maspack.geometry

Class InverseDistanceWeights

java.lang.Object

maspack.geometry.InverseDistanceWeights

All Implemented Interfaces:

PointInterpolationWeights

public class InverseDistanceWeights
extends java.lang.Object
implements PointInterpolationWeights

Computes normalized weights for point-based interpolation based on inverse-distance weighting, given a reference point p and a set of support points p_i.

Each weight w_i is generated according to

  w_i = f(p,p_i)/ sum_i f(p,p_i)
 

where

  f(p,p_i) = 1/ (d(p,p_i)^exp + lam)
 

and d(p,p_i) is the distance from p to p_i. If normalize is true, d(p,p_i) is normalized by dividing by the maximum distance.

Constructor Summary

Constructors

Constructor and Description
InverseDistanceWeights(double exp, double lam, boolean normalize) Creates an InverseDistanceWeightGenerator with specified parameters.

Method Summary

Modifier and Type	Method and Description
static boolean	adjustWeightsForPosition(VectorNd w, Vector3d p, java.util.Collection<Vector3d> support) Takes a set of desired weights, based on distance calculations, and adjusts them so that the weighted sum of the support points equals p, and the sum of the weights equals 1.
boolean	compute(VectorNd w, Vector3d p, java.util.Collection<Vector3d> support) Given a reference point p, computes and returns a set of weights w_i that can be used for interpolation of values associated with a set of support points p_i.

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

InverseDistanceWeights

public InverseDistanceWeights(double exp,
                              double lam,
                              boolean normalize)

Creates an InverseDistanceWeightGenerator with specified parameters.

Parameters:

exp - exponent for the inverse distance function

lam - regularization term for the inverse distance function

normalize - if true, use normalized distances (distance divided by the maximum distance)

Method Detail

adjustWeightsForPosition

public static boolean adjustWeightsForPosition(VectorNd w,
                                               Vector3d p,
                                               java.util.Collection<Vector3d> support)

Takes a set of desired weights, based on distance calculations, and adjusts them so that the weighted sum of the support points equals p, and the sum of the weights equals 1. We do this by minimizing the distance between the desired weights wd, and the computed weights w, subject to the above stated equality constraints. In other words,

     min || w - wd ||^2
 
 subject to

     G w = y
 

where the columns of G and y are given by

            [ p_i ]         [ p ]
     G(j) = [     ]     y = [   ]
            [  1  ]         [ 1 ]
 

This problem can be solved by computing

     w = G^T inv(G G^T) (y + G wd) - wd
 

Since G may be rank deficient, we compute the inverse using a singular value decomposition of G^T, noting that if

     G^T = U S V^T
 

then

     G^T inv(G G^T) = U inv(S) V^T
 

Rank deficiency can be determined by examining the singular values.

compute

public boolean compute(VectorNd w,
                       Vector3d p,
                       java.util.Collection<Vector3d> support)

Given a reference point p, computes and returns a set of weights w_i that can be used for interpolation of values associated with a set of support points p_i. The weights are normalized so that they sum to unity. The weights should also be computed such that

 p = sum w_i p_i
 

If the support points do not have sufficient coverage, the last condition may not be possible. In that case, the method should return false.

Specified by:

compute in interface PointInterpolationWeights