v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
314 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXx 1x 4x 6x 3x 5x 2Figure 79: Incomplete EDM corresponding to this dimensionless EDMgraph provides unique isometric reconstruction in R 2 . (drawn freehand, nosymmetry intended)ˇx 4x 1x 2ˇx 3ˇx 5Figure 80: Two sensors • and three anchors ◦ in R 2 . (Ye) Connectingline-segments denote known absolute distances. Incomplete EDMcorresponding to this dimensionless EDM graph provides unique isometricreconstruction in R 2 .
5.4. EDM DEFINITION 315105ˇx 4ˇx 50ˇx 3−5x 2x 1−10−15−6 −4 −2 0 2 4 6 8Figure 81: Given in red are two discrete linear trajectories of sensors x 1and x 2 in R 2 localized by algorithm (756) as indicated by blue bullets • .Anchors ˇx 3 , ˇx 4 , ˇx 5 corresponding to Figure 80 are indicated by ⊗ . Whentargets and bullets • coincide under these noiseless conditions, localizationis successful. On this run, two visible localization errors are due to rank-3Gram optimal solutions. These errors can be corrected by choosing a differentnormal in objective of minimization.
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- Page 289 and 290: Chapter 5Euclidean Distance MatrixT
- Page 291 and 292: 5.2. FIRST METRIC PROPERTIES 291cor
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- Page 297 and 298: 5.4. EDM DEFINITION 297The collecti
- Page 299 and 300: 5.4. EDM DEFINITION 2995.4.2 Gram-f
- Page 301 and 302: 5.4. EDM DEFINITION 301D ∈ EDM N
- Page 303 and 304: 5.4. EDM DEFINITION 3035.4.2.2.1 Ex
- Page 305 and 306: 5.4. EDM DEFINITION 305ten affine e
- Page 307 and 308: 5.4. EDM DEFINITION 307spheres:Then
- Page 309 and 310: 5.4. EDM DEFINITION 309By eliminati
- Page 311 and 312: 5.4. EDM DEFINITION 311whereΦ ij =
- Page 313: 5.4. EDM DEFINITION 3135.4.2.2.5 De
- Page 317 and 318: 5.4. EDM DEFINITION 317corrected by
- Page 319 and 320: 5.4. EDM DEFINITION 319aptly be app
- Page 321 and 322: 5.4. EDM DEFINITION 321As before, a
- Page 323 and 324: 5.4. EDM DEFINITION 323where ([√t
- Page 325 and 326: 5.4. EDM DEFINITION 325because (A.3
- Page 327 and 328: 5.5. INVARIANCE 3275.5.1.0.1 Exampl
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- Page 337 and 338: 5.7. EMBEDDING IN AFFINE HULL 337Fo
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- Page 361 and 362: 5.11. EDM INDEFINITENESS 361(confer
- Page 363 and 364: 5.11. EDM INDEFINITENESS 363we have
5.4. EDM DEFINITION 315105ˇx 4ˇx 50ˇx 3−5x 2x 1−10−15−6 −4 −2 0 2 4 6 8Figure 81: Given in red are two discrete linear trajectories of sensors x 1and x 2 in R 2 localized by algorithm (756) as indicated by blue bullets • .Anchors ˇx 3 , ˇx 4 , ˇx 5 corresponding to Figure 80 are indicated by ⊗ . Whentargets and bullets • coincide under these noiseless conditions, localizationis successful. On this run, two visible localization errors are due to rank-3Gram optimal solutions. These errors can be corrected by choosing a differentnormal in objective of minimization.
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