Abstract
We present here one particular wavelet method, which was developed by the author. This is certainly not the only wavelet method for tomographic problems on the 3D ball. There exist alternatives, where at least [60, 175, 176] should be mentioned here.
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- 1.
For this example, we assume that \({\tau }_{m,n}\neq 0\) for all \(m,n \in {\mathbb{N}}_{0}\).
- 2.
In the case of tensorial functions, that is, \({b}_{m,n,j} : D \rightarrow {\mathbb{R}}^{3\times 3}\), we can use the isomorphism of \({\mathbb{R}}^{3\times 3}\) and \({\mathbb{R}}^{9}\) and interpret this case as q = 9.
- 3.
“SB” refers here to “scaling function” and “ball.”
- 4.
This is, for example, caused by the following fact: Typically, g is noisy, where the relative contribution of the noise to the Fourier coefficients \(\langle g,\,{b{}_{m,n,j}\rangle }_{\mathrm{{L}}^{2}(D,\,{\mathbb{R}}^{q})}\) increases with increasing degree m or n. On the one hand, we get that \({\Phi }_{J}^{\wedge }(m,n)\) tends to zero as \(m \rightarrow \infty \) or \(n \rightarrow \infty \) from (SB4). On the other hand, (SB2) and (SB3) require that \(\vert {\Phi }_{J}^{\wedge }(m,n)\vert \) increases with increasing scale J and approaches the (absolute value of the) reciprocal of the singular value. The former causes that Fourier coefficients corresponding to high degrees are equipped with small factors due to the convolution \({\Phi }_{J} {_\ast} g\)—which is good due to the noisy character of these coefficients. The latter, however, causes that this smoothing effect gets weaker and weaker with increasing scale J such that finally the noise is not sufficiently attenuated any more. An optimal scale \({J}^{{_\ast}}\) corresponds to a trade-off between a notable attenuation and a low approximation error.
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Michel, V. (2013). Wavelets for Inverse Problems on the 3D Ball. In: Lectures on Constructive Approximation. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8403-7_11
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