ w T i ðjÞS 2 ½l i ðjÞS 2  S 1  w i ðjÞ Dw i ðjÞ¼mB qðjÞ Simulation results: The adaptive algorithm was tested using IR data. This data consists of an IR video sequence containing 388 frames, each of size 126  126, of a target (tank) as a camera approaches it. A sample frame is shown in Fig. 1. Also, M, n and k are 350, 256 and 6, respectively. The matrix A is computed according to (1). The EVD of ELECTRONICS LETTERS 28th September 2006 Vol. 42 No. 20