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1380 20 40 60 80 100 1200 20 40 60 80 100 120Figure 5.22: Segmentation of the dog lung image into 4 segments after morphological smoothing(opening and closing, conditional on the edge pixels).
1395.3.5 Washington CoastFigure 5.23 is a satellite image of a section of the Pacific coast of Washington state’sOlympic peninsula; this image, provided by the USGS as part of the NationalAerial Photography Program (NAPP), was obtained from the Terraserver webpage, terraserver.microsoft.com. The resolution is 8 meters per pixel, and it is a256-level greyscale image.Table 5.8 shows that the first maximum in BIC P L occurs at K = 6 segments,so we regard this as the optimal choice of K for this model. Parameter estimatesare given in table 5.9. The marginal histogram of this image is shown in figure5.24, along with the estimated mixture density for K = 6. The large spike is dueto the small variance in the greyscale value of the water.For comparison, BIC IND values are also shown in the table. The first localmaximum occurs at 4 segments. The BIC IND values are computed based onlyon marginal (without spatial information) greyscale data from the image. Twoof the table entries, denoted by †, are missing from the table. The entry for 7segments is missing because the EM algorithm converges to a classification whichcontains fewer segments than the number of components in the mixture. In otherwords, we do not obtain a valid segmentation into 7 segments from the final EMclassification. The entry for 8 segments is missing due to slow convergence, whichis not surprising in light of the result for 7 segments.Figures 5.25 to 5.28 illustrate the steps of the segmentation algorithm for thisimage. An initial segmentation is created by using Ward’s method to cluster themarginal greyscale values of the image; this is shown in figure 5.25. The wateris quite well classified as a single segment, while the land is mostly containedin two segments. The tideline accounts for most of the variability in the image.Starting from the initial segmentation, the EM algorithm is used to fit a Gaussianmixture; the resulting segmentation is displayed in figure 5.26. Most of the pixels
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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
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4.1 (a) Signal generated by an AR(1
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5.29 Aerial image of a buoy, before
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DEDICATIONThis work is dedicated to
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20 10 20 30 40 50 600 10 20 30 40 5
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4fast because they can be implement
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6multidimensional observations at e
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••••8Figure 2.1(a) is a sim
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10Principal Curve•••••Dat
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12based on the estimates of the par
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14first of these can be done by a h
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162.4 Examples2.4.1 A Simulated Two
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18Figure 2.5: HPCC applied to the t
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21Table 2.2: BIC results for simula
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232.4.3 New Madrid Seismic RegionDa
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25Table 2.3: BIC results for New Ma
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27Latitude35.5 36.0 36.5 37.0 37.5
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29set, so we want to avoid assumpti
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31The examples we have presented in
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Chapter 3MARGINAL SEGMENTATIONIn th
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35about ˜θ; this is a good approx
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373.2 Mixture ModelsThe mixture den
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39for estimating the model paramete
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41The ith pixel of X or C is denote
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43In componentwise classification,
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45ponent. This classification proce
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Chapter 4ADJUSTING FOR AUTOREGRESSI
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49Dependence CaseWhen |β| < 1, the
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52For practical purposes, equation
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54g ′′ (θ) ≈⎛⎜⎝∑ Ni=
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564.1.3 Computing BIC with the AR(1
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58upper, left, and right borders of
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60L(Y −B |M, B) = L IND (Y −B |
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62We now return to equation 4.56 an
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64and then computing a mean-correct
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66regression, which can be written
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68resulting R 2 values are 0.93 for
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700 5 10 15 200 5 10 15 20Figure 4.
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72negative value would mean that ne
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74Once each pixel has been updated,
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76The basic idea of this psuedolike
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78The consistency result presented
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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
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
Page 116 and 117: 96ization to initialize (see sectio
Page 118 and 119: 98ˆσ 2 j =∑ Ni=1 ˆQ ij (Y i
Page 120 and 121: 100one particular data value, which
Page 122 and 123: 102classification they are not. Eac
Page 124 and 125: 1045.2.6 Morphological Smoothing (O
Page 126 and 127: 1065.3 Image Segmentation Examples5
Page 128 and 129: 1080 10 20 30 400 10 20 30 40Figure
Page 130 and 131: 1100.0 0.005 0.010 0.015 0.020 0.02
Page 132 and 133: 112three pixels remain incorrectly
Page 134 and 135: 114Table 5.3: EM-based parameter es
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Page 140 and 141: 1205.3.3 Ice FloesFigure 5.10 shows
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Page 150 and 151: 1305.3.4 Dog LungFigure 5.17 presen
Page 152 and 153: 132Table 5.6: Logpseudolikelihood a
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Page 160 and 161: 140interior to the land which had i
Page 162 and 163: 142Table 5.9: EM-based parameter es
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Page 168 and 169: 1485.3.6 BuoyFigure 5.29 is an aeri
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Page 178 and 179: Chapter 6CONCLUSIONSThis dissertati
Page 180 and 181: 160of subsampling would have to be
Page 182 and 183: REFERENCESAllard, D., and Fraley, C
Page 184 and 185: 16494, 555-568.Fraley, C., and Raft
Page 186 and 187: 166Maps,” Biological Cybernetics,
Page 188 and 189: 168Zahn, C. (1971), “Graph-Theore
Page 190 and 191: 170structuring element file.Segment
Page 192 and 193: 172A.3 Splus codehpcc - Hierarchica
Page 194: VITA1993 B.S. Mathematics, Harvey M
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