Scattered data reconstruction by regularization in B-spline and associated wavelet spaces

Abstract

The problem of fitting a nice curve or surface to scattered, possibly noisy, data arises in many applications in science and engineering. In this paper we solve the problem using a standard regularized least square framework in an approximation space spanned by the shifts and dilates of a single compactly supported function φ. We first provide an error analysis to our approach which, roughly speaking, states that the error between the exact (probably unknown) data function and the obtained fitting function is small whenever the scattered samples have a high sampling density and a low noise level. We then give a computational formulation in the univariate case when φ is a uniform B-spline and in the bivariate case when φ is the tensor product of uniform B-splines. Though sparse, the arising system of linear equations is ill-conditioned; however, when written in terms of a short support wavelet basis with a well-chosen normalization, the resulting system, which is symmetric positive definite, appears to be well-conditioned, as evidenced by the fast convergence of the conjugate gradient iteration. Finally, our method is compared with the classical cubic/thin-plate smoothing spline methods via numerical experiments, where it is seen that the quality of the obtained fitting function is very much equivalent to that of the classical methods, but our method offers advantages in terms of numerical efficiency. We expect that our method remains numerically feasible even when the number of samples in the given data is very large.

An analysis of wavelet frame based scattered data reconstruction

“…Similar has been considered in [29] for the model min g∈S h (φ) ξ∈Ξ (g(ξ) − y(ξ)) 2 + ν|g| 2 H m , whose minimizer can be seen as an approximation to the thin plate smoothing spline. Hence gives a smooth approximation to the scattered data, meaning that discontinuities are not displayed very well.…”

“…Besides its structural simplicity shift invariant spaces have the beneficial property that for special choices of φ they provide good approximation orders to sufficiently smooth functions, see [25,26,29]. The compact support of φ also results in sparse system matrices which is of computational interest [29,40]. Such scaling functions φ are for instance B-splines, which in turn give rise to associated wavelet frame systems, as we discuss below.…”

Approximation of frame based missing data recovery

“…However, this requires the underlying function has a high order of smoothness. In this case, minimizing the 2 norm of the canonical coefficients of the framelet system will work and the error analysis can be done similarly as that of [45]. In this paper, the underlying function we are interested in does not meet certain order of smoothness.…”

“…The task is to recover the missing data on Ω\Λ. There are many methods to deal with this problem under many different settings, e.g., [3,4,10,25,38] for image inpainting, [11,14,15] for matrix completion, [29,55] for regression in machine learning, [8,9,20,23,24] for framelet-based image deblurring, [41,45] for surface reconstruction in computer graphics, and [16,22,26] for miscellaneous applications. We forgo to give a detailed survey on this fast developing area and the interested reader should consult the references mentioned above for the details.…”

Small Support Spline Riesz Wavelets in Low Dimensions

“…These wavelets can be useful when the fast reconstruction is needed, while the decomposition can be done "off line". Another such an example is that small support spline wavelets in [28] are used to derive fast algorithms for smooth surface fitting from scattered noisy data in [36].…”