adaptive whitening filter for small target detection

基于白虎滤波器的红外小目标检测 国外全英文论文 经典 值得研究
where j is the iteration number. The image is scanned from top to bottom and left to right (lexicograph ically)and hence j is found as j=mMn. lere X is the input image, Y the predicted image and w the weight matrix at the jn iteration. ' The images are assumed to be square of size MM pixels and the weight matrix (hence the window size of the pixels being used to predict) is taken to be square of SIZe This predicted value is compared with a reference and the residual is the error between the i wo given by E(m,)=;=D(m,7)Y(m,7) (3) where d is the reference image. In the line enhancer configur ation this reference is a shifted version of the input image. The weights arc updated by using instantancous estimates of thc autocorrelation of th noise and the cross correlation between the reference and the input and t hen using a steepest descent direction. 16 The update equations then become W;+1(k)=W+F;X(1,mk) wherc u is the adaptation parameter. Undcr stationary conditions. it can be shown that the weights of this filter converge to the solution of the two dimensional wienerHopf equations, provided u is chosen to bc within the bounds rcquircd for stability Knowing the difference in the correlation lengths of the signal of interest and the clutter in the image it is possible to find bounds on u such that this tdlMs based adaptive filter predicts the clutter but not the signal of intcrcst. c Thus a scparation of thc signal and thc clutter is obtaincd by thc usc of such an adaptive filter. The error channel output of this filter(which now ideally contains only the signal of intcrcst and white noisc)can be uscd for a matched filtering opcration and thus the dctection statistic can be generated 2.2. Filtering performance The performance of such a adaptive clutter whitener(acw) based detection system was studied. Both the filtering and detection performance for structured back ground clutter and real infrared data were studied. The filtering performance was characterized by the local signal to noise ratio(LSCR) defined in a window of interest. Thus, if the window of interest were a rectangular region defined from L to(Hr, Hy), then the lsCr (in dB) is defined as H ∑;=x,(s(i,j)m LsCR=10lo where, mu is the mean of the noise in the window and ou is the variance. And the gain(in dB)provided by such a filter is then given by Gain= lscroutlscrin Fig. la shows the input image used for some of the simulations. This image is part of a 6 channel multispectral data set collected by a Nasa Thermal Infrared Multispectral Scanner(TIMS)sensor and is of a rural back ground over the hills of Adelaide, Australia. An artificial target 2x2 pixels in extent. was inserted into this back ground For a vcry high lscr the output of thc ACW is shown in Fig. lb It is sccn that most of the corrclatcd back ground clutter is predicted and hence the residual contains very little correlated background clutter The target is not prcdictcd and r out put of thc ACw The performance of this ACW depends on u. Fig. 2 shows the lscr gain obtained as a function of the daptation parameter. It is seen that as u is increased beyond an optimum and approaches the stability bounds, the performance of thc acw degrades vcry fast. This is a rcsult of two factors. First, incrcasing PlEaS (a) Input Innage (b)Oulput froin 3x3 TDLMS FilLer Figure 1: Adelaide Infrared Scene with target inserted at (100, 100), Input LSCR= 25.8 dB; Output LSCR= 26.4 dB LTTTTT gure 2: Gain vs. u to appear in SPIE 92 4 Figure 3: Gain vs. Input LSCR for target position 100, 100 the a allows the filter to predict the signal of interest and thus some of the target energy is cancelled in the oulpul. Second, as u approaches the stability bound of the filler the weight InlisadjustInlenl noise Increases It is also seen that as u is decreased below the optimum, the performance of the acw degrades, though not as drastically as that in the upper bound case. This degradation is because, with the slower daptation parameter, the filter is unable to accurately predict the clutter in the environment 2.3. Comparison with Local Demeaning It has bccn shown that the rcmoval of the local mcan from an image causcs the imagc statistics to bccomo Gaussian and also removes any spatial correlation, thus "whitening"the image 7121 Consequently, the local demeaning filter is an alternative tcchniquc for whitening thc imagc. In this scction, the comparative performance of the adaptive clutter whitening filter and the local demeaning filter will be obtained The amount of lscr gain obtained by such a system can be calculated theoretically since the target inscrtcd in the image is known complctcly. If m is the mcan of thc background clutter using a window of the same size as that used for the local demeaning and ma is the value of back ground clutter at the pixel of intcrcst. Then the output of the local demeaning filter at that pixel will be sn m1, and t hat of the TDL MS will be sn ma, where st is the intensity of the pixel with the si.I present. The theoretical plots are based on the assumption that the tdl ms has conver ged fully and is able to predict t he hackground clutter wit hout any leakage of the clutter into the error channel Fig 3 shows the gain in the output as a function of the lscr in the input for a 2 2 target located at(100, 100). At this position, mt< ma. and TDLMS shows morc gain than the local demeaning al gorath However Fig 4 shows the gain in the output as a function of the input LsCR for a target position of (105, 105 where m> ma, and the local demeaning filter is seen to perform better than the TDLMS. This s primarily because the dominant component of the clutter is a d. c.(in the spatial sense) componen hich can then bc rcmoved by a local demeaning opcration across the imagc. Howevcr for images that do not havc a strong d.c. component but have a highly corrclatcd clutter to appear in SPIE.92 Figure 4: Gain vs. Input LSCR for target position 105, 105 igure 5: The Input Image with the target at(115,45) Input lscr=15.3 dB and Color=1 to appear in SPIE.92 (a)(ur put from 'I'I)IMS (b)Ont put from Local I)emeanin Figure 6: The PerformanCe of the tDlms and the Local Deinealing filter for structured sinusoidal clutter structure, the local demeaning filter fails to remove the back ground clutter, while the TdlMs is able to predict the correlated clutter. This is seen in Figure 5 which shows the input being a two dimensional sinusoid with a 2x 2 target. The TDLMs output shown in Fig 6a is seen to have only the target, while the output of the local demeaning is seen to contain sinusoidal clutter as well (fig. 6b) Thus we see that the tdlms based adaptive filter can whiten clutter without also removing the target, when the clutter is structured. However the performance depends on the local statistics of the image. The detection performance of such ACW augmented receivers in the presence of highl structured noise has been characterized and ROC curves plotted using MontCarlo methods, 17 It has been shown that as the amount of color in the back ground clutter increases, the detection performance of the augmented matched filter does not degrade as much as that of the conventional matched filter 3. ADAPTIVE RECURSIVE DETECTION 3. 1. Weighted Estimation Another approach to the target detection problem consists of estimating the noise autocorrelation matrix and calculating the detection statistic directly from Eq I. The sampled matrix inversion, the modified sampled matrix inversion23 and the generalized likelihood ratio4, 25 detectors belong in this class.The estimation is usually done as a maximum likelihood estimation using 重=∑x to appear in SPIE.92 Since the detection is to be done recursively on a stream of data, a detection statistic has to be found at each iteration point, viz (n) However this estimation procedure assumes the clutter to be stationary and the performance of such a s yslelnl would degrade if the clutter were nonlsLalionary. In infrared inage dala. the clutter is known to be nonstationary and a weighted estimation procedure can be used26 for the autocorrelation as 重m)=∑Xx(k)x(k) where a is a weighting factor.a=0 corresponds to an instantaneous estimate of the noise and x= 1 corresponds to the maximum likelihood estimate for a stationary noise process. The tracking behaviour of such an estimation process for 0<a< 1, for one dimensional adaptive prediction has been studied. 2? With an estimation scheme given by Eq 9, recursive update equations for g(n)and p(m), can be found 重(m)=1重1( )x(7)x()重 1+Ax7(n)重1(m1)x(m) (10) and 入s7重1(n1)x(n) 9(m)= 1+1x7()重1(n1)x (11) This es lina lion techinique windows Che dala used for Che es tiration of the noise autocorrelation, chus permitting a matched filter based on such an estimation scheme to change with nonstationary input However, it is no longer a maximum likelihood estimate and the probability of detection will now change with every iteration 3.2. Detection probabilities If we assume the input data to be gaussian under both hypotheses( H1: signal present and ho: signal absent), with zero mean noise, the mean under Hi becomes m=s gs. The probability of detection is then given by18 (12) where the variance of the process, m is the mean under h1. go is the threshold used, and erfcO is efined as ∫c(m) Similarly the probability of false alarm is defined as (14 For a nonstationary process, if we assume that at the(n1)th iteration the autocorrelation was known exactly(*(n1)), an optimum value of Pa(n1) and Pfa(n1)can he found 26 The mean and the variance for the (n1th iteration can then be found as 1)=s西“(1) (15) (?1) 更“(71) (16 to appear in SPIE.92 (a)'l'arget F'mhedlded in Infrared Image I )ata b)Ietection Statist ic Figure 7: Typical Input and Output for Recursive Estimation Based Detection with A=0.99 Now if the detector is designed for a constant false alarm rate operation(CFAR), Pra(n)= Pfa(n1) Further if the rate of nonsta.tionarity is sma ll we can show (1 (?2 )6a*(71) (18) 27^(m 1c(n1(6 (6+△) (19 A257p(m1)1(m)面“(m1) (0 and&=s Ads. Here Ad is the change in the inverse of the autocorrelation of the nonstationary noise input and is given by △重=重(m)重(m1) (21) Equations 17 to 21 thus define the relationship between the probability of detection(Pd), the rate of non stationarity of the noise(A)and the weighting factor of the weighted update(). A proper choice of a de pends on the autocorrelation matrix of the noise, and can influence the probability of detection a two dimensional version of the recursive detection procedure can be formulated by ordering the data matrix into a vector lexicographically. Such a matched filter was used in a noise canceller structure for multispectral images. a typical input image(with an artificially inserted target )and the resulting detection statistic are shown in Figures 7a and 7b. It is seen that due to the adaptive estimation of the autocorrelation matrix, the matched filter is able to follow the changes in the statistics of the background utter to appear in SPIE.92 4. CONCLUSIONS We have shown that the use of adaptive whitening filters improves the performance of detection systems without direct modeling of the input data characteristics. The performance of the lms based adaptive filter depends on the local statistics of the image. In cases where the back ground clutter is non stationary only because of a changing local mean, these filters may not do as well as the local demeaning algorithm. However in cases where the clutter is highly correlated but does not have a spatial d. c componcnt, thosc filters arc still able to prcdict and cancel the clutter which the local demeaning filtor is unable to do. Further, in case of detection svstems which est imat.e the noise statistics, the adaptive recursive estimation procedure described in section 3 permits a relationship between the windowing parameter and the probabilities of detection The computational complexity of the TDlMs based adaptive filter is o(w )where w is the size of the window being used. This makes such systems very attractive for real time detection systems. The cursive estimation scheme descrihed in section 3 needs more computations than the TDL MS filter. With the recursive form being used its computational complexity is on the order of O(wi)since th size of the ant correlation matrix itself is n)2x w?. However since this technique ses a very structred autocorrelation matrix, QR based algorithms can be applied to speed it up. Further due to the structure in the problem it is possible to formulate a parallel implementation of this algorithm which will lead to very fast throughput 5. REFERENCES 1. A. K. Jain. Fundamentals of Digital Image Processing. Englewood cliffs, NJ. Prentice Hall, 1989 2. A. K.Jain, Advances in Mathematical Modcls for Image Proccssing, Proccedings of the IEEE vol.69,pp.502528.1981 3. C. W. Therrien, T.F. Quatieri, and D. E. Dudgeon, "Statistical ModelBased Algorithms for Image Analysis, Proccedings of the IeeE, vol. 74, pp. 532551, Apr. 1986 4. R. Nitzbcrg, E. H. Takkcn, D. Fricdman, and A. F. Milton, " Spatial Filtering Techniques for IR Sensors, in Proc. of sptE, Technical Sampo sim on Smart. Sensors, vol.178,1979 5. F.H. Takken. T. Friedman. A. F. Milton. andR. Nitze Leastmeansquare Spatial Filter for IR Sensors, Applied optics, vol. 18, no. 24, pp 4210 4222, 197 6.M.S. Longmire and E. H. Takken, "LMS and Matched Digital Filters for Optical Clutter Suppres sion, Applied Optics, vol 27, no 6, pp. 11411159, 1988 7.J.Y. Chen and l.s. Reed, "A Detection Algorithm for Optical Targets in Clutter, IEEE Trans actions on Aerospace and Electronic Systems, vol 23, no. 1, pp. 4659, 1987 8. D. Wang, "Adpative Spatial Temporal Spectral Filters for Background Clutter Suppression and Target Detection, Optical Engineering, vol. 21, pp. 10331038, Dec. 1982 9. A. Adridges, G. Cook, S. Mansur, and K. Zonca, "Correlated Background Adaptive Clutter Sup pression and Normalisation Techniques, in Proc. of sPl, multispectral Image processing and Enhancement. vol 933. 1988 10. L.E. Hoff,. R. Evans, and L. E. Bunney, "Detection of Targets in Terrain Clutter by tsing Mullis pectral Infrared Linage Processing, Technical Reporl for Naval Ocean SysleTTs Cenler, Dec 1990 to appear in SPIE 92
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