Moving least-squares Moving least-squares for surfaces for surfaces David Levin – Tel David Levin – Tel Aviv University Aviv University Auckland, New Auckland, New Zealand 2005 Zealand 2005 • The MLS for functions • The projection concept • Local coordinate systems • The projection by MLS • Interpolating projection
17
Embed
Moving least-squares for surfaces David Levin – Tel Aviv University Auckland, New Zealand 2005 Moving least-squares for surfaces David Levin – Tel Aviv.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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