Regular Tessellation

A regular tessellation is a partitioning of space into mutually exclusive cells that together make up the complete study area and in which the cells have the same shape and size. A simple example of this is a rectangular raster of unit squares, represented in a computer in the 2D case as an array of n x m elements. 

All regular tessellations have in common that the cells have the same shape and size, and that the field attribute value assigned to a cell is associated with the entire area occupied by the cell. Square cell tessellation is commonly used, mainly because georeferencing of such a cell is straightforward. This type of tessellation is known under various names in different GIS packages: e.g. “raster” or “raster map”.

There are some issues related to cell-based partitioning of the study space. The field value of a cell can be interpreted as one for the complete tessellation cell, in which case the field is discrete, not continuous or even differentiable. Some convention is needed to state which value prevails on cell boundaries. With square cells, this convention states that lower and left boundaries belong to the cell. There are two approaches to refining the solution of this continuity issue: make the cell size smaller, so as to make the “continuity gaps” between the cells smaller; and/or assume that a cell value only represents elevation for one specific location in the cell, and to provide a good interpolation function for all other locations that have the continuity characteristic. If one wants to use rasters for continuous field representation, one usually uses the first approach but not the second, as the second technique is usually considered computationally too intensive for large rasters.

The location associated with a raster cell is fixed by convention: it may be the cell centroid (mid-point) or, for instance, its left lower corner. Values for other positions are computed using an interpolation function applied to one or more nearby field values. This allows us to represent continuous, even differentiable, functions. An important advantage of regular tessellations is that we know how they partition space, and that we can make our computations specific to this partitioning. This leads to fast algorithms. An obvious disadvantage is that they are not adaptive to the spatial phenomenon we want to represent. The cell boundaries are both artificial and fixed: they may or may not coincide with the boundaries of the phenomena of interest. If we use any of the above regular tessellations to represent an area with minor elevation differences, then, clearly we would need just as many cells as in a strongly undulating terrain: the data structure does not adapt to the lack of relief. We would, for instance, still use the m × n cells for the raster, even though variations in elevation are irrelevant.