From the analytic viewpoint, the reconstruction problem is mathematically ill-posed [Tikhonov and Arsenin 1977 ; Poggio et al. 1985] in at least that the solution is not unique ( cf. Section 1.5.3). Regularization methods can be used to convert the problem into a well-posed one.
In reconstruction from sparse data, the regularizer, i.e. the smoothness term, have to involve the second or higher order derivatives. This is because regularizers with first order derivatives can be creased, e.g. in the absence of data at lattice points, without any increase in the associated energy. Regularizers with second or higher order derivatives are sensitive to creases. The smoothness term with the second order derivative is
Its 2D counterpart, the plate with quadratic variation, is
For spatially discrete computation where the continuous domain is sampled at sites a regular lattice, the derivatives are approximated by the finite differences. A discrete form for (2.33) is
For piecewise smoothness, the prior terms may be defined with a g function having the property (2.16). In the 1D case,
In the 2D case,
They are called weak rod and weak plate, respectively. The latter has been used by [Geman and Reynolds 1992] to restore damaged film of movies.
Compared with the first order models like strings and membranes, the second order models are more capable of interpolating under sparse data; but they are less flexible in dealing with discontinuities and more expensive to compute. It is also reported that the mixture of 1st and 2nd order prior energies, which is akin to ``spring under tension'' [Terzopoulos 1983b ; Terzopoulos 1986b], performs poorly [Blake and Zisserman 1987].
