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

2.1 Array Databases 21
• Bilinear interpolation is used to interpolate cell values at direct positions within
a quadrilateral grid. It assumes that feature attribute values vary as a bilinear
function of position within the grid cell: v = a + bx + cy + dxy. Given a
direct position, p, in a grid cell whose vertices are V, V + V 1 , V + V 2 , and
V + V 1 + V 2 , where V 1 and V 2 are offset vectors of the grid, and with cell values
at vertices v 1 , v 2 , v 3 , and v 4 , respectively, there are unique numbers i and j, with
0 ≤ i ≤ 1, and 0 ≤ j ≤ 1 such that p = V + iV 1 + jV 2 . The cell value at p is:
v = (1 − i)(1 − j)v 1 + i(1 − j)v 2 + j(1 − i)v 3 + ijv 4 .
Since the values for output cells are calculated according to the relative positions
and values of input cells, bilinear interpolation is preferred for data where the
location from a known point or phenomenon determines the value assigned to
a cell (that is, continuous surfaces). Elevation, slope, intensity of noise from an
airport, and salinity of groundwater near an estuary are phenomena represented
as continuous surfaces and are most appropriately resampled using bilinear interpolation.
• Quadratic interpolation is used to interpolate cell values along curves. It assumes
that cell values vary as a quadratic function of distance along a value
segment: v = a + bx + cx 2 , where a is the value of a cell at the start of a value
segment and v is the value of a cell at distance x along the curve from the start.
Three point value pairs are needed to provide control values for calculating the
coefficients of the function.
• Cubic interpolation is used to interpolate cell values along curves. It assumes
that cell values vary as a cubic function of distance along a value segment:
v = a + bx + cx 2 + dx 3 where a is the value of a cell at the start of a value
segment and v is the value of a cell at distance x along the curve from the start.
Four point value pairs are needed to provide control values for calculating the
coefficients of the function.
Cubic convolution has a tendency to sharpen the edges of the data more than bilinear
interpolation since more cells are involved in the calculation of the output
values.
Pyramid Evaluation
During the evaluation of a scaling operation with a target scale factor s, the pyramid
level with the largest scale factor s ′ with s ′ < s is determined. This level is loaded and
then an adjustment is made by scaling the resulting image by a factor of s/s ′ . If, for
example, scaling by s = 11 is required, then pyramid level 3 with scale factor s ′ = 8
is chosen, requiring a rest scaling of 11/8 = 1.375, thereby touching only 1/64 of
what is read without a pyramid.
The computation complexity of a scaling operation depends on the chosen resampling
method. For example, nearest neighbor resampling considers the closest cell
center of the input raster and assigns the value of that cell to the corresponding cell