Preface |
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xiii | |
Contributors |
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xvii | |
I Foundations |
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1 | (188) |
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Discrete Tomography: A Historical Overview |
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3 | (32) |
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3 | (3) |
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Foundations and algorithms |
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6 | (20) |
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Definitions, notations, and basic problems |
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6 | (4) |
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Reconstruction on binary matrices from two projections |
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10 | (16) |
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26 | (3) |
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Data compression, data coding, and image processing |
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26 | (1) |
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27 | (1) |
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27 | (1) |
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28 | (1) |
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29 | (6) |
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29 | (6) |
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Sets of Uniqueness and Additivity in Integer Lattices |
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35 | (24) |
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35 | (2) |
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Uniqueness and additivity |
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37 | (6) |
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43 | (2) |
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45 | (1) |
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Nonadditive sets of uniqueness |
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46 | (4) |
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50 | (5) |
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55 | (1) |
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56 | (3) |
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57 | (2) |
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Tomographic Equivalence and Switching Operations |
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59 | (26) |
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Ryser's theorem for binary pictures on the square grid |
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59 | (11) |
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Binary and ternary pictures on the square grid |
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60 | (1) |
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Four statements of Ryser's theorem |
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61 | (2) |
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Metropolis algorithms based on Ryser graphs |
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63 | (5) |
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A proof of Ryser's theorem |
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68 | (2) |
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Nonexistence of analogs on m-grids with m > 2 |
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70 | (2) |
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70 | (1) |
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Validity of Ryser's theorem for arbitrary 2-grids |
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71 | (1) |
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72 | (1) |
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A proof of the main result |
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72 | (10) |
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Derivation of the main result from the Main Claim |
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73 | (2) |
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Justification of the Main Claim |
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75 | (7) |
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82 | (3) |
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83 | (2) |
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Uniqueness and Complexity in Discrete Tomography |
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85 | (78) |
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85 | (1) |
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Definitions and preliminaries |
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86 | (2) |
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88 | (9) |
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97 | (11) |
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Further extensions and variations |
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108 | (7) |
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Higher-dimensional X-rays |
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108 | (1) |
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109 | (1) |
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110 | (1) |
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111 | (4) |
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Reconstructionm of Plane Figures from Two Projections |
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115 | (1) |
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Review of the problem and known results |
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115 | (8) |
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Choice of good reconstruction for the discrete case |
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123 | (5) |
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Passage to continuous limit |
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128 | (9) |
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134 | (3) |
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Reconstruction of Two-Valued Functions and Matrices |
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137 | (1) |
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137 | (1) |
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138 | (1) |
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139 | (4) |
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Reconstruction of two-value functions |
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143 | (4) |
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147 | (6) |
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Reconstruction of two-valued matrices |
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153 | (4) |
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Uniqueness in the case of two-valued matrices |
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157 | (6) |
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160 | (3) |
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Reconstruction of Connected Sets from Two Projections |
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163 | (26) |
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163 | (1) |
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164 | (3) |
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167 | (5) |
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The complexity of RSHVP on (h), (p,h), (v), and (p,v) |
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167 | (2) |
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The complexity of RSHVP on (p) and (h,v) |
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169 | (3) |
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Reconstruction of convex polyominoes |
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172 | (17) |
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173 | (2) |
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The spine and its reconstruction |
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175 | (4) |
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The reconstruction algorithm |
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179 | (8) |
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187 | (2) |
II Algorithms |
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189 | (154) |
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Binary Tomography Using Gibbs Priors |
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191 | (22) |
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192 | (1) |
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Gibbs distributions on the hexagonal grid |
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193 | (2) |
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Reconstruction algorithms |
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195 | (3) |
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198 | (6) |
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Finding additional invariant elements |
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204 | (3) |
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Gibbs prior definition using a look-up table |
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207 | (2) |
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209 | (4) |
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211 | (2) |
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Probabilistic Modeling of Discrete Images |
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213 | (24) |
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213 | (2) |
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Image-modeling distributions for discrete images |
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215 | (2) |
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217 | (2) |
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217 | (1) |
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Combining global and local characteristics |
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217 | (2) |
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Recovering images corrupted by additive noice |
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219 | (7) |
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Adaptation to discrete tomographic reconstruction |
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226 | (6) |
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The posterior model and optimization criteria |
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226 | (2) |
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An approximate two-step reconstruction approach |
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228 | (1) |
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229 | (3) |
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232 | (5) |
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232 | (5) |
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Multiscale Bayesian Methods for Discrete Tomogaphy |
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237 | (28) |
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237 | (3) |
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Stochastic data models for tomography |
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240 | (3) |
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Markov random field prior models |
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243 | (2) |
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245 | (2) |
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Estimation of discrete levels |
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247 | (3) |
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250 | (1) |
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251 | (1) |
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252 | (2) |
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254 | (6) |
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260 | (5) |
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261 | (4) |
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An Algebraic Solution for Discrete Tomography |
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265 | (20) |
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265 | (2) |
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265 | (1) |
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266 | (1) |
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Problem formulation and nonuniqueness |
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267 | (1) |
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267 | (1) |
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267 | (1) |
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Reformulation and insights using the DFT |
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268 | (2) |
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Reformulation using the DFT |
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268 | (1) |
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Ambiguity insights using the DFT |
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269 | (1) |
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Relation to phase retrieval |
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269 | (1) |
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Agarwal-Cooley fast convolution |
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269 | (1) |
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Solution using DFT reformulation |
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270 | (2) |
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Solving for X (k1, k2, k3) |
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270 | (1) |
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Linear systems of equations for X (k1, k2, k3) |
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271 | (1) |
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Solving for x (i1, i2, i3) |
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271 | (1) |
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272 | (2) |
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2 x 2 x 2 general problem |
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272 | (1) |
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2 x 2 x 2 general solution |
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272 | (1) |
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273 | (1) |
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Closed-form solution to limited-angle discrete tomography |
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274 | (2) |
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274 | (1) |
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Limited-angle and discrete tomography |
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275 | (1) |
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An explicit formula for bandwidth extrapolation |
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276 | (1) |
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276 | (1) |
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Application to tomography |
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277 | (2) |
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Discussion of application |
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277 | (1) |
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278 | (1) |
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278 | (1) |
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A simple, illustrative numerical example |
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279 | (3) |
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279 | (1) |
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280 | (1) |
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Extrapolation of unknown F(m,n) |
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281 | (1) |
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Application to large images |
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282 | (1) |
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282 | (3) |
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283 | (2) |
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Binary Steering of Nonbinary Iterative Algorithms |
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285 | (12) |
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Introduction: Problem definition, approach, and motivation |
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285 | (1) |
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286 | (3) |
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289 | (5) |
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294 | (3) |
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295 | (2) |
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Reconstuction of Binary Images via the EM Algorithm |
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297 | (46) |
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297 | (4) |
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301 | (2) |
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Properties of the algorithm |
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303 | (1) |
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Implementations and experiments |
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304 | (5) |
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309 | (1) |
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310 | (3) |
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Characterization of local MLEs |
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310 | (2) |
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Convergence of the algorithm |
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312 | (1) |
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313 | (4) |
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315 | (2) |
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Compact Object Reconstruction |
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317 | (1) |
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317 | (1) |
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318 | (4) |
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Classical pixel representation approach |
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319 | (1) |
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320 | (1) |
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Shape reconstruction approach |
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321 | (1) |
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Polygonal and polyhedral shape modeling |
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322 | (8) |
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Exact reconstruction of polygonal shapes in 2D |
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323 | (2) |
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Exact reconstruction of polyhedral shapes in 3D |
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325 | (5) |
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Desciption of the proposed method |
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330 | (2) |
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332 | (7) |
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332 | (2) |
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334 | (5) |
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339 | (4) |
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339 | (4) |
III Applications |
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343 | (129) |
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CT-Assisted Engineering and Manufacturing |
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345 | (18) |
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345 | (2) |
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Agile manufacturing: A new manufacturing paradigm |
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347 | (2) |
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Obtaining the digital model |
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349 | (2) |
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The role of computed tomography: Selected examples |
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351 | (7) |
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The role of discrete tomography in improving the digital model |
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358 | (1) |
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359 | (1) |
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360 | (3) |
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360 | (3) |
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3D Reconstruction from Sparse Radiographic Data |
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363 | (22) |
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363 | (2) |
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Radiographic/tomographic data |
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365 | (4) |
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Idealized photon counting model |
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365 | (1) |
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Compton scattering effects |
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366 | (3) |
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3D maximum a posteriori reconstruction |
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369 | (7) |
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369 | (3) |
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Reconstruction from simulated radiographs |
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372 | (2) |
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Optimization considerations |
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374 | (2) |
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Physical radiographic experiments |
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376 | (5) |
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381 | (4) |
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382 | (3) |
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Heart Chamber Reconstruction from Biplane Angiography |
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385 | (20) |
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385 | (1) |
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Cardiac X-ray angiography |
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386 | (3) |
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Image recording and processing |
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386 | (1) |
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Quantitative biplane angiography |
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387 | (2) |
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389 | (3) |
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389 | (1) |
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Probability-driven reconstruction |
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390 | (2) |
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392 | (2) |
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394 | (3) |
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394 | (1) |
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Ventricular reconstruction |
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395 | (1) |
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396 | (1) |
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397 | (2) |
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397 | (2) |
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399 | (1) |
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399 | (6) |
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401 | (4) |
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Discrete Tomography in Electron Microscopy |
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405 | (12) |
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406 | (1) |
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407 | (3) |
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Discrete tomography in electron microscopy |
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410 | (1) |
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Considering experimental applications: The problems |
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410 | (3) |
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The Contrast Transfer Function (CTF) |
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411 | (1) |
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Discretization of the electron density of specimens |
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412 | (1) |
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413 | (1) |
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414 | (3) |
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414 | (3) |
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Tomography on the 3D-Torus and Crystals |
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417 | (18) |
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417 | (1) |
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418 | (4) |
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422 | (3) |
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Binary tomography on lattices |
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425 | (3) |
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Binary tomography and crystals |
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425 | (1) |
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Stating the binary tomography problem |
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426 | (2) |
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The reconstruction algorithm |
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428 | (1) |
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Final remarks and examples |
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429 | (6) |
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432 | (3) |
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A Recursive Algorithm for Diffuse Planar Tomography |
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435 | (20) |
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435 | (2) |
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437 | (4) |
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440 | (1) |
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440 | (1) |
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441 | (2) |
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Elimination of parameters |
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443 | (4) |
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447 | (1) |
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Appendix I --- Matrix identities |
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448 | (2) |
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448 | (1) |
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449 | (1) |
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449 | (1) |
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Appendix II --- Enforcing range conditions |
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450 | (5) |
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453 | (2) |
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From Orthogonal Projections to Symbolic Projections |
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455 | (17) |
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455 | (2) |
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2D string representations of symbolic pictures |
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457 | (2) |
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459 | (2) |
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Computer-aided design database |
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461 | (3) |
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Geographical information systems |
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464 | (3) |
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Retrieval of similar Chinese characters |
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467 | (1) |
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Three-dimensional image database querying |
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468 | (1) |
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Medical image database system |
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469 | (3) |
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470 | (2) |
Index |
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472 | |