The Art of Solving Minimal Problemscmp.felk.cvut.cz/.../SMP-2015-Warm-Start-Kneip.pdf · Tricks...

The Art of Solving Minimal ProblemsTricks session

Laurent Kneip

Research School of EngineeringCentre of Excellence for Robotic Vision

The Australian National University

11th December, 2015

Tricks sessionWarm startIdeal case assumption

Warm startMinimal Relative Rotation with Central Camera

L. Kneip, R. Siegwart, and M. Pollefeys. Finding the Exact Rotation Between Two Images Independently of the Translation.ECCV, Florence, Italy, 2012.

Solving Minimal Problems – http://cmp.felk.cvut.cz/minimal-iccv-2015/index.html 2/12

Minimal Relative Rotation with Central CameraEpipolar Constraint

Pure translation: epipolar plane normals coplanar

Goal: constrain rotation

Minimal Relative Rotation with Central CameraTranslation Independent Computation of Rotation

Idea: Unrotate f′i such that ni coplanar

Epipolar plane normal: ni = fi ×Rcc′f′i

Coplanarity given if three ni linearly dependentAnalyze determinant of 3×3 matrix with three ni

3 DoF⇒ 3 constraints needed

Formulation 2: Minimal relative rotation|(f1×Rc

c′f′1) (f2×Rc

c′f′2) (f3×Rc

c′f′3)|= 0

|(f1×Rcc′f′1) (f2×Rc

c′f′2) (f4×Rc

c′f′4)|= 0

c′f′2) (f5×Rc

c′f′5)|= 0

Solving Grobner basis⇒ 20 rotation matrices8000 code lines

c′f′1) (f2×Rc

c′f′2) (f3×Rc

c′f′3)|= 0

c′f′2) (f4×Rc

c′f′4)|= 0

c′f′2) (f5×Rc

c′f′5)|= 0

c′f′1) (f2×Rc

c′f′2) (f3×Rc

c′f′3)|= 0

c′f′2) (f4×Rc

c′f′4)|= 0

c′f′2) (f5×Rc

c′f′5)|= 0

c′f′1) (f2×Rc

c′f′2) (f3×Rc

c′f′3)|= 0

c′f′2) (f4×Rc

c′f′4)|= 0

c′f′2) (f5×Rc

c′f′5)|= 0

c′f′1) (f2×Rc

c′f′2) (f3×Rc

c′f′3)|= 0

c′f′2) (f4×Rc

c′f′4)|= 0

c′f′2) (f5×Rc

c′f′5)|= 0

c′f′1) (f2×Rc

c′f′2) (f3×Rc

c′f′3)|= 0

c′f′2) (f4×Rc

c′f′4)|= 0

c′f′2) (f5×Rc

c′f′5)|= 0

Minimal Relative Rotation with Central CameraWarm Start

A first solution:Standard Parametrization:Rc

c′ =(c1 c2 c3

r1r2r3

=r11 r12 r13r21 r22 r23r31 r32 r33

Equations:

|(f1 ×Rcc′

f′1) (f2 ×Rcc′

f′2) (f3 ×Rcc′

f′3)| = 0|(f1 ×Rc

c′f′1) (f2 ×Rc

c′f′2) (f4 ×Rc

c′f′4)| = 0

|(f1 ×Rcc′

f′1) (f2 ×Rcc′

f′2) (f5 ×Rcc′

f′5)| = 0|(f1 ×Rc

c′f′1) (f3 ×Rc

c′f′3) (f4 ×Rc

c′f′4)| = 0

|(f1 ×Rcc′

f′1) (f3 ×Rcc′

f′3) (f5 ×Rcc′

f′5)| = 0|(f1 ×Rc

c′f′1) (f4 ×Rc

c′f′4) (f5 ×Rc

c′f′5)| = 0

|(f2 ×Rcc′

f′2) (f3 ×Rcc′

f′3) (f4 ×Rcc′

f′4)| = 0|(f2 ×Rc

c′f′2) (f3 ×Rc

c′f′3) (f5 ×Rc

c′f′5)| = 0

|(f2 ×Rcc′

f′2) (f4 ×Rcc′

f′4) (f5 ×Rcc′

f′5)| = 0|(f3 ×Rc

c′f′3) (f4 ×Rc

c′f′4) (f5 ×Rc

c′f′5)| = 0

r1Rc Tc′

=(1 0 0

(rT2 rT

Macaulay output{2}(9)mmmmmmmmm{3}(28)mmmmmmmmmmmmmmmmmmmmmmmm{4}(122)mmmomomooooooomommooooooomoooooooomoooommmmmmommmoommmmmmmooooooooooomoomoommmoommmmmmmmmmmoooommmmmmmmoooo{5}(330)oooooooooooooooooooooooooooomoomooooooooooommmmmmmmmmooooommmmmmmmmmmmmooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo{6}(141)ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooonumber of (nonminimal) gb elements = 131number of monomials = 3294#reduction steps = 42277#spairs done = 3866ncalls = 11190nloop = 46519nsaved = 130

Trick: add more equations!

|(f1 ×Rcc′

f′1) (f2 ×Rcc′

f′2) (f3 ×Rcc′

f′3)| = 0|(f1 ×Rc

c′f′1) (f2 ×Rc

c′f′2) (f4 ×Rc

c′f′4)| = 0

|(f1 ×Rcc′

f′1) (f2 ×Rcc′

f′2) (f5 ×Rcc′

f′5)| = 0|(f1 ×Rc

c′f′1) (f3 ×Rc

c′f′3) (f4 ×Rc

c′f′4)| = 0

|(f1 ×Rcc′

f′1) (f3 ×Rcc′

f′3) (f5 ×Rcc′

f′5)| = 0|(f1 ×Rc

c′f′1) (f4 ×Rc

c′f′4) (f5 ×Rc

c′f′5)| = 0

|(f2 ×Rcc′

f′2) (f3 ×Rcc′

f′3) (f4 ×Rcc′

f′4)| = 0|(f2 ×Rc

c′f′2) (f3 ×Rc

c′f′3) (f5 ×Rc

c′f′5)| = 0

|(f2 ×Rcc′

f′2) (f4 ×Rcc′

f′4) (f5 ×Rcc′

f′5)| = 0|(f3 ×Rc

c′f′3) (f4 ×Rc

c′f′4) (f5 ×Rc

c′f′5)| = 0

r1Rc Tc′

=(1 0 0

(rT2 rT

3 = 1cT

1 Rcc′

=(1 0 0

(c2 c3

)c1 ×c2 −c3 = 0c2 ×c3 −c1 = 0c3 ×c1 −c2 = 0

New Macaulay output{2}(20)mmmmmmmmmmmmmmmmmmmm{3}(74)oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooommmmmmmmmm{4}(62)mmmmmmmmmmmmmmmmmmmmommmmmmooooooooooooooooooooooooooooooooooo{5}(144)oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooonumber of (nonminimal) gb elements = 56number of monomials = 873#reduction steps = 12993#spairs done = 1261ncalls = 3059nloop = 13367nsaved = 63

Minimal Relative Rotation with Central CameraWarm Start: Comparison{2}(9)mmmmmmmmm{3}(28)mmmmmmmmmmmmmmmmmmmmmmmm{4}(122)mmmomomooooooomommooooooomoooooooomoooommmmmmommmoommmmmmmooooooooooomoomoommmoommmmmmmmmmmoooommmmmmmmoooo{5}(330)oooooooooooooooooooooooooooomoomooooooooooommmmmmmmmmooooommmmmmmmmmmmmooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo{6}(141)ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooonumber of (nonminimal) gb elements = 131number of monomials = 3294#reduction steps = 42277#spairs done = 3866ncalls = 11190nloop = 46519nsaved = 130

{2}(20)mmmmmmmmmmmmmmmmmmmm{3}(74)oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooommmmmmmmmm{4}(62)mmmmmmmmmmmmmmmmmmmmommmmmmooooooooooooooooooooooooooooooooooo{5}(144)oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooonumber of (nonminimal) gb elements = 56number of monomials = 873#reduction steps = 12993#spairs done = 1261ncalls = 3059nloop = 13367nsaved = 63

Absolute Pose with Non-Central Cameran-Point Case: NPnP

L.Kneip, H. Li, and Y. Seo. UPnP: An optimal O(n) solution to the absolute pose problem with universal applicability.

ECCV, Zurich, Switzerland, 2014.

Problem formulation:

minimize E =∑

εi = sT

i Ai ATi βi

βTi Ai βT

] s = sT Ms.

subject to qT q = 1 (1)

Lagrange multiplier method:

∂E∂q1...∂E∂q4

= −λ ·

2q1...

qT q = 1

80 solutionsSolving Minimal Problems – http://cmp.felk.cvut.cz/minimal-iccv-2015/index.html 11/12

Assume ideal case:Some stationary points of E fulfil norm constraint exactly!Set Lagrange multiplier λ to 0⇒16 solutions!

∂E∂q1...∂E∂q4

qT q = 1

The Art of Solving Minimal Problemscmp.felk.cvut.cz/.../SMP-2015-Warm-Start-Kneip.pdf · Tricks...

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