Both the smoothing and boundary detection procedures are instances of
the convolution operation. The discrete case of 2-D convolution is done
by taking two data matrices, one the original data set and the other a
``mask'', and effectively ``sliding'' one across the other, matching
all the points of one against all the points of the other. The
result is a new data matrix that has a maximum range in any given
dimension that is the range of the original data set for that
dimension plus the range of the mask, minus one. For example, if a
data set with the range 
were convolved with a mask with a
range of 
, the new resulting data space would have a range
of 
. The convolution process multiplies corresponding elements
between the mask and the original matrix and sums all of the products.
Different configurations of masks produce different types of results.
This concept is easily extendible from two-dimensions into
N-dimensions. The algorithm is exactly the same, extending over all
the dimensions in the space. For example in four dimensions a data
set of cardinality
convolved with a mask of
cardinality
produces a new data set with a range of
. From this, the operations of smoothing and
boundary detection can be implemented. Figures 6a and 6b
show the
effects of a using directional Sobel operator to detect boundaries on
a 4-D hypersurface. The first figure shows that, while the boundaries
have been detected, they do not appear to be cohesive. Manipulating the
data as discussed in section 3.2 shows that what appear to be uniformly
scattered points actually do form a connected boundary.
