Biomedical Engineering Reference
In-Depth Information
did not use the multiscale structure possessed by the wavelet bases, nor the
Mallat algorithm to speed up the computation. Since the selected wavelet bases
are time-limited (therefore it is not band-limited), it may be not the best choice
for approximating differential operators.
At this point, we would like to mention the idea of regularization. The shape
from shading problems can be regarded as inverse problems since they attempt
to recover physical properties of a 3D surface from a 2D image associated with
the surface. Therefore, the Tikhonov regularization approach can be applied to
this problem. The time-limited filters, such as the difference boxes [22] or the
Daubechies wavelets used in Section 5.4.2, do not satisfy one of the conditions re-
quested by the Tikhonov regularization [61]. In contrast with time-limited filters,
band-limited filters are commonly used for regularizing differential operators,
since the simplest way to avoid harmful noise is to filter out high frequencies that
are amplified by differentiation. Meyer wavelet family constitutes an interesting
class of such type of band-limited filters. The ill-posedness/ill-conditioness of the
SFS model and its connection to the regularization theory have been discussed
in [7]. Minimization (5.21) will lead to a smoother solution (the regularization
solution). In some cases, the Lagrange multipliers are the “regularizers.” How-
ever, the numerical experiments presented in Section 5.3 are treated by choosing
those regularizers equal to 1. The nonlinear ill-posed problems are quite difficult
and basically no general approaches seem to exist [7]. For the classic theory of
regularization, we highly recommend Tikhonov
et al.
[60].
A 2D basis constructed from the tensor product of 1D wavelet basis is much
easier to compute than the nonseparable wavelets. There is also some ongoing
research on nonseparable wavelets for use in image processing. For a detailed
discussion on nonseparable wavelets, we recommend [37,38,40] and references
therein.
The development of a wavelet-based method which reflects the multiscale
nature with an effective algorithm, namely, using Mallat algorithm, is still an
open problem.
5.5 Concluding Remarks
In this chapter, we have given a super brief introduction of the shape from shad-
ing problems. A variety of elementary numerical techniques related to solution
