An Interpolated Volume Data Model

Representing volume data is an important task in many fields, from medicine [1] to physics and geology. Volumes are generated by collecting discrete measurements over a finite region of space, and this collection process leads naturally to two observations: first, what is usually called a volume is in reality a function f : V → M from a volume V to a measurement space M ; second, the volume—which is commonly understood to be a continuum—is in reality represented as a discrete (finite, in fact) set of samples. Most volume data models carry the discreteness of the measurements all the way to the level of the abstract data type. Many a model, for instance, consider a volume as a rectangular arrangement of cubic elements called “voxels” that is, essentially, as a three-dimensional array [2]. From the point of view of storing volume data into a database and querying them, this solution has the obvious advantage of relying on a data type (the array) that is already available in commercial databases and for which a sizable literature exists on issues like their effects on query optimization [3, 4]. We argue, however, that a discrete data type is not the best way to model a continuum such as a volume at an abstract level: the finiteness of the sample should be confined to the internal representation, while the abstract data model should be continuous. Consider, as a simple example, a “one dimensional volume” that is, an interval on a line. Assume that the measurements available for this volume consist of a single real