Multivariate interpolation with RBF

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19.06.2022

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Python, VEX, Environment

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Houdini 19.0

1 Introduction

Making a grid flow through guide points using multivariate interpolation with the Python module Numpy / Scipy.

2 Initialization

The f@height attribute is going to be interpolated based on common X and Z positions between the grid and the guide points.

// Grid
f@px = v@P.x;
f@pz = v@P.z;

f@height = 0.0;
// Guide points
f@px = v@P.x;
f@pz = v@P.z;

f@height = v@P.y;

3 Interpolation

Python's NumPy/SciPy library offers multivariate interpolation. It's making sure the grid will always flow smoothly through our guide points even if they are positioned irregularily.

There are various interpolation methodes provided: multiquadric, inverse multiquadric, gaussian, linear, cubic, quintic and thinplate.

import numpy as np
from scipy.interpolate import griddata
import scipy.interpolate as interp

node = hou.pwd()
geo1 = node.geometry()
inputs = node.inputs()
geo2 = inputs[1].geometry()

method_nr = node.evalParm('method')
method_names = 'multiquadric,inverse_multiquadric,gaussian,linear,cubic,quintic,thin_plate'.split(',')
method_str = method_names[method_nr]

grid_x = np.array(geo1.pointFloatAttribValues('px'))
grid_z = np.array(geo1.pointFloatAttribValues('pz'))
pos_x = np.array(geo2.pointFloatAttribValues('px'))
pos_z = np.array(geo2.pointFloatAttribValues('pz'))
height = np.array(geo2.pointFloatAttribValues('height'))

rbf_height = interp.Rbf(pos_x, pos_z, height, function=method_str)
smooth_rbf_height = rbf_height(grid_x, grid_z)

geo1.setPointFloatAttribValuesFromString("height", smooth_rbf_height.astype(np.float32))
v@P.y = f@height;

4 Conclusion

The radial basis function (RBF) in Scipy.interpolate does an excellent job at interpolating between irregular points. RBF looks smoother than what we´d get from the Attribute Transfer SOP, too.

5 Contribution

Adrian Pan contributed a corresponding example file that works directly with heightfields.

6 VEX version

rbf.hip contains a simplified VEX version of multivariate interpolation using radial basis functions. The screenshot compares the attribute transfer node's result (left) with RBF.

int mode = chi('mode');
float k = chf('radius');

float w    = 0.0;
vector clr = 0.0;

for(int i = 0; i < npoints(1); i++){
    vector pos = point(1, 'P', i);
    float d = distance2(v@P, pos);
    vector clr_pt = point(1, 'Cd', i);
    
    float weight = 0.0;
    if     (mode == 0) weight = 1.0 / sqrt(d);                // inverse multiquadratic
    else if(mode == 1) weight = pow(1.0 + (d * d), -0.5) / d; // multiquadric
    else if(mode == 2) weight = exp(-d / (2.0 * k * k));      // gaussian
    else if(mode == 3) weight = 1.0 / (d * d * d);            // cubic
    w += weight; 
    
    clr += clr_pt * weight;
}

v@Cd = clr / w;
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