H. L. V. Trees. Detection, Estimation, and Modulation Theory. John Wiley and Sons, Inc., 1968.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithm. Given a set of observations Z = z, we would ideally like to nd the maximum likelihood estimate (MLE) # #L = argmax # ln fZ## (z) The MLE has many attractive properties such as consistency, asymptotic unbiasedness, and asymptotic minimum variance among unbiased estimates [14]. However, it is not always be possible to nd # #L directly, and this turns out to be the case with the fZ## (z) In order to achieve this goal, we will derive a reconstruction algorithm which ts into the broad framework of Expectation Maximization (EM) algorithms. The algorithm proceeds as ....

H. L. V. Trees. Detection, Estimation, and Modulation Theory, Part I. John Wiley & Sons, New York, 1968.

....Carlo simulations, using both the 3D to 3D algorithm and the 2D to 3D algorithm. 3.2 Interpretation of Covariance A useful interpretation of the covariance matrix can be obtained by assuming that the errors are jointly Gaussian. The joint probability density for Ndimensional error vector DX is [8]: X C X C X p X T X N D D = D 1 2 1 1 2 1 2 exp 2p (6) If we look at surfaces of constant probability, the argument of the exponent is a constant, given by the relation 2 1 z X C X X T = D D . This is the equation of an ellipsoid in N dimensions. For a given value of ....

....covariance matrix C X . Assume that we apply a transformation, represented by a sixelement vector W, to X to create a new pose Y. Denote Y = g(X, W) A Taylor 2 The exact formula for the cumulative probability in N dimensions is ( G = z X N N N dX e X P N 2 1 1 2 2 2 2 1 [8]. 7 series expansion yields X J Y D = D , where J = g X ) The covariance matrix C Y is found by: T X T T T T Y J C J J X X E J X J X J E Y Y E C = D D = D D = D D = 8) A variation on this method is to assume that the transformation W also has an associated covariance ....

H. L. V. Trees, Detection, Estimation, and Modulation Theory, New York, Wiley, 1968.

.... image data by utilizing the Cram er Rao bound g = 2 n m C 1 g ; 10) where 2 n denotes the variance of additive white Gaussian image noise, m the number of voxels in a local 3D window, and C g = rg (rg) T is the averaged dyadic product of the image gradient (Rohr [19] van Trees [20]) Note, that the Gaussian noise model is an approximation and that we assume that the dependence of the noise on the signal can be neglected (but see Abbey et al. 1] In Fig. 1 the landmark localization uncertainties are represented by error ellipses (note, that the ellipses have been enlarged ....

van Trees, H.L.: Detection, Estimation, and Modulation Theory, Part I. John Wiley and Sons, New York London 1968

....D jH i ) is defined via the integral, p(I D jH i ) Z S p(I D js; H i )p(sjH i )ds where s is the nuisance parameter. In most practical situations, the integrand is too complicated to be computed analytically. A non Bayesian solution is to formulate a generalized likelihood ratio test ([19]) according to the rule, p(I D jH 1 ; s 1 ) p(I D jH 0 ; s 0 ) H 1 H 0 ; for some ; and s i = argmax s2S p(I D jH i ; s) 2) s i s are the maximum likelihood estimates under the two hypotheses. A Bayesian solution involves computing the integral using some numerical ....

H.L. Van Trees. Detection, Estimation, and Modulation Theory, vol. I. John Wiley, N.Y., 1971.

....likelihood detector that chooses the direction of the shift that is most likely to have caused the observed U 2 Gamma U 1 and U 3 Gamma U 2 . It minimizes the average probability of decision error when the middle block is equally likely to have been shifted up or down a priori; see, e.g. [12]. It is convenient to use as decision variable the differences Gamma u : U 2 Gamma U 1 ) Gamma (c 2 Gamma c 1 ) Gamma l : U 3 Gamma U 2 ) Gamma (c 3 Gamma c 2 ) of the corrupted centroid separations and the uncorrupted separations. Gamma u is the change in the distance of the ....

Harry Van Trees. Detection, Estimation, and Modulation Theory, volume I. John Wiley & Sons, 1968.

....is at least as great as that for the original signal. 2 We make some final observations. First, in the above proof we assume that the fading is cyclically stationary. This is not restrictive since any wide sense stationary fading process asymptotically becomes cyclically stationary as T 1 [21]. Second, the role of the block length T is secondary to that of the coherence time . We impose the constraint that blocks of T symbols be independent because it allows 20 us to use the standard notions of mutual information and channel capacity per block of T symbols. When T AE , the ....

....d 1 2 1=4 M Y m=1 2 4 1 1 (aeT =M) 2 (1 Gammad 2 m ) 4(1 aeT=M) 3 5 N = 1 2 M Y m=1 2 4 1 1 (aeT =M) 2 (1 Gammad 2 m ) 4(1 aeT=M) 3 5 N ; B.11) which is (18) It turns out that (B. 11) is, in fact, exactly the Chernoff bound obtained by computing (see, e.g. [21]) P e 1 2 e ( where ( ln E fexp [ ln p(X j Phi 2 ) Gamma ln p(X j Phi 1 ) j Phi 1 g ; and where 0 1 is a free parameter that is chosen to minimize ( To help see this, we note that ( is merely the logarithm of the previously computed characteristic function for = i, and ....

H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: John Wiley and Sons, 1968.

....bound. In particular the quantity ln R dX p p(X j S 1 ) Delta p(X j S 2 ) is the so called Bhattacharyya distance between the two conditional densities, and the Chernoff bound (with the tilting parameter set equal to 1 2) on the probability of error for deciding between the two signals is [17] P e fS 1 vs: S 2 g 1 2 Z dX q p(X j S 1 ) Delta p(X j S 2 ) 30) The expectation of the above bound with respect to S 1 and S 2 gives the pairwise probability of error between any two members of the randomly chosen constellation, which when multiplied by the number of signals in 12 the ....

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I, New York: John Wiley, 1968.

....description, then interference avoidance can be brought to bear. In this context we consider optimal waveform selection to maximize the signal to interference ratio (SIR) for a power constrained user in the presence of interference. Starting from the essentially classical approach of whitening [12] and showing the relation to modern methods (typified by [13] we then consider ensembles of users and describe a class of distributed greedy algorithm which can optimize their shared use of the medium. That is, through local self interested action, a social optimum can usually be reached. ....

....combine these either to detect the bit b or to derive an estimate of b. When Z(t) is Gaussian, these projections would indeed be independent Gaussian random variables and the optimal detection problem would be easily solved. A complete and rigorous development of the ideas can be found in [12]. Here we provide a brief recapitulation. In general, given a stochastic process, Z(t) we seek an orthonormal representation Z(t) lim N N i=1 a i F i (t) 1) with a i = R T 0 Z(t)F i (t)dt. Note that in Equation (1) the convergence requirement is not the usual pointwise limit, but ....

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H.L. Van Trees. Detection, Estimation, and Modulation Theory, Part I. Wiley, New York, 1968.

....structure is illustrated in Figure 3.3. Observe that this structure corresponds to standard linear beamforming followed by quadratic TFR TSR based detection for each hypothesized angle of arrival. Also note that this structure is consistent with the idea of coherent combining from detection theory [13]. x (t) 1 x (t) 2 x (t) M L opt max (t,a,q) Align q TFR TSR Figure 3.3 Optimal Array Processor in a Coherent Environment 3.3 Detection in a Noncoherent Array If the array environment is perfectly noncoherent, the signals between two different sensors are completely uncorrelated. ....

....hypothesized angle of arrival, sum them together, and use the maximum value in the time frequency space or time scale space distribution. The noncoherent detection structure is illustrated in Figure 3.4. Observe that this is consistent with the idea of noncoherent combining from detection theory [13]. x (t) 1 x (t) 2 x (t) M L opt max (t,a,q) TFR TSR TFR TSR TFR TSR Align q Figure 3.4 Optimal Array Processor in a Noncoherent Environment 19 3.4 Near Field Processing In our use of time frequency space and time scale space, we have used the term space to refer to the ....

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H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: John Wiley and Sons, 1968. 54

.... between two images is the well known result from classical detection and estimation theory that, under certain conditions, an optimum decision rule for detecting the presence of a known signal in an observed noisy waveform can be obtained from the cross correlation of the signal with the waveform [61]. The decision is based on whether the value of the correlation function is above a threshold determined from either a Bayes cost function or a desired false alarm rate in a Neyman Pearson test. The position of the maximum correlation value gives the most likely position of the signal, under the ....

....t p(V H ) 1 p(V H ) 0 V H V H 1 P F P D 11 0 Figure 6 1: Threshold selection for a decision rule that chooses between two hypotheses H 0 and H 1 . PD = Pr(V ijk jH 1 ) 6:5) while the false alarm rate is the probability that the test will be passed given that the match is incorrect [61]. P F = Pr(V ijk jH 0 ) 6:6) As indicated by the diagram in Figure 6 1, it is not necessary to explicitly compute PD and P F to determine , but only to find the mean and variance of V ijk under the hypotheses H 0 and H 1 . The Chebyshev inequality [87] ensures that PD = 1 Gamma Pr(V ijk ....

Harry L. Van Trees. Detection, Estimation, and Modulation Theory. John Wiley & Sons, New York, NY., 1968.

....via the integral p(I D jH i ) R S p(I D js; H i )p(sjH i )fl(ds) where s is a nuisance parameter. In most practical situations, the integrand is too complicated to be computed analytically. One common solution is to formulate a generalized likelihood ratio test or pseudo likelihood test [29] according to the rule p(I D jH1 ;s 1 ) p(I D jH0 ;s 0 ) for some ; and s i = argmax s2S p(I D jH i ; s) s i s are the maximumlikelihood estimates (MLEs) of the unknown target parameter s, under the two hypotheses. A similar ratio, of the maximized posterior densities under ....

....is to decide which target ff 2 A best describes that I D . Associate with each target ff i 2 A, a hypothesis H i which selects ff i as the best match. H 0 is the null hypothesis signifying that no target is present. A Bayesian approach is to solve a series of binary likelihood ratio tests (see [29]) P (H i jI D ) P (H j jI D ) 1 or equivalently, L ij (I D ) p(I D jH i ) p(I D jH j ) P (H j ) P (H i ) j ij : 4) The likelihood of I D given that a target ff i is present is, using (2) p(I D jH i ) 1 Z(oe) Z S expf Gamma 1 2oe 2 E ff i (s; ....

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H.L. Van Trees. Detection, Estimation, and Modulation Theory, vol. I. John Wiley, N.Y., 1971. 15

....description, then interference avoidance can be brought to bear. In this context we consider optimal waveform selection to maximize the signal to interference ratio (SIR) for a power constrained user in the presence of interference. Starting from the essentially classical approach of whitening [11] and showing the relation to modern methods (typified by [12] we then consider ensembles of users and describe a class of distributed greedy algorithms which can optimize their shared use of the medium. That is, through local self interested action, a social optimum can usually be reached. ....

....Z(t) F i (t) Z(t) yields uncorrelated projections. Precisely, we require Z T 0 F j (t) Z T 0 R Z (t, t)F i (t)dt dt = l j d i j (1) The solution to this integral equation requires l i F i (t) Z T 0 R Z (t, t)F i (t)dt (2) The properties of (2) are discussed in detail in [11]. Since integral equations are in general difficult to solve, it is useful to derive an equivalent discrete representation of (2) This will allow us to use simple methods from linear algebra. So let us assume that Z(t) and therefore the function set F i (t) can be well approximated by a ....

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H.L. Van Trees. Detection, Estimation, and Modulation Theory, Part I. Wiley, New York, 1968.

....noise, and jamming. TFRs and TSRs provide a natural detection framework for such hypothesis testing problems for two main reasons: first, optimal detection of a second order signal (such as a Gaussian signal) in the presence of Gaussian noise involves a quadratic function of the observations [7, 11], and bilinear TFRs and TSRs are quadratic in the observations; second, TFRs and TSRs possess additional degrees of freedom provided by the TFR and TSR parameters (time and frequency for Cohen s class and time and scale for the affine class) Because of our assumptions, the dependence of the ....

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: John Wiley and Sons, 1968.

...., L exactly spans the set of codewords s k , k = 1,2, K which share the same eigenvalue l, SS # W)s k = ls k . If we define a k# = s # k f # , then for r i j = s # i s j we must have K j=1 a i# a j# r i j a i# a j# = 0 (31) Proof: Theorem 3 By Theorem 2 and Mercer s theorem [18] we have SS # W= l L i=1 f i f # i N i=L 1 l i f i f # i (32) where the f i is the complete eigenvector set for W. Since the s k , k = 1,2, K are exactly spanned by the f i , i = 1,2, L we then have l L i=1 f i f # i = K k=1 s k s # k L i=1 ....

H.L. Van Trees. Detection, Estimation, and Modulation Theory, Part I. Wiley, New York, 1968.

....and have very small variance unless edges or strong textures are present. Since we are attempting to detect W b in a test image, X b is treated as uncorrelated noise and the binary hypothesis test is H 0 : Observe X b versus H 1 : Observe W b X b . From classical detection theory [15], the optimum detectors are correlation based. If some assumptions about W b and X b were made, explicit expressions for the probability of false detection and miss can be derived. However, one can immediately see that the detector will perform well in smooth regions of the image and not as ....

H. Van Trees, Detection, Estimation, and Modulation Theory, John Wiley & Sons, 1968.

....that is parameterized and A that is unstructured. In essence, the roles of time and space have thus been reversed. The classical approach to estimate k and D k is to form a matched filter that correlates the received signal, x(t) with a delayed and frequency shifted version of the known signal [VT68, Hel95]. However, this estimate is efficient only if a single signal is received. In Chapter 5 we make use of the modified data model in (1.52) to develop efficient subspacebased estimators for the case when multiple arrivals are present. 14 Introduction 1.6 Contributions and thesis outline This ....

....Soderstrom. Maximum Likelihood Estimation of the Parameters of Multiple Sinusoids from Noisy Measurements . IEEE Trans. on ASSP, 37(3) 378 392, March 1989. Swi97] A. Swindlehurst. Time Delay and Spatial Signature Estimation Using Known Asynchronous Signals . to appear in IEEE Trans. SP, 1997. [VT68] H. L. Van Trees. Detection, Estimation, and Modulation Theory, Part I. John Wiley and Sons, Inc. 1968. Chapter 2 Matrix Optimization Result We derive the matrix that minimizes a quadratic criterion subject to linear equality constraints. Since we do not assume that the matrix of the quadratic ....

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H. L. Van Trees. Detection, Estimation, and Modulation Theory, Part I. John Wiley and Sons, Inc., 1968.

....recognition. Let us first assume that the HMM state sequence is given. Then, the maximum likelihood channel estimate is given by (9) where Y is the collection of observations, S is the state sequence, are the HMM parameters and h is the channel. For Gaussian output distributions, it can be shown [3] that this estimate is given by (10) where the HMM output distribution (11) is a multivariate normal distribution with a state dependent mean and covariance . Hence, when the state HMM sequence is given, the channel estimate can be obtained as a weighted combination of the deviations of the ....

H.L. Van Trees, Detection, Estimation, and Modulation Theory, John Wiley & Sons, 1968.

....goal of the present work was to recast the EMAP estimation procedure as an adaptive filtering problem to obtain a more efficient implementation. From estimation theory it is well known that MAP and minimum mean square error (MMSE) estimation techniques are equivalent for normally distributed data [25]. In adaptive filter theory, the computationally efficient least mean square (LMS) transversal filter is derived as an approximation of MMSE formulations. Expressing the Bayesian estimate as an MMSE adaptive filter led to the development of an estimation method (called LMS C) which was more ....

....density, i.e. the conditional mean of p( c) The EMAP estimate is the value of at which p( c) has its maximum. If the a posteriori pdf is a unimodal function which is symmetric about the conditional mean, as is the case with the multivariate normal density, then these two estimates are equivalent [25]. To demonstrate this equivalence, F aa and F must be expressed in terms of S, S , and . It was also found to be necessary for a bias a o o term to be incorporated into the MMSE estimate to represent the contribution of the term in the o EMAP expression. Without this bias, it is not possible ....

Van Trees,H.L., Detection, Estimation, and Modulation Theory, Part I, Wiley, New York, 1968.

....both depend upon the radar s CNR. The preceding material provides a complete statistical characterization for the following laser radar range imaging problem: given the range data vector r, find the maximumlikelihood (ML) estimate of the true range vector r # , i.e. the range estimate satisfying [13] r # ML (R)# arg max R # h p r r #( R R # ) i . 6) Unfortunately, the probability density function (pdf) from Eq. 1 implies that r # ML (R) R, i.e. the ML range image equals the raw data. Mathematically, it is the pixel independence combined with the unimodal single pixel range pdf ....

....range data which demonstrate the soundness of the preceding multiresolution imaging approach. Before that, however, it is germane to present a theoretical result on the ultimate performance limit for ML range imaging. 3. 3 Performance of the Parametric ML Range Imager Following standard practice [13], we shall use the bias and the error covariance matrix to assess the performance of our ML parameter vector estimate, xPML . These can be converted, in turn, to corresponding performance results for the ML range estimate via 9 r # PML =H P xPML . In keeping with our interest in ....

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H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 2.

.... former is due to the rough surfaced nature of most reflecting surfaces measured on the scale of a laser wavelength [3] the latter is the fundamental noise encountered in optical heterodyne detection [4] Range measurement is a nonlinear estimation problem, hence it is subject to a threshold e#ect [5], even in the absence of speckle. Thus, at high enough carrier to noise ratio (CNR) the range measurement accuracy will be better than the resolution limit set by the pulse bandwidth, but an abrupt and severe performance degradation is incurred as CNR decreases, owing to range anomalies. An ....

....to a measured range far removed from the true range value. Target speckle severely exacerbates the range anomaly problem, because the concomitant Rayleigh fading of the video detected target return transforms the anomaly probability from a function that decreases exponentially with increasing CNR [5] to one that is inversely proportional to CNR [6] We have long been interested in the statistics of peak detecting coherent laser radars, starting from the fundamentals of single pixel statistics [6] through target detection studies for 2 D imagers [7] 9] to planar range profiling and target ....

H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 4.

....[ Delta] denotes complex conjugation. B. Joint Maximum Likelihood Estimation of Channel and Data The system model is here identical to that of the previous section, except for the assumption that the transmission begins at epoch 0. Given the observation sequence fy n g k n=0 , it is wellknown [37] that the joint Maximum Likelihood (ML) estimate of data sequence and discrete channel response is obtained by maximizing the likelihood function over fa n g k n=0 and ff n g L n=0 according to max fang k n=0 max ffng L n=0 0 Gamma k X n=0 fi fi fi fi fi y n Gamma L X ....

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: John Wiley & Sons, 1968.

....equal, one can combine (19) and (20) to obtain the following asymptotic approximation to P e : P e # 1 2s(1 s) # 2# ## (s) e (s) 21) For the Gaussian model above, maximization of (18) with respect to s gives the optimal Cherno# exponent s = 0. 5, so (s) SNR 8, ## (s) SNR, and [17] P e # # 2 # SNR e SNR 8 , 22) which holds for large SNR. The applicability of these asymptotic conditions to target recognition is examined next. 5 Bounds for Target Detection Without Nuisance Parameters We first consider Detection Problem 1 in Sec. 2.6, and derive performance bounds for ....

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Wiley, New York, 1968.

....many of the ideas described herein also apply to the second class. The classical approach to the design of a signal detection system is to optimize a timeinvariant (i.e. fixed) receiver structure for a known class of signals with unknown parameters that are corrupted by noise of known statistics [1 3]. The parameter used in the design is the likelihood ratio of the received signal. When, however, the noise is nonstationary, but of known form, the receiver must take on a time varying structure, making its design more difficult [4] The design becomes even more difficult when the statistics of ....

....criterion for radar detection is the Neyman Pearson criterion [51] According to this latter criterion, the probability of detection is maximized subject to a prescribed upper bound imposed on the probability of false P D 0.91 for the NN receiver 0. 71 for the Doppler CFAR receiver = 38 alarm [1 3]. Unfortunately, minimization of the mean square error does not guarantee fulfillment of the Newman Pearson criterion. Recognizing that the basic idea of the Newman Pearson criterion is to treat the two kinds of error (i.e. missed detection and false alarm) differently, Principe et al. 52] have ....

Van Trees, H.L. Detection, Estimation, and Modulation Theory, Part 1, Wiley, 1968.

....features vectors and their characterization with probability distributions, Gaussian density functions, Bayesian decision, the Perceptron Algorithm, the Nearest neighbor classifier and its analysis. 3. Lindenbaum) 2D object recognition: Matched filter design from several perspectives [1], limitations and variations, hierarchical implementation. 4. Lindebaum) 2D binary object (shape) recognition: projections, moments, moment invariants, boundary descriptions and recognition. 5. Guy Levanon) Karhunen Loeve based recognition [2] 6. Lindenbaum) Hough transform with variations ....

H.L. Van Trees. Detection, Estimation, and Modulation Theory, Part I. Wiley, NewYork, 1965.

....Digital images and audio signals are then extracted from these movie files. PALM decodes the audio signals using an algorithm (Fig. 3.5, Fig. 3.6) that is based on correlation. The correlation method is used because it corresponds to match filtering which maximizes the output signal to noise ratio [67]. The digitized audio signal is correlated with two stored templates: one corresponding to the output of the hardware encoder when its input is HIGH; and the other one corresponding to the output when its input is LOW. These templates were collected during the building of the hardware encoder ....

H. L. Van Trees. Detection, Estimation, and Modulation Theory. John Wiley & Sons, 1968.

....1g. In general, these prior probabilities and cost functions are unknown. In the Bayes decision, these unknown terms are usually collected as a new term, call the the likelihood threshold Gamma. For a detailed analysis of how the likelihood threshold Gamma is obtained, the reader can refer to [9]. The likelihood Gamma is: Gamma = P rjH 1 (RjH 1 ) P rjH 0 (RjH 0 ) where P rjH 1 (RjH 1 ) and P rjH 0 (RjH 0 ) are the probability densities of an observation R, given that a pixel is within a character (H 1 hypothesis) or within the background (H 0 hypothesis) respectively. The way to ....

H.L. Van Trees (1968): Detection, Estimation, and Modulation Theory, part 1.

.... for doing so would provide important benefits for critical military and commercial systems (e.g. helicopter gearboxes, shipboard fire pumps, motors, and generators) For these problems, recognition based learning systems such as Single Hypothesis Testing [Fukunaga1990] Signal Detection [Van Trees1968], Example based Learning [Sung and Poggio1994] View Based Eigen Spaces [Pentland et al..1994] Turk and Pentland1991] PARSNIP [Hanson and Kegl1987] are much more appropriate than discrimination based learning systems because they do not use counter examples in the concept learning phase and ....

....in the absence of counterexamples, or recognition based learning, is not a novel enterprise since it has been considered previously in the context of hypothesis testing. Practically speaking, techniques for recognition based learning were used in the fields of Signal Detection Theory [Van Trees1968] and Pattern Detection in Images which rely on hypothesis testing and other techniques that require a different type of preliminary setting phase [Sung and Poggio1994] Various methods designed within these two fields are now described, followed by a discussion of their limitations with respect ....

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Harry L. Van Trees. Detection, Estimation, and Modulation Theory. Wiley, New York, 1968.

....the variance of additive white Gaussian image noise and m the number of voxels in a local 3D window. Then, we can relate the matrix C g to the minimal localization uncertainty of the center of the window x = x; y; z) The minimal localization uncertainty is given by the Cram er Rao bound (e.g. [18]) and is represented by the covariance matrix Sigma g = oe 2 n m C Gamma1 g : 2) We see, that Sigma g is proportional to the inverse of C g . From (2) we can derive the 3D error ellipsoid of the position estimate with semi axes oe x ; oe y ; and oe z . A quantitative measure for the ....

H.L. van Trees, Detection, Estimation, and Modulation Theory, Part I, John Wiley and Sons, New York London 1968

....problem x3.4 becomes the linear parameter estimation problem of finding point and interval estimators for a given noisy observations r satisfying r = Xa n; 3.3) as discussed in the next section. Chapter 4 Linear Parameter Estimation As motivated in the previous section, and following [VT] consider the problem of recovering a fixed but arbitrary non random parameter a 2 R k from noisy observations r 2 R j , j k, which are generated by r = Xa n (4.1) where X is a non random known injective j Theta k matrix, and n is a zero mean jointly Gaussian random R j vector with ....

....t Gamma1 n ] t = X t Gamma1 n X) Gamma1 = a Consequently, aML Gamma a N(0; a ) 4.14) Conclusion (C2) follows. Conclusion (C3) From (4.14) we have that E[a ML ja] a V ar(a ML ja) a : Thus aML is unbiased. To asses the efficiency of the MLE, we show, following [VT] that aML attains equality in the Cramer Rao inequality: V ar(a) GammafE[r 2 a log p rja (rja) g Gamma1 for every unbiased estimator a for a. Indeed, from (4.10) I(a) is actually nonrandom so GammafE[r 2 a log p rja (rja) g Gamma1 = Gamma( GammaX t Gamma1 n X) Gamma1 ....

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H. Van Trees (1968) Detection, Estimation, and Modulation Theory, Part I. New York: John Wiley & Sons.

....projections of I(t) onto the various Phi i (t) on (0; T ) should be uncorrelated. With I(t) assumed Gaussian, then the projections are indeed independent Gaussian random variables and the optimal detection problem is easily solved. A complete and rigorous development of the ideas can be found in [11]. Here we provide a brief recapitulation. 3.2 Orthonormal Representation of a Noise Process In general, given a stochastic process, I(t) we seek an orthonormal representation I(t) lim N 1 N X i=1 a i Phi i (t) 1) with a i = R T 0 I(t) Phi i (t)dt. Our convergence requirement is not the ....

....requirement is not the usual pointwise limit, but what is called a limit in the mean. Specifically, we require that lim N 1 E 2 4 I(t) Gamma N X i=1 a i Phi i (t) 2 3 5 = 0 (2) Here we simply assume that such an expansion exists and converges. The interested reader is referred to [11, 12] for further detail. Now we seek a special set of orthonormal Phi i which produce uncorrelated projections. Writing this condition out (and assuming zero mean I(t) we have E[ Z T 0 Z T 0 Phi j (t)I(t) Phi j ( I( dtd ] i ffi ij (3) Propagating the expectation and simplifying we obtain ....

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H.L. Van Trees. Detection, Estimation, and Modulation Theory, Part I. Wiley, New York, 1968.

....of algorithms in the subsequent chapters, we briefly present here some fundamental facts relevant to our visual reconstruction problems as well as establish some notation that will be used frequently in the sequel. For derivations and other details, introductory texts on estimation theory (e.g. [16, 45, 74]) should be consulted. The maximum likelihood (ML) estimation problem deals with estimation of an unknown vector x based on an observed sample of a random vector y whose distribution is parameterized by x. Specifically, let P y ( Deltajx) be the probability density function for such a ....

H. L. van Trees. Detection, Estimation, and Modulation Theory (Part I). John Wiley & Sons, New York, 1968.

.... by the number of possible false acceptances when all the remaining speakers pose as impostors for each speaker (100 speakers x 99 impostors in the case of a population size of 100) For each feature set, we found 14 nearest neighbors (cohorts) for each speaker using the Mahalanobis distance metric [40]. Specifically, 1 and 2 , oe 2 1 and oe 2 2 , are d dimensional mean vectors and dxd dimensional covariance matrices for two speaker models, respectively. The Mahalanobis distance squared, D 2 , between the two speakers is then D 2 (1; 2) d X i=1 ( 1i Gamma 2i ) 2 oe 2 i ....

....(MAP) adaptation procedure, a common method for adapting all the statistics of models. MAP provides a way to incorporate prior information into the estimation process, by assuming an a priori distribution of the parameters that are being estimated. Details on the MAP technique can be found in [40, 23]. 5.2.6 Incorporating into GALAXY Finally, we plan to incorporate our speaker verification system into the GALAXY conversational system [13] GALAXY is a system currently under development in our group that enables information access using spoken dialogue. Presently, GALAXY can access the ....

H. Van Trees. Detection, Estimation, and Modulation Theory (Part I). John Wiley and Sons, Inc., 1968.

....each random variable has probability density function (pdf) either f(y; 0 ) or f(y; 1) where 0 and 1 are known. It is also known that the number of changes between the two pdf s does not exceed ; however, it is not known where these changes occur. To obtain the maximum likelihood hypothesis [10], we have to maximize the log likelihood function X i:h i =1 log f(y i ; 1) X i:h i =0 log f(y i ; 0 ) where the hypothesis h is such that the sample y i is hypothesized to be from the pdf f(y; h i ) Note that by making the definitions u0;i = log f(y i ; 1) Gamma log f(y i ; 0) ....

....with the following discrete Gaussian probability mass function: pr(K = k) C exp ( Gamma 1 2 k Gamma N 2 oe 1 2 ) k = 1; N Gamma1; where C is a normalization constant. Given an observation y of Y, we seek the best hypothesis in the generalized likelihood ratio sense [10]: the maximum likelihood estimates of the hypothesis and 0 are calculated for each value of K, and these estimates are then used in a multiple hypothesis testing procedure to estimate K. In other words, we seek ( h; 0 ; K) arg max h; 0 ;k (log f1 (yjh; 0 ; k) pr(K = k) where ....

H. van Trees. Detection, estimation, and modulation theory, volume 1. Wiley, 1968.

.... vision [6, 7, 8, 9] Systematic errors, or biases, have been examined experimentally for LADAR in [10] and for stereo in [11] For the quality of obstacle detection, a relevant theoretical framework exists for modeling the probability of detection and false alarms in classical detection problems [12]. This framework has been applied to obstacle detection with LADAR for a spacecraft landing application [13] however, no work has been done yet on applying this to obstacle detection for ground vehicles. In this paper, we focus on obstacle detection with range images produced by stereo vision. ....

.... Gamma 1 2 ( DeltaH Gamma H) 2 oe 2 DeltaH d DeltaH This conditional probability is a key tool for analyzing obstacle detection performance, because it embodies both the probability of detecting an obstacle that is actually present (P d ) and the probability of a false alarm (P f ) [12]. For example, the conditional probability P d of detecting an obstacle of a size H 1 , given that such an obstacle is actually present (i.e. has its base at pixel p 1 ) is the integral p( DeltaH tjH 1 ) Figure 8a shows P d versus range for stepheight S = 30 cm, H = 30 cm, and a threshold t = ....

H. L. Van Trees. Detection, Estimation, and Modulation Theory, volume Part I. John Wiley and Sons, New York, 1968.

....determine which composite hypotheses to discard We need to address this question in light of the fact that computational complexity is a key concern. In particular, if computational load were not an issue, then we could solve the problem optimally using a generalized likelihood ratio test (GLRT) [14] as follows. At a coarse level in the hierarchy, we would enumerate all hypotheses, Hm , process the data to choose among these optimally, and then choose the composite hypothesis, e Hn , that contains this optimal choice. Since an underlying assumption for problems of interest here is that ....

....Rn and the hypothesis that no anomaly exists. To derive the coarse scale likelihood statistic, we associate with each e Hn a coarse scale anomaly: f a = e bn where e bn , as defined in Section II D, is the indicator function for e Rn . From equation (7) and standard results in hypothesis testing [14], we find that the log likelihood for each of these four hypotheses is given by an affine operation on the observed data, g, namely, e n (g) T e bn ) T Gamma1 g g Gamma 1 2 (T e bn ) T Gamma1 g T e bn ; n = 1; 2; 3; 4 (11) and the resulting decision rule consists of choosing the ....

H. L. Van Trees. Detection, Estimation, and Modulation Theory, Part I. John Wiley and Sons, Inc., 1968.

....use mean squared error (MSE) to assess the estimation accuracy of the parameters d # , d # , f # , and f # . From each parameter estimate, Figures 4.4 and 4.5 show the MSE vs. SNR plots of the ve methods as well as the Cramer Rao Bound (CRB) which is the lower bound of the Cramer Rao inequality [102]. Each data point is measured based on 500 independent runs of the random noise process. From Figures 4.4 and 4.5, all ve methods yield about the same results at high SNRs but quite di erent results at low SNRs. The MMP1 method achieves comparable performance in estimating the frequencies as the ....

H. L. V. Trees, Detection, Estimation, and Modulation Theory. John Wiley & Sons, 1968.

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H. L. V. Trees. Detection, Estimation, and Modulation Theory. John Wiley and Sons, Inc., 1968.

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H. V. Trees, Detection, estimation, and modulation theory, Wiley, New York, 1968.

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H. L. V. Trees, Detection, Estimation, and Modulation Theory, vol. I. John Wiley & Sons, NY, 1968.

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H. L. V. Trees, Detection, Estimation, and Modulation Theory, vol. III. John Wiley & Sons, NY, 1968.

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H. L. Trees, Detection, Estimation, and Modulation Theory, Part I. John Wiley and Sons Inc., 1968.

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H. L. V. Trees. Detection, Estimation, and Modulation Theory. John Wiley & Sons, Dec. 1968.

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H. L. Trees, Detection, Estimation, and Modulation Theory, Part I. John Wiley and Sons Inc., 1968.

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H. L. Van Trees, Detection, Estimation, and Modulation Theory, vol. 1. John Wiley & Sons, 1968.

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H. L. Van Trees, Detection, Estimation, and Modulation Theory, Wiley, New York, 1968.

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H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York, NY: J. Wiley, 1968.

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H. L. Van Trees, Estimation, and Modulation Theory, Part I, Wiley, 1968.

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H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part II, New York: John Wiley & Sons, Inc., 1971.

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H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part III, New York: John Wiley and Sons, Inc., 1971.

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H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part I, New York: John Wiley and Sons, Inc., 1968.

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