Applying OLAP Pre-Aggregation Techniques to ... - Jacobs University

38 3. Fundamental Geo-Raster Operations
Arrays are represented as functions mapping n-dimensional points from discrete
Euclidean space to values. The spatial domain of an array is defined as a finite set of
n-dimensional points in Euclidean space forming a hypercube with boundaries parallel
to the coordinate system axes.
Let X ⊆ Z d be a spatial domain and F a value set i.e., a homogeneous algebra.
Then, an F-valued d-dimensional array over the spatial domain X(multi-dimensional
array) is defined as:
a : X → F (i.e., a ∈ F X ),
a = {(x, a(x)) : x ∈ X, a(x) ∈ F }
The array elements a(x) are referred to as cells. Auxiliary function sdom(a) denotes
the spatial domain of some array a.
3.1.1 Constructor
The MARRAY array constructor allows arrays to be defined by indicating a spatial
domain and an expression evaluated for each cell position of the array. An iteration
variable bound to a spatial domain is available in the cell expression so that the cell
value depends on its position. Let X be a spatial domain, F a value set, and v a free
identifier. Let e v be an expression with result type F containing zero or more free occurrences
of v as placeholder(s) for an expression with result type X. Then, an array
over spatial domain X with base type F is constructed through:
MARRAY X,v (e v ) = {(x, a(x)) : a(x) = e x , x ∈ X}
A straightforward application of MARRAY is spatio-temporal sub-setting by simply
changing its domain.
Example: For some 2-D grey-scale image a, its cutout to domain [x0:x1,y0:y1] (assumed
to lie inside the array) is given by:
MARRAY [x0:x1,y0:y1],p (a[p])
Similarly, trimming produces a cutout of an array of lower volume, but unchanged
dimensionality, and section cuts out a hyperplane with reduced dimensionality.
We can also change an array’s values by changing the e v expression. In the simplest
case this expression takes the cell value and modifies it. The following expression adds
the values in the cells of two raster images, regardless of their extent and dimension:
a + b = MARRAY X,p (a[p] + b[p])
If we allow the use of all operations known on the base algebra, i.e., on the pixel
type, we immediately obtain a cohort of the following useful operations.