Closed-form detector for solid sub-pixel targets in multivariate t-distributed background clutter
CCLOSED-FORM DETECTOR FOR SOLID SUB-PIXEL TARGETSIN MULTIVARIATE T -DISTRIBUTED BACKGROUND CLUTTER James Theiler a , Beate Zimmer b , and Amanda Ziemann aa Space Data Science and Systems Group, Los Alamos National Laboratory, Los Alamos, NM b Department of Mathematics and Statistics, Texas A&M University–Corpus Christi, Corpus Christi, TX
ABSTRACT
The generalized likelihood ratio test (GLRT) is used toderive a detector for solid sub-pixel targets in hyperspectralimagery. A closed-form solution is obtained that optimizesthe replacement target model when the background is a fat-tailed elliptically-contoured multivariate t -distribution. Thisgeneralizes GLRT-based detectors that have previously beenderived for the replacement target model with Gaussian back-ground, and for the additive target model with an elliptically-contoured background. Experiments with simulated hyper-spectral data illustrate the performance of this detector invarious parameter regimes. Index Terms — Adaptive signal detection, algorithms,data models, detectors, multidimensional signal processing,pattern recognition, remote sensing, spectral image analysis.
1. INTRODUCTION
To detect small targets in cluttered backgrounds requiresmodels of the target, of the background, and of how the twointeract. Although target variability models are important,particularly for solid targets [1], we will take the targetsignature as a single vector. However, we will consider arange of elliptically-contoured background models, fromGaussian to very fat tailed, and we will consider two differenttarget-background interaction models – the additive modeland the replacement model – that incorporate variability intothe strength of the target.
The importance of background modeling has recently beenemphasized [2], and although the Gaussian model is oftensurprisingly effective, a useful extension is the multivariate t -distribution, which has in particular been proposed for hyper-spectral imagery [3]. This is similar to the Gaussian in that itis defined by a mean and a covariance matrix, which impliesthat the distribution is uni-modal with ellipsoidal contours of This work was supported in part by the United States Department ofEnergy NA-22 project on Hyperspectral Advanced Research and Develop-ment for Solid Materials (HARD Solids). constant density; this density decreases with distance from themean, but the decrease can be much slower than the exp( − r ) decay exhibited by Gaussians, leading to heavier tails that areoften more representative of observed data. In many signal detection applications, the signal of interest isassumed to be additive with respect to a background that isgenerally characterized in some statistical way. We can write x = z + α t (1)where x ∈ R d is the measured signal, z ∈ R d is the back-ground signal, t ∈ R d is the signal of interest, d is the numberof spectral channels, and α is a scalar quantity that character-izes the strength of the signal. The additive model is the basisfor many traditional target detection algorithms, including theadaptive matched filter (AMF) [4] and the adaptive coherenceestimator (ACE) [5]. When the background is multivariate t , then the GLRT solution for the additive model leads to adetector we will here call the elliptically-contoured adaptivematched filter (EC-AMF) [6]; it is given by D ( x ) = (cid:112) ( ν − t T R − ( x − µ ) (cid:113) ( ν −
2) + ( x − µ ) T R − ( x − µ ) , (2)where µ is the mean and R is the covariance matrix of thebackground distribution. In Eq. (2), ν → ∞ leads to theAMF detector, and ν → leads to the ACE detector.While the additive model is the basis for many targetdetection algorithms, it has limitations [7], and in particulardoes not account for the occlusion of the background bya solid target. In the replacement model [8, 9], we treat ≤ α ≤ as the target area (fraction of a pixel), and write x = (1 − α ) z + α t . (3)This is called the replacement model because a fraction α ofthe background signal z is replaced with target signal t . The In [6], it is given a different name (EC-GLRT) that is not as consistentwith the naming conventions used in this paper. a r X i v : . [ c s . C V ] A p r nite target matched filter (FTMF), derived by Schaum andStocker [10], is the replacement-model version of the AMF:it is the GLRT solution to Eq. (3) in the situation that thebackground z is Gaussian. Although this detector is some-what more complicated than the AMF, or even the EC-AMFin Eq. (2), it can be written as a closed-form expression.Closed-form generalizations of the FTMF have beenderived for Gaussian target variability [11] and alternativemodels of covariance scaling [12]. Here, we derive a closed-form GLRT solution when the background is a general classof elliptically-contoured distribution. In the special case thatthe background is Gaussian, we obtain the FTMF solution.We treat the target detection problem in a hypothesistesting framework, with the null hypothesis correspondingto α = 0 and the alternative associated with α > . Sincethe nonzero α is unspecified, this is a composite hypothesistesting problem [13], and we use the generalized likelihoodratio test (GLRT) to derive our detector. The detector is afunction of x given by the logarithm of this ratio D ( x ) = log max α p x ( x | α ) p x ( x | . (4)The expression in Eq. (4) is written in terms of p x , which isthe probability density function for x . We can express thisfunction in terms of p z ( z ) , the probability density associatedwith the background z . We have p x ( x | α ) = (1 − α ) − d p z (( x − α t ) / (1 − α )) (5)where the argument ( x − α t ) / (1 − α ) is obtained by solvingEq. (3) for z , and where the prefactor (1 − α ) − d arises fromthe Jacobian of the transformation of variables from p x to p z .Taking z to have mean µ and covariance R , the multi-variate t distribution is given by p z ( z ) = c | R | − d/ (cid:32) z − µ ) T R − ( z − µ ) ν − (cid:33) − d + ν (6)where d is number of spectral channels, ν is a parameter thatspecifies how fat-tailed the distribution is (larger ν is less fat-tailed, with the ν → ∞ limit corresponding to a Gaussiandistribution), and the normalizing constant c depends only on d and ν . Thus, p x ( x | α ) =(1 − α ) − d p z (( x − α t ) / (1 − α )) (7) = c | R | − d/ (1 − α ) d (cid:32) w T R − w (1 − α ) ( ν − (cid:33) − d + ν (8)where w = ( x − µ ) − α ( t − µ ) . To find the value of α thatmaximizes Eq. (8), we can take the derivative of log p ( x | α ) with respect to α , set that expression to zero, and solve for α .For Gaussian p z ( z ) , that approach was found [10] to producea quadratic equation in α . For the more general multivariate t -distribution, we also obtain a quadratic equation, though with Table 1 . Taxonomy of detection algorithms. The EC-FTMF(and its special case FTCE) are introduced in this paper, toextend the FTMF algorithm to non-Gaussian backgrounds.Gaussian Multivariate t Fat-tailedTarget model ν → ∞ ≤ ν ≤ ∞ ν → Additive AMF [4] EC-AMF [6] ACE [5]Replacement FTMF [10] EC-FTMF FTCEmodified coefficients. The solution to that quadratic equationis given by (cid:98) α = 1 − − B + (cid:112) B − AC A (9)where A = ( t − µ ) T R − ( t − µ ) + ( ν − , (10) B = (1 − ν/d )( x − t ) T R − ( t − µ ) , (11) C = − ( ν/d )( x − t ) T R − ( x − t ) . (12)This value of α satisfies p x ( x | (cid:98) α ) = max α p x ( x | α ) . Thus, ourdetector, the elliptically-contoured finite target matched filter(EC-FTMF), is given by D ( x ) = log p x ( x | (cid:98) α ) − log p x ( x | (13)with p x ( x | α ) given in Eq. (8) and (cid:98) α given by Eqs. (9-12).In the ν → ∞ limit, the multivariate t becomes Gaussian,and the expressions in Eqs. (10-12) diverge. But in Eq. (9)it is only the relative values that matter; thus we can expressthis limit with the expressions B/A = − ( x − t ) T R − ( t − µ ) /d, (14) C/A = − ( x − t ) T R − ( x − t ) /d. (15)These values recapitulate the FTMF result obtained for aGaussian background [10].For ν → , we have the heavy-tailed limit A = ( t − µ ) T R − ( t − µ ) , (16) B = (1 − /d )( x − t ) T R − ( t − µ ) , (17) C = − (2 /d )( x − t ) T R − ( x − t ) , (18)which we call the finite target coherence estimator (FTCE).We remark that these three replacement-model detectors,the general EC-FTMF and the special cases FTMF and FTCE,have corresponding detectors associated with the additivemodel in Eq. (1), as shown in Table 1. These additive-modeldetectors are the EC-AMF and its special cases, AMF andACE. For very small α and very large target magnitude | t | , we expect these replacement-model detectors to be wellapproximated by their associated additive-model detectors. Inthis sense, we can argue that the EC-FTMF detector describedin Eq. (13) covers all six cases. M a t c h e d F il t e r MFR -8 -7 -6 -5 -4 -3 -2 -1 False alarm rate0.00.20.40.60.81.0 D e t e c t i o n r a t e ROCEC-FTMFEC-AMFFTMFAMFFTCEACE
Fig. 1 . Top panel is Matched-Filter Residual (MFR) plot ofsimulated data, showing both background (blue) and target(red) pixels, along with the contours associated with severaldifferent detection algorithms. The contours are chosen sothat the detection rate is exactly 0.5; the better detectors arethose with fewer false alarms, which are associated with bluepixels that are “above” the contours. Bottom panel is ReceiverOperating Characteristic (ROC) curves for these detectors.Here, ν = 10 , d = 90 , T = 3 , and α = 0 . .
2. SIMULATION
We can illustrate the performance of the EC-FTMF detectoron simulated data. In this simulation we draw N samplesfrom a d -dimensional multivariate t -distribution parameter-ized by ν , with (for simplicity) zero mean and unit covari-ance. These N samples are representative of background pix-els from a multi- or hyper-spectral image that have been de-meaned and whitened.For each background sample, we used the matched-pairformulation [14, 15] to produce an associated target pixel,produced by the replacement model in Eq. (3) using a fixedvalue of α (which we know, but the algorithm does not). Ourtarget signature t is given by a vector of magnitude T .The background and target pixels are d -dimensional M a t c h e d F il t e r MFR -8 -7 -6 -5 -4 -3 -2 -1 False alarm rate0.00.20.40.60.81.0 D e t e c t i o n r a t e ROCEC-FTMFEC-AMFFTMFAMFFTCEACE
Fig. 2 . Here, ν = 10 , d = 10 , T = 30 , and α = 0 . . Thetarget is strong ( T (cid:29) ) and small ( α (cid:28) ), so the effectdue to occlusion is limited, but still discernible. Here, theadditive-model detectors do almost (but not quite) as well asthe corresponding replacement-model detectors.vectors x , but are presented in a two-dimensional matched-filter-residual (MFR) plot [16] in which the matched-filtermagnitude MF is plotted on the y -axis and the residual R ison the x -axis. In this zero-mean unit-covariance case:MF = t T x /T (19)R = (cid:113) x T x − ( MF ) . (20)Fig. 1 illustrates these pixels as points in a scatter-plot. Thereason for choosing this representation is that all of thedetectors we consider here have contours that can be plot-ted in this two-dimensional space. Fig. 1 also compares theperformance of various detectors on this simulated data, andfor these parameters, we see that the new EC-FTMF detec-tor does well. The original FTMF is confounded because itincorrectly assumes the background is Gaussian; the ACE,AMF, and EC-AMF detectors are confounded because theyincorrectly assume the additive target model.In the regime of very large T and very small α , theeplacement model is “nearly” additive. In Fig. 2, we observethat the replacement-model and additive-model variants ofthe same detectors are similar, although EC-FTMF is stilldiscernibly better than EC-AMF, and FTMF is substantiallybetter than AMF. Interestingly, FTCE is no better than ACE.We have observed (results not shown here) that larger T and smaller α lead to a regime in which replacement-modeland additive-model variants are virtually identical.
3. DISCUSSION
In introducing the EC-FTMF detector, and showing that theGLRT solution can be expressed in closed form, we obtain atarget detection algorithm that is both convenient and adapt-ive to a range of conditions. In practice, using EC-FTMF(just as in using EC-AMF or other EC-based algorithms), onemust estimate the multivariate t -distribution parameter ν thatdescribes the fatness of the tails. To keep things simple, oursimulations employed the same ν in the algorithm that wasused for the simulation. But estimation of the single scalar pa-rameter ν from a large dataset is not that difficult; one simpleapproach employs higher moments of the whitened data [17].Finally, we remark that the GLRT – although widelyused, and very often with good results – is not the only ornecessarily the optimal solution to the composite hypothesistesting problem. One may prefer Bayesian approaches [13]or the recently introduced clairvoyant fusion [18, 19].
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