In the standard regularization [Tikhonov and Arsenin 1977 ; Poggio et al. 1985], the potential function takes the pure quadratic form
With 
, the more irregular 
is at x, the larger
, and consequently the larger potential 
contributed to 
. The standard quadratic regularizer can have a
more general form
where 
are the pre-specified non-negative continuous functions
[Tikhonov and Arsenin 1977]. It may also be generalized to multi-dimensional
cases and to include cross derivative terms.
The quadratic regularizer imposes the smoothness constraint
everywhere. It determines the constant interaction between
neighboring points and leads to smoothing strength proportional to
, as will be shown in the next section. The homogeneous or
isotropic application of the smoothness constraint inevitably leads to
oversmoothing at discontinuities at which the derivative is infinite.
If the function 
can be pre-specified in such a way that 
at x where 
is infinite, then the oversmoothing
can be avoided. In this way, 
act as
continuity-controllers [Terzopoulos 1983b]. It is further
suggested that 
may be discontinuous and not pre-specified
[Terzopoulos 1986b]. For example, by regarding 
as
unknown functions, one could solve these unknowns using variational
methods. But how well 
can thus be derived remains unclear. The
introduction of line processes
[Geman and Geman 1984 ; Marroquin 1985] or weak continuity
constraints [Blake and Zisserman 1987] provides a solution to this problem.
