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32curve. Extending the method to accommodate this possibility explicitly wouldalso be worthwhile.Splus source files for HPCC and CEMPCC are available athttp://www.stat.washington.edu/stanford/princlust.html. Statlib has Splus functionsavailable for fitting principal curves, Kth nearest neighbor denoising, andmodel-based clustering (http://lib.stat.cmu.edu/S/principal.curve,http://lib.stat.cmu.edu/S/NNclean, andhttp://lib.stat.cmu.edu/S/emclust).
Chapter 3MARGINAL SEGMENTATIONIn this chapter I present a discussion of image segmentation based on themarginal (without spatial information) pixel values. I begin by presenting somebackground on the Bayesian Information Criterion (BIC), and then I introduce themixture model. I discuss the use of BIC for model selection in this context, andI examine two schemes for classification of pixels after model selection has beendone.3.1 BICThe Bayesian Information Criterion (BIC) was first given by Schwarz (1978) in thecontext of model selection with IID observations from a certain class of densities,and Haughton (1988) extended the class of densities to curved exponential families.The difference in the BIC value between two models is an approximation of twicethe log of the Bayes factor comparing the two models; the BIC has the advantageof being relatively easy to compute. The basic idea of the BIC is to use Laplace’smethod to approximate the integrated likelihood in the Bayes factor, and thenignore terms which do not increase quickly with N. In this section I present aderivation of the BIC, which largely follows the discussions found in Kass andRaftery (1995) and Raftery (1995). The following sections show how the BIC canbe adjusted for use with the AR(1) model and for the raster scan autoregression(RSA) model.A common approach to comparing two models or hypotheses, say M 2 vs. M 1 ,
Page 1: Fast Automatic Unsupervised Image S
Page 5: In presenting this dissertation in
Page 9 and 10: TABLE OF CONTENTSList of FiguresLis
Page 11 and 12: 4.1.2 Penalty Adjustment . . . . .
Page 13 and 14: 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: 31The examples we have presented in
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 and 66: 45ponent. This classification proce
Page 67 and 68: Chapter 4ADJUSTING FOR AUTOREGRESSI
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
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82| log(f(Y i |X i = S, θ 1 ))|
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84⇒ (L ˆX (Y |K)) exp(−(D K
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86the object of the expectation is
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88Thus, equation 5.42 holds, so BIC
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90the inequality in equation 5.55 h
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92N log(σ K ) − N log(σ 1 ) −
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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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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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