**Registration of Point Cloud Data from a Geometric** Optimization Perspective Niloy J. Mitra1, Natasha Gelfand1, Helmut Pottmann2, Leonidas J. Guibas1 1 Stanford University 2 Vienna University of Technology

**Q** P data model Registration Problem Given Two point cloud data sets P (model) and Q (data) sampled from surfaces P and Q respectively. Assume Q is a part of P.

**Q** P data model Registration Problem Given Two point cloud data sets P and Q. Goal Register Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms.

**Registration Problem** Given Two point cloud data sets P and Q. Goal Register Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms. Contributions • use of second order accurate approximation to the squared distance field. • no explicit closest point information needed. • proposed algorithm has good convergence properties.

**Related Work** • Iterated Closest Point (ICP) • point-point ICP [ Besl-McKay ] • point-plane ICP [ Chen-Medioni ] • Matching point clouds • based on flow complex [ Dey et al. ] • based on geodesic distance [ Sapiro and Memoli ] • MLS surface for PCD [ Levin ]

**Notations**

**Notations**

**Squared Distance Function (F)** x

**Squared Distance Function (F)** x

**Rigid transform thattakes points** Registration Problem Our goal is to solve for, An optimization problem in the squared distance field of P, the model PCD.

**Registration Problem** Our goal is to solve for, Optimize for R and t.

**Overview of Our Approach** • Construct approximate such that, to second order. • Linearize a. • Solve to get a linear system. • Apply a to data PCD (Q) and iterate.

**Registration in 2D** • Quadratic Approximant

**Registration in 2D** • Quadratic Approximant • Linearize rigid transform

**Registration in 2D** • Quadratic Approximant • Linearize rigid transform • Residual error

**Registration in 2D** • Minimize residual error Depends on F+ and data PCD (Q).

**Registration in 2D** • Minimize residual error • Solve for R and t. • Apply a fraction of the computed motion • F+ valid locally • Step size determined by Armijo condition • Fractional transforms [Alexa et al. 2002]

**Registration in 3D** • Quadratic Approximant • Linearize rigid transform • Residual error Minimize to get a linear system

**valid in the neighborhood of x** Approximate Squared Distance • Two methods for estimating F • d2Tree based computation • On-demand computation

**F(x, P) using d2Tree** • A kd-tree like data structure for storing approximants of the squared distance function. • Each cell (c) stores a quadratic approximant as a matrix Qc. • Efficient to query. [ Leopoldseder et al. 2003]

**F(x, P) using d2Tree** • A kd-tree like data structure for storing approximants of the squared distance function. • Each cell (c) stores a quadratic approximant as a matrix Qc. • Efficient to query. • Simple bottom-up construction • Pre-computed for a given PCD. Closest point information implicitly embedded in the squared distance function.

**Example d2trees** 2D 3D

**Y** Approximate Squared Distance For a curve Y, [ Pottmann and Hofer 2003 ]

**Approximate Squared Distance** For a curve Y, For a surface F, [ Pottmann and Hofer 2003 ]

**On-demand Computation** • Given a PCD, at each point p we pre-compute, • a local frame • normal • principal direction of curvatures • radii of principal curvature (r1and r2)

**On-demand Computation** • Given a PCD, at each point p we pre-compute, • a local frame • normal • principal direction of curvatures • radii of principal curvature (r1and r2) Estimated from a PCD using local analysis • covariance analysis for local frame • quadric fitting for principal curvatures

**d/(d-j) if d < 0** 0 otherwise. where j = On-demand Computation Given a point x, nearest neighbor (p) computed using approximate nearest neighbor (ANN) data structure

**Iterated Closest Point (ICP)** • Iterate • Find correspondence between P and Q. • closest point (point-to-point). • tangent plane of closest point (point-to-plane). • Solve for the best rigid transform given the correspondence.

**ICP in Our Framework** • Point-to-point ICP (good for large d) • Point-to-plane ICP (good for small d)

**Convergence Properties** • Gradient decent over the error landscapeGauss -Newton Iteration • Zero residue problem (model and data PCD-s match)Quadratic Convergence • For fractional steps, Armijo condition usedDamped Gauss-Newton IterationLinear convergence • can be improved by quadratic motion approximation (not currently used)

**Convergence Funnel** Set of all initial poses of the data PCD with respect to the model PCD that is successfully aligned using the algorithm. Desirable properties • broad • stable

**Convergence Funnel** Translation in x-z plane. Rotation about y-axis. Converges Does not converge

**Convergence Funnel** Plane-to-plane ICP Our algorithm

**Convergence Rate I** Bad Initial Alignment

**Convergence Rate II** Good Initial Alignment

**Partial Alignment** Starting Position

**Partial Alignment** After 6 iterations

**Partial Alignment** Different sampling density After 6 iterations

**Future Work** • Partial matching • Global registration • Non-rigid transforms

**Questions?**