PROCEEDINGS ARTICLE | January 29, 2007

Proc. SPIE. 6499, Vision Geometry XV

KEYWORDS: Mathematical modeling, Data modeling, Control systems, Computer science, Data centers, Electronics engineering, 3D vision, Systems modeling, 3D image processing, Vision geometry

Scattered data is defined as a collection of data that have little specified connectivity among data points. Trivariate
scattered data interpolation from <i>R<sup>3</sup></i> --> <i>R</i> consists of constructing a function <i>f</i> = (x, y, z) such
that <i>f</i>(x<sub>i</sub>, y<sub>i</sub>, z<sub>i</sub>) = <i>F</i><sub>i</sub>, <i>i</i>=1, <i>N</i> where V = {v<sub>i</sub> = (x<sub>i</sub>, y<sub>i</sub>, z<sub>i</sub>) &egr; <i>R</i><sup>3</sup>, <i>i</i>=1,....<i>N</i>} is a set of distinct and non-coplanar
data points and <i>F</i> = (<i>F</i><sub>1</sub>, ......, <i>F</i><sub>N</sub>) is a real data vector. The weighted alpha shapes method is defined for a finite set
of weighted points. Let <i>S</i> &subuline; <i>R</i><sup>d</sup> x <i>R</i> be such a set. A weighted point is denoted as p=(p', &ohgr;) with <i>p</i>' &egr; <i>R</i><sup>d</sup> its location and
&ohgr; &egr; <i>R</i> its weight. For a weighted point p and a real &agr; define P<sub>+&agr;</sub>=(<i>P</i>', &ohgr; + &agr;). So <i>p</i> and P<sub>+&agr;</sub> share
the same location and their weights differ by &agr;. In other words, it is a polytope uniquely determined by the points, their
weights, and a parameter &agr; &egr; <i>R</i> that controls the desired level of detail.
Therefore, how to assign the weight for each point is one of the main tasks to achieve the desirable volumetric scattered
data interpolation. In other words, we need to investigate the way to achieve different levels of detail in a single shape
by assigning weights to the data points. In this paper, Modified Shepard's method is applied in terms of least squares
manner.