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

• The MLS for functions• The projection concept• Local coordinate systems• The projection by MLS• Interpolating projection

The problem and goalsThe 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

The Moving Least-Squares - MLSThe 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 .

ii fx ,

x

Cx

To evaluate the MLS approximation at a point To evaluate the MLS approximation at a point we first find a local polynomial approximation we first find a local polynomial approximation such thatsuch that::

where as increaseswhere as increases..

Then we define the approximation at the point asThen we define the approximation at the point as

Choosing the resulting Choosing the resulting function interpolates the datafunction interpolates the data..

x

min||)(||])([ 2 xxwfxp ii

iix

x0)( tw t

).()( xpxM x,..2,1,)( 2 kttw k

M

mxp

It can be shown that for interpolating MLSIt can be shown that for interpolating MLS

In particular, if , this implies the flat spots In particular, if , this implies the flat spots property of Shepards interpolationproperty of Shepards interpolation . .

This property is also the basis for showing that This property is also the basis for showing that Hermite type MLS interpolation existsHermite type MLS interpolation exists::

Given data one finds such thatGiven data one finds such that

Then, as in ordinary MLS, defineThen, as in ordinary MLS, define

0xp

}',,{ iii ffx xp

min||)(||]')('[

||)(||])([

2

2

xxufxp

xxwfxp

iiii

x

iiii

x

).()( xpxH x

)(')(' ixi xpxMi

The Projection conceptThe Projection concept

Let N be a neighborhood of the data set.

We look for a projection P such 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 } !

Projection mesh - independenceProjection 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 . . .

Defining the projection P(r)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 3 e 1 e 2

An example of projection on a curve An example of projection on a curve

We want thatWe 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

Defining C( r ) = { q , e Defining C( r ) = { q , e 1 1 , e , e 2 2 , e , e 3 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

Denoting the data points by , the local Denoting the data points by , the local reference plane related to the point is defined reference plane related to the point is defined by minimizing the quantityby minimizing the quantity::

subject to andsubject to and

Note that this step of the process is NON-LINEARNote that this step of the process is NON-LINEAR..

}{ irr

||)(||),( 23 qrweqr i

ii

3|| erq .1|||| 3 e

Defining the projection P(r)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

approximation p(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 3 e 1 e 2

The local polynomial approximation is defined as the The local polynomial approximation is defined as the polynomial of degree m minimizingpolynomial of degree m minimizing::

where are the where are the projections of the data points onto the plane spanned projections of the data points onto the plane spanned

byby . .

||)(||)],()([ 23 qrweqrxp ii

ii

33),( eeqrqrx iii

21,ee

Smoothing and InterpolationSmoothing 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

Recall that the local polynomial approximation is Recall that the local polynomial approximation is defined as the polynomial of degree m minimizingdefined as the polynomial of degree m minimizing::

wherewhere . .

For interpolation we must of course choose a For interpolation we must of course choose a singular weight function, but we should also replacesingular weight function, but we should also replace

the weights by , where is the the weights by , where is the origin of the local coordinate system related to the origin of the local coordinate system related to the data point . Thus, all the points having a data point . Thus, all the points having a coordinate system with origin will be projected coordinate system with origin will be projected

to the data pointto the data point. .

||)(||)],()([ 23 qrweqrxp ii

ii

33),( eeqrqrx iii

||)(|| qqw i iq

ir riq

ir

Surface approximation example:Surface approximation example: Projection of a rectangular net Projection of a rectangular net