The Hough transform is well known as a method of determining the parameters
of one or more straight lines which pass through a set of points. The
lines have parameters 
and p, and any points 
on them
must satisfy
However, this equation can also be interpretted as a constraint on the
parameters 
to ensure the line passes through the point 
.
Any line passing through 
must have parameters 
which
lie on the sinusoid in parameter space determined by the equation above.
The Hough approach involves creating an accumulator array in the
space. For every point 
we increment the bins in the
array lying on the corresponding sinusoid. We then scan through the
array to find the bins with the most entries, which give the parameters
of lines through the original points. If suitable noise models are considered
when bins are incremented then the distribution of hits in the array about the
best entries gives the distribution of parameter values [1]
[7].
This is a special case of a general Hough approach. In the general case
we have a set of N observation vectors, 
, some of
which are assumed to satisfy a model equation of the form
, where a specify some model parameters
which we would like to determine. Each observation, 
, imposes
a constraint on the parameters a, forcing them to lie on a
particular surface in the parameter space. Thus we can determine optimal
parameters and their distributions by incrementing the bins of accumulator array
which lie under each such surface.
This approach is particularly useful for problems with many observations which are not from the true model, and where there may be multiple instances of the desired model (each leads to a different peak in the accumulator array).