Summary

The wavelet-based method we demonstrated in this section is based on the approximation of the objective function in V0. It should be pointed out that it 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 requested 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 ofthe 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." However, 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.

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