Robust Global Registration Natasha Gelfand Niloy Mitra Leonidas

Robust Global Registration Natasha Gelfand Niloy Mitra Leonidas Guibas Helmut Pottmann

Registration Problem Given: Two shapes P and Q which partially overlap. Goal: Using only rigid transforms, register Q against P by minimizing the squared distance between them.

Applications • Shape analysis – similarity, symmetry detection, rigid decomposition [Koller 05] [Pauly 05] • Shape acquisition [Anguelov 04]

Approaches I [Besl 92, Chen 92] • Iterative minimization algorithms (ICP) 1. Build a set of corresponding points 2. Align corresponding points • Properties – Dense correspondence sets – Converges if starting positions are “close” 3. Iterate

Approaches II [Stockman 87, Hecker 94, Barequet 97] • Voting methods – Geometric hashing, Hough transform, alignment method • Rigid transform can be specified with small number of points – Try all possible transform bases – Find the one that aligns the most points • Guaranteed to find the correct transform – But can be costly

Approaches III [Mokh 01, Huber 03, Li 05] • Use neighborhood geometric information • Descriptors should be – Invariant under transform – Local – Cheap • Improves correspondence search or used for feature extraction

Registration Problem • Why it’s hard – Unknown areas of overlap – Have to solve the correspondence problem • Why it’s easy – Rigid transform is specified by small number of points – Prominent features are easy to identify We only need to align a few points correctly.

Method Overview 1. Use descriptors to identify features – Integral volume descriptor 2. Build correspondence search space – Few correspondences for each feature 3. Efficiently explore search space – – Distance error metric Pruning algorithms

Integral Descriptors • • Multiscale • Inherent smoothing [Manay 04] Br(p) f(x) p

Integral Volume Descriptor • 0. 20 • Relation to mean curvature

Descriptor Computation • Approximate using a voxel grid – Convolution of occupancy grid with ball

Feature Identification • Pick as features points with rare descriptor values – Rare in the data [ rare in the model [ few correspondences • Works for any descriptor Features

Multiscale Algorithm • Features should be persistent over scale change r=0. 5 r=1 r=2 r=4 r=8 Y Y Y N N

Feature Properties • Sparse • Robust to noise • Non-canonical

Correspondence Space • Search the whole model for correspondences – Range query for descriptor values – Cluster and pick representatives P Q

Evaluating Correspondences • Coordinate root mean squared distance – Requires best aligning transform – Looks at correspondences individually

Rigidity Constraint • Pair-wise distances between features and correspondences should be the same P Q

Evaluating Correspondences • Distance root mean squared distance – Depends only on internal distance matrix

Search Algorithm • Few features, each with few potential correspondences – Minimize d. RMS – Exhaustive search still too expensive P Q

Search Algorithm • Branch and bound – Initial bound using greedy assignment • Discard partial correspondences that fail thresholding test – • Prune if partial correspondence exceeds bound Rc – Spacedpout features make incorrect qi correspondences fail i quickly • Since we pexplore the entire search space, we are j guaranteed to find optimal alignment qj – Up to cluster size

Search Algorithm • Branch and bound – Initial bound using greedy assignment • Discard partial correspondences that fail thresholding test – • Prune if partial correspondence exceeds bound – Spaced out features make incorrect correspondences fail quickly • Since we explore the entire search space, we are guaranteed to find optimal alignment – Up to cluster size

Greedy Initialization • Good initial bound is essential • Build up correspondence set hierarchically …

Greedy Initialization • Good initial bound is essential • Build up correspondence set hierarchically … …

Greedy Initialization • Good initial bound is essential • Build up correspondence set hierarchically …

Partial Alignment • Allow null correspondences, while maximizing the number of matches points

Alignment Results Input: 2 scans Our alignment Refined by ICP

Alignment Results Our alignment Input: 10 scans Refined by ICP

Symmetry Detection • Symmetry detection – Match a shape to itself

Rigid Decomposition • Repeated application of partial matching

Conclusions • Simple feature identification algorithm – Used integral volume descriptor – Applicable for any low-dimensional descriptor • Correspondence evaluation using d. RMS error – Look at pairs of correspondences, instead of individually • Efficient branch and bound correspondence search – Finds globally best alignment

Future Work • Linear features

Future Work • Linear features • Fast rejection tests • More descriptors

Acknowledgements • • Daniel Russel An Nguyen Doo Young Kwon Digital Michelangelo Project • NSF CARGO-0138456, FRG-0454543, ARO DAAD 19 -03 -1 -033, Max Plank Fellowship, Stanford Graduate Fellowship, Austrian Science Fund P 16002 N 05

Questions?