v2004.06.19 - Convex Optimization

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174 CHAPTER 4. EUCLIDEAN DISTANCE MATRIX(a)(c)(b)(d)(f)(e)Figure 4.6: Map of United States of America showing some state boundariesand the Great Lakes. All plots made using 5020 connected points. Anydifference in scale in (a) through (d) is an artifact of plotting routine.(a) shows original map made from decimated (latitude, longitude) data.(b) shows original map data rotated to highlight curvature of Earth.(c) Map isometrically reconstructed from the EDM.(d) Same reconstructed map illustrating curvature.(e)(f) Two views of one isotonic reconstruction; problem (506) with no sortconstraint Ξd (and no removal of hidden lines).
4.11. RECONSTRUCTION EXAMPLES 1754.11 Reconstruction examples4.11.1 Isometric reconstruction4.11.1.0.1 Example. Map of the USA.The most fundamental application of EDMs is to reconstruct relative pointposition given only inter-point distance information. Drawing a map of theUnited States is a good illustration of isometric reconstruction from completedistance data. We obtained latitude and longitude information for the coast,border, states, and Great Lakes from the usalo atlas data file within theMATLAB Mapping Toolbox; the conversion to Cartesian coordinates (x,y,z)via:φ ∆ = π/2 − latitudeθ ∆ = longitudex = sin(φ) cos(θ)y = sin(φ) sin(θ)z = cos(φ)(497)We used 64% of the available map data to calculate EDM D from N = 5020points. The original (decimated) data and its isometric reconstruction areshown in Figure 4.6(a)-(d). The MATLAB code is in §F.3.1. The eigenvaluescomputed for (486) areλ(−V T NDV N ) = {199.8, 152.3, 2.465, 0, 0, 0, ...} (498)The 0 eigenvalues have numerical error on the order of 2E-13 ; meaning, theEDM data indicates three dimensions (r = 3) are required for reconstructionto nearly machine precision.4.11.2 Isotonic reconstructionSometimes only comparative information about distance is known (e.g., theEarth is closer to the Moon than it is to the Sun). Suppose, for example,the EDM D for three points is unknown:D = [d ij ] =⎡⎣0 d 12 d 13d 12 0 d 23d 13 d 23 0⎤⎦ ∈ S 3 0 (323)
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Euclidean Distance GeometryviaConve
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for Jennie Columba♦♦Antonio♦a
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illustrations: Mayura Drawtypesetti
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8 CONTENTS2.7 Convex polyhedra . .
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10 CONTENTS5.3 EDM cone by inverse
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12 CONTENTSB.2 Doublet . . . . . .
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14 CONTENTSG Notation and some defi
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16 CHAPTER 1. PRELUDEThese pages co
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18 CHAPTER 1. PRELUDEChapter 8 inve
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20 CHAPTER 2. CONVEX GEOMETRY2.1 Co
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22 CHAPTER 2. CONVEX GEOMETRY(a)R(b
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24 CHAPTER 2. CONVEX GEOMETRYTogeth
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26 CHAPTER 2. CONVEX GEOMETRYand wh
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28 CHAPTER 2. CONVEX GEOMETRY2.1.2
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30 CHAPTER 2. CONVEX GEOMETRYwhere
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32 CHAPTER 2. CONVEX GEOMETRYLike b
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34 CHAPTER 2. CONVEX GEOMETRYGiven
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36 CHAPTER 2. CONVEX GEOMETRYGiven
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38 CHAPTER 2. CONVEX GEOMETRYThis r
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42 CHAPTER 2. CONVEX GEOMETRY(a)Yy
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44 CHAPTER 2. CONVEX GEOMETRY1) If
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46 CHAPTER 2. CONVEX GEOMETRY2.4.1.
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48 CHAPTER 2. CONVEX GEOMETRYABCDFi
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50 CHAPTER 2. CONVEX GEOMETRYAn aff
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52 CHAPTER 2. CONVEX GEOMETRYXXFigu
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54 CHAPTER 2. CONVEX GEOMETRYcone S
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58 CHAPTER 2. CONVEX GEOMETRYBCADFi
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64 CHAPTER 2. CONVEX GEOMETRY√2β
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66 CHAPTER 2. CONVEX GEOMETRYwhich
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68 CHAPTER 2. CONVEX GEOMETRYequals
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70 CHAPTER 2. CONVEX GEOMETRYAssumi
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74 CHAPTER 2. CONVEX GEOMETRYFrom t
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76 CHAPTER 2. CONVEX GEOMETRYFor an
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80 CHAPTER 2. CONVEX GEOMETRY• (S
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82 CHAPTER 2. CONVEX GEOMETRYmerely
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84 CHAPTER 2. CONVEX GEOMETRYy ∈
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88 CHAPTER 2. CONVEX GEOMETRYthat f
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90 CHAPTER 2. CONVEX GEOMETRYWhen X
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92 CHAPTER 2. CONVEX GEOMETRY2.9 Fo
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94 CHAPTER 2. CONVEX GEOMETRYFor γ
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96 CHAPTER 2. CONVEX GEOMETRYConver
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98 CHAPTER 2. CONVEX GEOMETRYAssume
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100 CHAPTER 2. CONVEX GEOMETRYwhere
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102 CHAPTER 2. CONVEX GEOMETRYbiort
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X 1 =4X †T1 =⎡⎢⎣⎡⎢⎣10
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106 CHAPTER 2. CONVEX GEOMETRYFigur
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108 CHAPTER 2. CONVEX GEOMETRY∂K
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110 CHAPTER 2. CONVEX GEOMETRYΓ 4
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112 CHAPTER 2. CONVEX GEOMETRY∂K
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114 CHAPTER 3. GEOMETRY OF CONVEX F
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224 CHAPTER 5. EDM CONE
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226 CHAPTER 6. CONVEX OPTIMIZATIONT
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230 CHAPTER 6. CONVEX OPTIMIZATION6
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232 CHAPTER 6. CONVEX OPTIMIZATIONW
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234 CHAPTER 6. CONVEX OPTIMIZATION6
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236 CHAPTER 6. CONVEX OPTIMIZATIONF
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238 CHAPTER 6. CONVEX OPTIMIZATIONt
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242 CHAPTER 6. CONVEX OPTIMIZATION
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244 CHAPTER 7. EDM PROXIMITY7.0.1 L
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246 CHAPTER 7. EDM PROXIMITY......
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248 CHAPTER 7. EDM PROXIMITYparticu
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250 CHAPTER 7. EDM PROXIMITYmatrice
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252 CHAPTER 7. EDM PROXIMITY7.1.3 C
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254 CHAPTER 7. EDM PROXIMITYis dime
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256 CHAPTER 7. EDM PROXIMITYFor lac
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258 CHAPTER 7. EDM PROXIMITYH is no
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260 CHAPTER 7. EDM PROXIMITYwhere
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262 CHAPTER 7. EDM PROXIMITYbe zero
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264 CHAPTER 7. EDM PROXIMITYFor the
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266 CHAPTER 7. EDM PROXIMITYK ∗ 2
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K ρ+2λ268 CHAPTER 7. EDM PROXIMIT
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270 CHAPTER 7. EDM PROXIMITYsimply
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272 CHAPTER 7. EDM PROXIMITY
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274 CHAPTER 8. EDM COMPLETION(b)(c)
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276 CHAPTER 8. EDM COMPLETION8.1.0.
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278 CHAPTER 8. EDM COMPLETION8.2 Re
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280 CHAPTER 8. EDM COMPLETION8.2.2
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282 CHAPTER 8. EDM COMPLETIONR 2 ha
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284 CHAPTER 8. EDM COMPLETION0.50.4
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290 CHAPTER 9. PIANO TUNING
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292 APPENDIX A. LINEAR ALGEBRAmatri
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294 APPENDIX A. LINEAR ALGEBRACorol
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296 APPENDIX A. LINEAR ALGEBRA...an
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298 APPENDIX A. LINEAR ALGEBRAλ (
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300 APPENDIX A. LINEAR ALGEBRA• M
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302 APPENDIX A. LINEAR ALGEBRA• F
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304 APPENDIX A. LINEAR ALGEBRA• F
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306 APPENDIX A. LINEAR ALGEBRAA.3.1
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310 APPENDIX A. LINEAR ALGEBRAThe o
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314 APPENDIX A. LINEAR ALGEBRAThe p
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316 APPENDIX A. LINEAR ALGEBRAR{u i
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320 APPENDIX A. LINEAR ALGEBRANow s
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322 APPENDIX A. LINEAR ALGEBRAwhich
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324 APPENDIX A. LINEAR ALGEBRA
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326 APPENDIX B. SIMPLE MATRICESB.1
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328 APPENDIX B. SIMPLE MATRICESB.1.
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330 APPENDIX B. SIMPLE MATRICESby t
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332 APPENDIX B. SIMPLE MATRICESN(u
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334 APPENDIX B. SIMPLE MATRICESthe
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340 APPENDIX B. SIMPLE MATRICESThis
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342 APPENDIX B. SIMPLE MATRICES
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344 APPENDIX C. SOME OPTIMAL ANALYT
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352 APPENDIX D. MATRIX CALCULUSwhil
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354 APPENDIX D. MATRIX CALCULUSa cu
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356 APPENDIX D. MATRIX CALCULUSprod
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358 APPENDIX D. MATRIX CALCULUS∇
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360 APPENDIX D. MATRIX CALCULUSin m
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362 APPENDIX D. MATRIX CALCULUSNoti
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364 APPENDIX D. MATRIX CALCULUS⎡
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366 APPENDIX D. MATRIX CALCULUSD.1.
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368 APPENDIX D. MATRIX CALCULUSExam
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370 APPENDIX D. MATRIX CALCULUSD.2
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372 APPENDIX D. MATRIX CALCULUS
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382 APPENDIX D. MATRIX CALCULUS
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384 APPENDIX E. PSEUDOINVERSE, PROJ
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436 APPENDIX F. MATLAB PROGRAMS...F
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438 APPENDIX F. MATLAB PROGRAMS...F
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440 APPENDIX F. MATLAB PROGRAMS...l
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442 APPENDIX F. MATLAB PROGRAMS...s
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444 APPENDIX F. MATLAB PROGRAMS...
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446 APPENDIX G. NOTATION AND SOME D
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456 BIBLIOGRAPHY[9] Günter M. Zieg
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458 BIBLIOGRAPHY[28] Roger A. Horn
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460 BIBLIOGRAPHY[54] Hong-Xuan Huan
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462 BIBLIOGRAPHY[76] Peter Gritzman
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464 BIBLIOGRAPHYtors, Handbook of S
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466 BIBLIOGRAPHY[117] Shao-Po Wu an
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468 BIBLIOGRAPHY[134] Maryam Fazel,
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470 BIBLIOGRAPHY[154] George P. H.
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472 BIBLIOGRAPHY[178] Jean-Baptiste
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474 BIBLIOGRAPHY
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v2004.06.19 - Convex Optimization

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