Geometric Registration for Deformable Shapes 2.2 Deformable Registration

Transcription

1 Geometric Registration or Deormable Shapes 2.2 Deormable Registration Variational Model Deormable ICP

2 Variational Model What is deormable shape matching?

3 Example? What are the Correspondences? Eurographics 2010 Course Geometric Registration or Deormable Shapes 3

4 What are we looking or? Problem Statement: Given: Two suraces S 1, S 2 R 3? S 1 S 2 We are looking or: A reasonable deormation unction : S 1 R 3 that brings S 1 close to S 2 Eurographics 2010 Course Geometric Registration or Deormable Shapes 4

5 Example? Correspondences? no shape match too much deormation optimum Eurographics 2010 Course Geometric Registration or Deormable Shapes 5

6 This is a Trade-O Deormable Shape Matching is a Trade-O: We can match any two shapes using a weird deormation ield We need to trade-o: Shape matching close to data Regularity o the deormation ield reasonable match Eurographics 2010 Course Geometric Registration or Deormable Shapes 6

7 Variational Model Components: Matching Distance: Deormation / rigidity: Eurographics 2010 Course Geometric Registration or Deormable Shapes 7

8 Variational Model Variational Problem: Formulate as an energy minimization problem: E = E match + E regularizer Eurographics 2010 Course Geometric Registration or Deormable Shapes 8

9 Part 1: Shape Matching Assume: Objective Function: E match = S, dist S 1,2 1 2 S 2 S 1 Example: least squares distance E match = 1 x S dist x 1 1, S 2 2 dx 1 Other distance measures: Hausdor distance, L p -distances, etc. L 2 measure is requently used models Gaussian noise Eurographics 2010 Course Geometric Registration or Deormable Shapes 9

10 Point Cloud Matching Implementation example: Scan matching Given: S 1, S 2 as point clouds S 1 = {s 1 1,, s 1 n } S 2 = {s 2 1,, s 2 m } Energy unction: m match S1 E = dist S1, s m i= 1 2 How to measure dist S 1,x? Estimate distance to a point sampled surace i 2 s i 2 i S 1 Eurographics 2010 Course Geometric Registration or Deormable Shapes 10

11 Surace approximation s i 2 S 1 Solution #1: Closest point matching Point-to-point energy E match = S1 m m i= 1 dist 2 2 s, NN s i in S 1 i 2 Eurographics 2010 Course Geometric Registration or Deormable Shapes 11

12 Surace approximation s i 2 S 1 Solution #2: Linear approximation Point-to-plane energy Fit plane to k-nearest neighbors k proportional to noise level, typically k 6 20 Eurographics 2010 Course Geometric Registration or Deormable Shapes 12

13 Surace approximation s i 2 S 1 Solution #3: Higher order approximation Higher order itting e.g. quadratic Moving least squares Eurographics 2010 Course Geometric Registration or Deormable Shapes 13

14 Variational Model Variational Problem: Formulate as an energy minimization problem: E = E match + E regularizer Eurographics 2010 Course Geometric Registration or Deormable Shapes 14

15 Part II: Deormation Model What is a nice deormation ield? Isometric elastic energies E regularizer Extrinsic volumetric deormation Intrinsic as-isometric-as possible embedding Thin shell model Preserves shape metric plus curvature Thin-plate splines Allowing strong deormations, but keep shape Eurographics 2010 Course Geometric Registration or Deormable Shapes 15

16 Elastic Volume Model Extrinsic Volumetric As-Rigid-As Possible Embed source surace S 1 in volume should preserve 3 3 metric tensor least squares E regularizer = [ ] T I 2 dx V 1 V 1 ambient space irst undamental orm I R 3 3 V 1 S 1 S 2 Eurographics 2010 Course Geometric Registration or Deormable Shapes 16

17 Volume Model Variant: Thin-Plate-Splines Use regularizer that penalizes curved deormation E regularizer = V 1 H x 2 dx second derivative R 3 3 V 1 ambient space H = V 1 S 1 S 2 Eurographics 2010 Course Geometric Registration or Deormable Shapes 17

18 How does the deormation look like? as-rigid-as possible volume original thin plate splines Eurographics 2010 Course Geometric Registration or Deormable Shapes

19 Isometric Regularizer Intrinsic Matching 2-Maniold Target shape is given and complete Isometric embedding E regularizer = S 1 [ ] T I 2 dx irst und. orm S 1, intrinsic tangent space S 1 S 2 Eurographics 2010 Course Geometric Registration or Deormable Shapes 19

20 Elastic Thin Shell Regularizer Thin Shell Energy Dierential geometry point o view Preserve irst undamental orm I Preserve second undamental orm II in a least least squares sense S 1 II II I I Complicated to implement S 2 Usually approximated Volumetric shells as shown beore Other approximation next slide Eurographics 2010 Course Geometric Registration or Deormable Shapes 20

21 Example Implementation Example: approximate thin shell model Keep locally rigid Idea Will preserve metric & curvature implicitly Associate local rigid transormation with surace points Keep as similar as possible Optimize simultaneously with deormed surace Transormation is implicitly deined by deormed surace and vice versa Eurographics 2010 Course Geometric Registration or Deormable Shapes 21

22 Parameterization Parameterization o S 1 Surel graph This could be a mesh, but does not need to edges encode topology surel graph Eurographics 2010 Course Geometric Registration or Deormable Shapes 22

23 Deormation rame t rame t+1 prediction A i A i Orthonormal Matrix A i per surel neighborhood, latent variable Eurographics 2010 Course Geometric Registration or Deormable Shapes 23

24 Deormation rame t rame t+1 prediction A i A i Orthonormal Matrix A i error per surel neighborhood, latent variable E regularizer = surels neighbors [ ] 2 t t t t + 1 t + A s s s s 1 i i i j i i j Eurographics 2010 Course Geometric Registration or Deormable Shapes 24

25 Eurographics 2010 Course Geometric Registration or Deormable Shapes 25 Unconstrained Optimization Orthonormal matrices Local, 1st order, non-degenerate parametrization: Optimize parameters α, β, γ, then recompute A 0 Compute initial estimate using [Horn 87] = γ β γ α β α t C i exp 0 0 t i i i I C A C A A + = =

26 Variational Model Variational Problem: Formulate as an energy minimization problem: E = E match + E regularizer Eurographics 2010 Course Geometric Registration or Deormable Shapes 26

27 Deormable ICP

28 Deormable ICP How to build a deormable ICP algorithm Pick a surace distance measure Pick an deormation model / regularizer E = E match + E regularizer Eurographics 2010 Course Geometric Registration or Deormable Shapes 28

29 Deormable ICP How to build a deormable ICP algorithm Pick a surace distance measure Pick an deormation model / regularizer Initialize S 1 with S 1 i.e., = id Pick a non-linear optimization algorithm Gradient decent easy, but bad perormance Preconditioned conjugate gradients better Newton or Gauss Newton recommended, but more work Always use analytical derivatives! Run optimization Eurographics 2010 Course Geometric Registration or Deormable Shapes

30 Example Example Elastic model Local rigid coordinate rames Align A B, B A Eurographics 2010 Course Geometric Registration or Deormable Shapes 30