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46fine tuning of this. For instance, if we know from the application that each featureshould have a very consistent and unique grey value, then we should split a segmentwhich contains two or more disjoint sets of grey values. This will not change themodel fitting process, only the final pixel classification will be changed.
Chapter 4ADJUSTING FOR AUTOREGRESSIVE DEPENDENCEIn this chapter I present a theoretical approach to model selection for data withautoregressive dependence. I show how the BIC should be modified to deal withcertain kinds of dependence in the data. First, I address the simplest case, whichis the the AR(1) model with one-dimensional data. I then present the raster-scanautoregression (RSA) model, which is a special case of an AR(P) model. I showthat for data with autoregressive dependence, a simple modification to the usualBIC formula is necessary, and the appropriate “boundary” data points must beexcluded from analysis.4.1 Adjusting BIC for the AR(1) ModelIn the following sections, I use an AR(1) model to examine the effect of spatial (ortemporal) dependence on the BIC (equation 3.12). I approach this in two parts,first considering the loglikelihood term and then the penalty term.4.1.1 Loglikelihood AdjustmentConsider data from the AR(1) modelY i = C + βY i−1 + ɛ i (4.1)where |β| < 1 and the ɛ i are IID N(0, σ 2 ɛ ). Note that it is important todistinguish between the variance of ɛ i , which is σ 2 ɛ , and the variance of Y i , which
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Fast Automatic Unsupervised Image S
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In presenting this dissertation in
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TABLE OF CONTENTSList of FiguresLis
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4.1.2 Penalty Adjustment . . . . .
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Appendix A: Software Discussion 169
Page 15 and 16: 4.1 (a) Signal generated by an AR(1
Page 17 and 18: 5.29 Aerial image of a buoy, before
Page 19: DEDICATIONThis work is dedicated to
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Page 24 and 25: 4fast because they can be implement
Page 26 and 27: 6multidimensional observations at e
Page 28 and 29: ••••8Figure 2.1(a) is a sim
Page 30 and 31: 10Principal Curve•••••Dat
Page 32 and 33: 12based on the estimates of the par
Page 34 and 35: 14first of these can be done by a h
Page 36 and 37: 162.4 Examples2.4.1 A Simulated Two
Page 38 and 39: 18Figure 2.5: HPCC applied to the t
Page 41 and 42: 21Table 2.2: BIC results for simula
Page 43 and 44: 232.4.3 New Madrid Seismic RegionDa
Page 45 and 46: 25Table 2.3: BIC results for New Ma
Page 47 and 48: 27Latitude35.5 36.0 36.5 37.0 37.5
Page 49 and 50: 29set, so we want to avoid assumpti
Page 51 and 52: 31The examples we have presented in
Page 53 and 54: Chapter 3MARGINAL SEGMENTATIONIn th
Page 55 and 56: 35about ˜θ; this is a good approx
Page 57 and 58: 373.2 Mixture ModelsThe mixture den
Page 59 and 60: 39for estimating the model paramete
Page 61 and 62: 41The ith pixel of X or C is denote
Page 63 and 64: 43In componentwise classification,
Page 65: 45ponent. This classification proce
Page 69 and 70: 49Dependence CaseWhen |β| < 1, the
Page 72 and 73: 52For practical purposes, equation
Page 74 and 75: 54g ′′ (θ) ≈⎛⎜⎝∑ Ni=
Page 76 and 77: 564.1.3 Computing BIC with the AR(1
Page 78 and 79: 58upper, left, and right borders of
Page 80 and 81: 60L(Y −B |M, B) = L IND (Y −B |
Page 82 and 83: 62We now return to equation 4.56 an
Page 84 and 85: 64and then computing a mean-correct
Page 86 and 87: 66regression, which can be written
Page 88 and 89: 68resulting R 2 values are 0.93 for
Page 90 and 91: 700 5 10 15 200 5 10 15 20Figure 4.
Page 92 and 93: 72negative value would mean that ne
Page 94 and 95: 74Once each pixel has been updated,
Page 96 and 97: 76The basic idea of this psuedolike
Page 98 and 99: 78The consistency result presented
Page 100 and 101: 80⎛∑ ⎞Kj=1f(Yg i (Y i ) = log
Page 102 and 103: 82| log(f(Y i |X i = S, θ 1 ))|
Page 104 and 105: 84⇒ (L ˆX (Y |K)) exp(−(D K
Page 106 and 107: 86the object of the expectation is
Page 108 and 109: 88Thus, equation 5.42 holds, so BIC
Page 110 and 111: 90the inequality in equation 5.55 h
Page 112 and 113: 92N log(σ K ) − N log(σ 1 ) −
Page 114 and 115: 94final number of clusters. The cho
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96ization to initialize (see sectio
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98ˆσ 2 j =∑ Ni=1 ˆQ ij (Y i
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100one particular data value, which
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102classification they are not. Eac
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1045.2.6 Morphological Smoothing (O
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1065.3 Image Segmentation Examples5
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1080 10 20 30 400 10 20 30 40Figure
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1100.0 0.005 0.010 0.015 0.020 0.02
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112three pixels remain incorrectly
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114Table 5.3: EM-based parameter es
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1160 10 20 30 40 50 600 10 20 30 40
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1180 10 20 30 40 50 600 10 20 30 40
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1205.3.3 Ice FloesFigure 5.10 shows
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1220 20 40 60 80 1000 20 40 60 80 1
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124Percent0.0 0.005 0.010 0.015 0.0
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1260 20 40 60 80 1000 20 40 60 80 1
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1305.3.4 Dog LungFigure 5.17 presen
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132Table 5.6: Logpseudolikelihood a
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134Percent0.0 0.05 0.10 0.15 0.200
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1360 20 40 60 80 100 1200 20 40 60
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1380 20 40 60 80 100 1200 20 40 60
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140interior to the land which had i
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142Table 5.9: EM-based parameter es
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1440 20 40 60 80 100 1200 20 40 60
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1460 20 40 60 80 100 1200 20 40 60
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1485.3.6 BuoyFigure 5.29 is an aeri
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150Table 5.10: Logpseudolikelihood
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1540 20 40 60 80 1000 20 40 60 80 1
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1560 20 40 60 80 1000 20 40 60 80 1
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Chapter 6CONCLUSIONSThis dissertati
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160of subsampling would have to be
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REFERENCESAllard, D., and Fraley, C
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16494, 555-568.Fraley, C., and Raft
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166Maps,” Biological Cybernetics,
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168Zahn, C. (1971), “Graph-Theore
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170structuring element file.Segment
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172A.3 Splus codehpcc - Hierarchica
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VITA1993 B.S. Mathematics, Harvey M
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