Registration and Alignment Speaker: Liuyu

The goal Form a 3D model of an object: –Data acquisition –Registration between views –Integration of views ICP algorithm

References A Method for Registration of 3-D Shapes –Paul J. Besl, Member, IEEE, and Neil D. McKay –IEEE Transaction on Pattern Analysis and Machine Intelligence,1992

Mathematic Preliminaries Let t be the triangle defined by the three points The distance between and t : Let T = {t i } for i =1,… ， N t,, then the distance between and T:

References Object Modeling by Registration of Multiply Range Images –Yang Chen and Gerard Medioni –Robotics and Automation, 1991, Proceedings –In Image and Visual Computer, 1992

Mathematic Preliminaries Point to Parametric Entity Distance – the parametric entity –let,use the Newton`s iteration method: Point to Implicit Entity Distance –Minimize the condition: –Update formula:

Mathematic Preliminaries Corresponding Point Set Registration –Let P = { p i } be a measured data, X be a model shape, C be the closest point operator: Y = C(P,X), where Y denote the resulting set of closest points.

Mathematic Preliminaries Corresponding Point Set Registration –The least squares registration (q, d) = φ (P,Y), where q is the registration state vector, and d is the mean square point matching error.

Get q The formulas to get q : –where is a unit rotation quaternion to generate the rotation matrix and is a translation vector. –Minimize the mean square objective function to get q : is 3*3 rotation matrix generated by

Get q corresponding to the maximum eigenvalue of the matrix : where is the cross-covariance matrix of P and X, The translation vector

Get d The mean square point matching error

ICP Algorithm Statement Input: the point set P = { p i } from the data shape and the model shape X( with N x supporting geometric primitives), a tolerance т Initialization of the iteration : P 0 = P, and k=0; then start the iteration: –Computer the closest points : Y k = C( P k, X )(cost:o( N p *logN x ) ) –Compute the registration : (q k,, d k ) = φ ( P k, Y k )(cost O( N p ) –Apply the registration : P k+1 = q k ( P 0 ) (cost: O( N p ) ) –Terminate the iteration when d k – d k+1 < т.

An Accelerated ICP Algorith For q is the angular tolerance For d : a linear approximation and a parabolic interpolant to the last three datas d1(v) = a1*v+b1; d2(v) = a2*v^2 + b2 *v + c2;

Resuts curve

Results triangular

Results triangular

Thank you!