CE 603 Photogrammetry II. Condition number = 2.7E+06

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1 CE 60 Phoogrammery II Condiion number.7e06

2 CE 60 Phoogrammery II Condiion number.8

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7 CE 60 Phoogrammery II Simulaed Block 6 phoos, 9 pass poins, conrol poins 4 equaions, 450 unknowns Phoo & poin layou very similar o he Purdue block Pba.m on sun ulra-0: ~/ minue per ieraion (all numerical parials, and full marix inverse) B Marix, from command: spy(b)

8 CE 60 Phoogrammery II Simulaed Block normal equaions, 450x450, pariions shown: camera inernal parameers (6), phoo exerior orienaion parameers (6 per phoo), and ground poins ( per poin) The off diagonal block is nonzero if ha poin occurs on ha phoo The marix is abou 85% zeros. Le s look a an efficien block gauss eliminaion mehod o derive a se of reduced normal equaions from he full normal equaions Figure from: spy()

9 CE 60 Phoogrammery II Technique o move from full normals o reduced normals: use block gauss eliminaion o eliminae one poin a a ime This figure is a schemaic represenaion of The off-diagonal pariion is ofen sparse bu we will assume full

10 CE 60 Phoogrammery II Block Gauss Eliminaion Make a new pariion corresponding he unknown(s) ha we wish o eliminae This is he parameer vecor pariion ha we wan o eliminae Recall: forward eliminaion followed by back subsiuion for complee soluion

11 CE 60 Phoogrammery II Label he pariions. oe ha and are zero.

12 CE 60 Phoogrammery II ( ) ( ) Eliminae ha 0 ow subsiue firs wo equaions from eqn., remember ha expression ino he Eliminaion Sep Remember o save he s and he from his sep so ha afer we have solved for dela- we can come back and solve for dela- oe only changes are in he pariion and in he pariion, ha is why i is so efficien oice ha he eliminaion of a poin does no involve he oher poins hence hey can be accumulaed and eliminaed righ away, never form full

13 CE 60 Phoogrammery II Afer Eliminaing All of he Poins When all poins have been eliminaed, one by one, by he procedure jus described, we are lef wih he reduced normal equaions in which now he only remaining unknowns are he phoo and camera parameers. The rules, now, for when a block is nonzero: (a) he diagonal blocks are nonzero as before, (b) any off-diagonal block (corresponding o wo phoos) is nonzero if hose wo phoos share a common poin. Wih large blocks and parallel fligh lines, even his reduced normal equaion marix is sill sparse. Is srucure is banded (or banded-bordered). A similar pariioning plan can be used o successively furher reduce his unil i is full (number of seps is relaed o he bandwidh). A famous phoogrammeris, Duane Brown, did much work on he efficien soluion of large phoogrammeric blocks, and referred o his procedure as recursive pariioning.

14 CE 60 Phoogrammery II w n Band Marices Soluion of Sysem wih full marix akes on he order of n operaions. Soluion of band sysem akes on he order of w n ha can be much less for large n and small w. Do i by special pariioning o creae a zero pariion.

15 CE 60 Phoogrammery II Make he pariion so ha is zero. Then we can efficienly eliminae dela

16 CE 60 Phoogrammery II Eliminae ha ( ) ow subsiue firs wo equaions 0 from eqn., ha expression ino he ( ) remember oice all aciviy akes place wihin he band Similar o previous eliminaion sep This is he forward eliminaion sep. Do i many imes, saving inermediae resuls. Then do back subsiiuion, recalling saved resuls unil he full banded sysem is solved. Tha ges you he phoo parameers, hen do back subsiuion for he eliminaed poins, and you are done for his ieraion!

17 CE 60 Phoogrammery II Process daa by poin Soluion Sraegy for Block Adjusmen When all conribuions (equaions) for a poin have been consruced, eliminae i, and save inermediae resuls Afer all poins have been processed and eliminaed, you have lef only he camera and phoo parameers, which are banded or bandbordered Use band marix processing o efficienly solve for camera/phoo parameers Rerieve saved inermediae resuls and solve for all of he ground poins Finished his ieraion keep going unil you converge!