Angle-Independent 3D Reconstruction. Ji Zhang Mireille Boutin Daniel Aliaga
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1 Angle-Independent 3D Reconstructon J Zhang Mrelle Boutn Danel Alaga
2 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
3 revous Work Bundle Adustment Mnmze: E N J omplety: O(JN JN E E Bundle p p
4 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.
5 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
6 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.
7 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.
8 hallenge Overcomng the fact that camera pose reconstructon s ll condtoned especally for camera orentaton Small α Bg E E α
9 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
10 Our Approach: c c c 3 Elmnate amera ose Gaussan Elmnaton (easy + c + c + c 3 + c + c + c c 4 + c + c 4 34 Nonlnear Elmnaton (hard F ( 3 F ( 3 F 3 ( 3 k + k + k k 4 k + k k 4 k k 34 G ( 3 G ( 3 G 3 ( 3
11 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
12 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
13 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
14 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
15 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.
16 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.
17 Two ossble Applcatons Obectve: To compare the robustness of our method wth SBA (Sparse Bundle Adustment.
18 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.
19 Eperments. heck Board number of ponts96 number of mages48
20 6 5 Reconstructon Error(mm 4 3 Our Method Lnear BA amera Orentaton Error (radan
21 Eperments. Graffe number of ponts48 number of mages36
22 No nose 6 degrees nose degrees nose Actual model Lnear wth 6 degrees nose BA wth degrees nose
23 Eperments 3. House number of ponts67 number of mages amera poston error: 4% of the model space dagonal amera orentaton error: 7 degrees.
24 Actual model Our method Lnear BA
25 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.
26 Eperments. heck Board number of ponts96 number of mages48 Tme cost: AIBAR: 838s. SBA: s.
27 8 7 AIBAR SBA 6 reconstructon error(% nput nose
28 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
29 Insde lookng out sequence Group Group Group 3
30 Outsde lookng n sequence Group Group Group 3
31 Eperments. heck Board number of ponts96 number of mages48 Tme cost: AIBAR (wthout partton: 838s. AIBAR (wth partton nto 8 subsets: 4s. SBA: s.
32 8 AIBAR(whole OT 7 AIBAR(partton OT SBA(whole OT 6 reconstructon error(% nput nose
33 Eperments. Graffe number of ponts48 number of mages36 Tme cost for dfferent parttons: AIBAR None Small Medum Large (s SBA None Small Medum Large (s
34 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(% nput nose
35 Actual Model SBA (medum nose SBA (large nose AIBAR (no nose AIBAR (medum nose AIBAR (large nose
36 Eperments 3. Floor number of ponts3688 number of mages644 artton: 55 subsets
37
38 5 reconstructon error(% 5 5 AIBAR(parton SBA(parton nput nose
39 oncluson Low dmenson:. Robust : angle ndependent. omplety: O(JN. Scalable: partton or usng anchor ponts.
40 Future Work Elmnate camera centers Lower the complety. Improve the parttonng.
41 Thank you!
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