On approximation of smooth functions from samples of partial derivatives with application to phase unwrapping

Abstract

This paper addresses the problem of approximating smooth bivariate functions from the samples of their partial derivatives. The approximation is carried out under the assumption that the subspace to which the functions to be recovered are supposed to belong, possesses an approximant in the form of a principal shift-invariant (PSI) subspace. Subsequently, the desired approximation is found as the element of the PSI subspace that fits the data the best in the 2 -sense. In order to alleviate the ill-posedness of the process of finding such a solution, we take advantage of the discrete nature of the problem under consideration. The proposed approach allows the explicit construction of a projection operator which maps the measured derivatives into a stable and unique approximation of the corresponding function. Moreover, the paper develops the concept of discrete PSI subspaces, which may be of relevance for several practical settings where one is given samples of a function instead of its continuously defined values. As a final point, the application of the proposed method to the problem of phase unwrapping in homomorphic deconvolution is described.

Blind Deconvolution of Medical Ultrasound Images: A Parametric Inverse Filtering Approach

“…In this case, the functionals (21) and (22) can be considered to be real-valued functions of a complex vector, whose first and second derivatives can be computed by the standard methods of calculus.…”

“…By analogy, the same convex term can be added to the ML cost functional (12) resulting in (22) However, while in the case of (21), the addition of was done in order to improve the numerical characteristics of the resulting minimization, in the case of (22) this becomes a necessity. The fact is that, for the PSF used in the present study, without adding this term, it was found impossible to achieve stable convergence to the local minima of (12) for any value of parameter λ.…”

“…The fact is that, for the PSF used in the present study, without adding this term, it was found impossible to achieve stable convergence to the local minima of (12) for any value of parameter λ. This issue of stability of the ML design of the inverse filter was recently addressed in [41], where it was shown that the cost functional (22) can be formally obtained via reformulating the problem of estimation of the inverse filter within the framework of MAP estimation.…”

“…It should be noted that the switch-over to a different reference method has been a necessary step, since for the data at hand, no parameters of the ML design (22) of the inverse filter could be found for which the latter would provide stable reconstruction. On the other hand, the process of "cepstrumbased" deconvolution consists of two stages: the PSF is first estimated using the homomorphic relationship (5) [14], followed by non-blindly deconvolving the related image.…”

“…Moreover, the assumption of minimum-phase (which could have been used to avoid the phase estimation had the PSF possessed this property) does not seem to be applicable in ultrasound imaging in general [6]. Although there are several other efficient ways to avoid the problem of phase unwrapping [20]- [22], estimation of the Fourier phase of the PSF from that of the RF-image is generally difficult because of the latter being an extremely noisy, nonsmooth, and severely aliased function. Consequently, alternative ways for estimating the PSF should be sought.…”