Angle-Independent 3D Reconstructon J Zhang Mrelle Boutn Danel Alaga
Goal: Structure from Moton To reconstruct the 3D geometry of a scene from a set of pctures (e.g. a move of the scene pont reconstructon
revous Work Bundle Adustment Mnmze: E N J omplety: O(JN JN E E Bundle p p
revous Work Bundle Adustment Is a global optmzaton tolerant to mssng samples and outlers. Optmzes both structure and moton even though structure may be the only obectve. Needs a suffcently good ntal guess. Usually used as a fnal refnement stage.
revous Work roectve Reconstructon + upgrade to Eucldean Reconstructon Use eppolar geometry to solve for a proectve reconstructon whch equals to Eucldean reconstructon E multpled by a lnear transform F. Use etra nformaton to get the lnear transform F and upgrade the proectve reconstructon to Eucldean reconstructon E. E F
revous Work roectve Reconstructon + upgrade to Eucldean Reconstructon The frst step s lnear whle the second one s nonlnear. Structure reconstructon depends on moton reconstructon. Senstve to nose need refnement (BA.
Tradtonal roectve Equatons for p cf... N and... J. F R T p y ( ( where p represents the D coordnates of the 3D feature pont observed on pcture c s a constant and F s a 3-by3 by-4 4 matr contanng the camera parameters correspondng to pcture.
hallenge Overcomng the fact that camera pose reconstructon s ll condtoned especally for camera orentaton Small α Bg E E α
Our Approach Varable elmnaton Elmnate camera poston and/or orentaton from the reconstructon process Ths s not self-calbraton whch computes the camera pose on ts own; rather camera poston and/or orentaton s mathematcally elmnated from the equatons
Our Approach: c c c 3 Elmnate amera ose Gaussan Elmnaton (easy + c + c + c 3 + c + c + c 3 3 33 3 3 3 + c 4 + c + c 4 34 Nonlnear Elmnaton (hard F ( 3 F ( 3 F 3 ( 3 k + k + k 3 3 + k 4 k + k 3 3 + k 4 k 33 3 + k 34 G ( 3 G ( 3 G 3 ( 3
Our Approach: Elmnate amera ose Gröbner Bases Degree ncreases rapdly when elmnatng varables. Invarant Based Elmnaton erre-lous Bazn Mrelle Boutn (3 Invarant theory Movng frame
Invarant Based Elmnaton Invarant Based Elmnaton N k N k N 3 k 3... ( (( (... ( (( ( ((... ( ( for for for ( (( y y k ( (( ( (( y y y y k ( (( ( y y y k 3 ( ( y w w w
Invarant Based Elmnaton Invarant Based Elmnaton Smplfy the equatons to make t work for least square Smplfy the equatons to make t work for least square refnement. We focus on lower degree and symmetry. refnement. We focus on lower degree and symmetry. Κ ( ( ( ( y y K ( p p p ( p
Invarant Based Elmnaton A new angle ndependent cost functon for Bundle Adustment Refnement (AIBAR. E N N J [ ( ( ] Κ Here N s the number of 3D ponts J s the number of mages. omplety O(JN hgher than tradtonal BA O(JN JN
Invarant Based Elmnaton Invarant Based Elmnaton J NJ 4 3 N 4 3 NJ J 4 J J 3J J N 4 3 N 4 3 B B B B B B V M V M V M V M V M V M V M V M V M M M M M O O M M O O O O ( ( ( ( ( ( M 3 k k k V ( (( ( ( (( ( ( ( B Observe the orgnal equatons f we know Observe the orgnal equatons f we know and and then the system s lnear. then the system s lnear.
Two ossble Applcatons Speed rorty: Select a number of pars to be use as anchor ponts.. Reconstructed anchor ponts va ntal guess and least squares mnmzaton.. Reconstructed then rest of ponts lnearly. Accuracy rorty: Use least square mnmzaton to refne all 3D ponts and camera centers (AIBAR.
Two ossble Applcatons Obectve: To compare the robustness of our method wth SBA (Sparse Bundle Adustment.
Applcaton Estmate camera center and camera orentaton as best as we can. Add nose n camera orentaton ncrementally. a Reconstruct structure usng lnear trangulaton to generate ntal guess. b Use the result of a as ntal guess for SBA and our method. c ompare the results.
Eperments. heck Board number of ponts96 number of mages48
6 5 Reconstructon Error(mm 4 3 Our Method Lnear BA...3.4.5.6 amera Orentaton Error (radan
Eperments. Graffe number of ponts48 number of mages36
No nose 6 degrees nose degrees nose Actual model Lnear wth 6 degrees nose BA wth degrees nose
Eperments 3. House number of ponts67 number of mages amera poston error: 4% of the model space dagonal amera orentaton error: 7 degrees.
Actual model Our method Lnear BA
Applcaton Estmate camera center camera orentaton structure as best as we can. Add error n camera center camera orentaton and structure ncrementally. a Refne structure+camera center+camera orentaton usng SBA. b Refne structure+camera center usng AIBAR.
Eperments. heck Board number of ponts96 number of mages48 Tme cost: AIBAR: 838s. SBA: s.
8 7 AIBAR SBA 6 reconstructon error(% 5 4 3 3 4 5 6 7 8 9 nput nose
Image Sequence arttonng omplety comparson: Our: O(JN BA: O(JN JN To compensate for the addtonal computatonal cost subdvde the mage sequence nto dsont subsets accordng to dfferent type of sequence. Insde lookng out sequence Outsde lookng n sequence
Insde lookng out sequence Group Group Group 3
Outsde lookng n sequence Group Group Group 3
Eperments. heck Board number of ponts96 number of mages48 Tme cost: AIBAR (wthout partton: 838s. AIBAR (wth partton nto 8 subsets: 4s. SBA: s.
8 AIBAR(whole OT 7 AIBAR(partton OT SBA(whole OT 6 reconstructon error(% 5 4 3 3 4 5 6 7 8 9 nput nose
Eperments. Graffe number of ponts48 number of mages36 Tme cost for dfferent parttons: AIBAR None Small Medum Large (s 88 53 96 SBA None Small Medum Large (s 698 7 43
4 AIBAR(small partton OT AIBAR(medum partton OT AIBAR(large partton OT SBA(whole OT SBA(small partton OT SBA(medum partton OT SBA(large partton OT reconstructon error(% 8 6 4 3 4 5 6 7 nput nose
Actual Model SBA (medum nose SBA (large nose AIBAR (no nose AIBAR (medum nose AIBAR (large nose
Eperments 3. Floor number of ponts3688 number of mages644 artton: 55 subsets
5 reconstructon error(% 5 5 AIBAR(parton SBA(parton 3 4 5 6 7 nput nose
oncluson Low dmenson:. Robust : angle ndependent. omplety: O(JN. Scalable: partton or usng anchor ponts.
Future Work Elmnate camera centers Lower the complety. Improve the parttonng.
Thank you!