Hausdorff Moments Method of Acoustical Imaging

Conclusions

The moments method has been shown to give accurate reconstructions in different types of model problems when exact but incomplete boundary spectral data is used. The majority of the error in reconstruction does not occur in the nonlinear step for the creation of the moments, but in the second step of the classical Hausdorff moments problem, which is linear and ill-posed. A Fourier technique for this problem is particularly prone to error when the function to be reconstructed is non-smooth. Knowledge of the value of the function on the boundary can be used to reduce these errors substantially. A Legendre technique is less susceptible to error due to non-smoothness of the function, such that it can be used to reconstruct a uniform field with isolated lumps in it, for which the Fourier technique is totally inappropriate. However the Legendre technique is more susceptible to error in the data than is the Fourier technique. The way forward would appear to be the development of some form of regularization procedure for error-prone data such that the Legendre technique remains accurate in such cases.