3.1.2
Grimson and Pavlidis (1985) propose an approach in which the degree of interaction between pixels across edges is adjusted in order to detect discontinuities. Lee and Pavlidis (1987) investigate a class of smoothing splines which are piecewise polynomials. Errors of fit are measured after each successive regularization and used to determine whether discontinuities should be inserted. This process iterates until convergence is reached. Besl et al. (1988) propose a smoothing window operator to prevent smoothing across discontinuities based on robust statistics. Liu and Harris (1989) develop, based on a previous work [Harris 1987], a computational network in which surface reconstruction, discontinuity detection and estimation of first and second derivatives are performed cooperatively.
Mumford and Shah (1985) define an energy on a continuous domain
where 
and 
are constants, k is the number of
discontinuities and the sequence 
indicates
the locations of discontinuities. The minimal solution 
minimizes 
over each value of the
integer k, every sequence 
, and every function 
continuously differentiable on each interval 
.
The minimization over k is a hard problem.
Using the minimal description length principle, Leclerc (1989) presents the following function for restoration of piecewise constant image f from noisy data d
where 
is the Kronecker delta function. To
minimize the function, he approximates the delta function with the
exponential function parameterized by 
and approaches the solution by continuation in 
toward 0.
