Moving least-squares for surfaces David Levin – Tel Aviv University Auckland, New Zealand 2005

1. Moving least-squares for surfacesDavid Levin – Tel Aviv University Auckland, New Zealand 2005 • The MLS for functions • The projection concept • Local coordinate systems • The projection by MLS • Interpolating projection

2. The problem and goals Given points on (or near) a surface we look for a mesh-independent method to define an interpolating (approximating) surface which is: • Smooth • Approximating • Locally dependent

3. The Moving Least-Squares - MLS The MLS idea (McLain ‘76): Given data { }, the approximation at a point is defined by a local least-squares polynomial approximation to the data, weighted according to distances from . The resulting approximation is .

4. To evaluate the MLS approximation at a point we first find a local polynomial approximation such that: where as increases. Then we define the approximation at the point as Choosing the resulting function interpolates the data.

5. It can be shown that for interpolating MLS In particular, if , this implies the flat spots property of Shepards interpolation. This property is also the basis for showing that Hermite type MLS interpolation exists: Given data one finds such that Then, as in ordinary MLS, define

6. The Projection concept Let Nbe a neighborhood of the data set. We look for a projection Psuch that the set S = { P( r ) | r N } is the desired surface (curve). P( u ) = P( r ) u L( r ) N L( r ) = { r +t P( r ) | t } S = { r | P( r ) = r } ! P(r) r

7. Projection mesh - independence The projection property of P implies That is, starting from any ‘triangulation’ T N, P( T ) S . How to realize this concept ? One way is by local coordinate systems . . . Mesh - independence

8. Defining the projection P(r) P( r )is defined via a local coordinate system C( r ) = { q , e 1 ,e 2 , e 3 }s.t. q - r || e 3 . In the coordinate system C( r ), we find a local polynomial approximation p(x,y) to the data. Then, P(r) e 3 q P( r ) = q + p(0 , 0 ) e 3e 1 e 2

9. An example of projection on a curve Local polynomial approximations r P(r) q Local coordinate system

10. We want that • the points P( r ) define a smooth surface. Thus: • the coordinate systems C( r ) should vary smoothly with r. • the coordinate systems should be the same for all points in L( r ) N . C( u ) = C( r ) u L( r ) N

11. Defining C( r ) = { q , e 1 , e 2 , e 3 } Given a point r we look simultaneously for a point q and a plane through q , n x = n q which is the best local least-squares approximation to the data, weighted according to the distances from q, so that q-r|| n=e 3. Note: q is the same for r all points r on the line:q

12. Denoting the data points by , the local reference plane related to the point is defined by minimizing the quantity: subject to and Note that this step of the process is NON-LINEAR.

13. Defining the projection P(r) In the local coordinate system C( r ) = { q , e 1 ,e 2 , e 3 } we find a local least-squares polynomial approximationp(s,t)to the projected data, weighted according to distances P(r) e 3 from q. Then q P( r ) = q + p(0 , 0 ) e 3e 1 e 2

14. The local polynomial approximation is defined as the polynomial of degree m minimizing: where are the projections of the data points onto the plane spanned by .

15. Smoothing and Interpolation The weight function for the weighted MLS coordinate systems and for the polynomial approximation may be chosen in many ways: • With finite support - or with fast decay • Smoothing or interpolating • Globally or locally defined

16. Recall that the local polynomial approximation is defined as the polynomial of degree m minimizing: where . For interpolation we must of course choose a singular weight function, but we should also replace the weights by , where is the origin of the local coordinate system related to the data point . Thus, all the points having a coordinate system with origin will be projected to the data point .

17. Surface approximation example: Projection of a rectangular net