The reconstmction of a function and its derivatives from a set of given samples of that function is a fidarnenti operation in many areas. Computer graphics, scientific visu-on, and image prc essing are just a few examples. k~these arm, a set of samples of nn unknown function is usufly W we know of that function. Hence, the rmnstruction of the tiction between sample points is rather arbitrary, and one cannot t~about an ided reconstruction function. The notion of ideal reconstruction is based on the assumption, that a given function is a member of a certain functional space, e.g. the L2 space or the bandlimited function space. This functional space (and therefore the ideal reconstruction :. method) is usually determined by the particular application. In visufization, and in other fields, we assure that the given function belongs to the space of smooth functions d, where n is assinteger. ,, Avery impo~t and often studid space in the class of rdl smooth func~ond spat=, is the space of bandimited fictions (a subclass ,, of C ). They are often studied in the frequency domain using a sign~processing approach. Nthough these methods are capable of mntrotig global errors such as blurring and tilasing, no local spatial assessment of their accuracy can be conducted directly. It turns out that the idd recons~ction filters for the space of bandhrnited functions are impractical to use. Hence research in this area has focused on finding efficient filters that approximate the iderd titer The Iocd error was measured and minimized using a Taylor series expansion. Since visual perception, judged by ringing, tiasing and blurring, was of concern, the frequency behavior of the resulting flter was discussed. h addition, spatird design gives an easy control over the size of the filter, and hence on the efficiency of the restiting filter, a property frequency-based methods do not have. However, it was found that the sole concern for numerical accuracy can lead to discontinuous filters, which can produce visual , artifacts that are easily detected [17]. The god of this paper is to overcome this problem by introducing a smoothness requirement into the Hter design proms. M flter designs in the spatial domain have built filters according to an accuracy criteria. In this paper we introduce, for the first time, filter d~ign criteria for interpolation and derivative filters ,. which yield functions with a minimal numerical error and still maintain good spectral properties. The ordy assumption that we 1. require of the original function (represented by the given samples) t is that it is smooth and a member of the functional space d. We rdso show how our design criteria relate to criteria in frequency domain. Our falter design is not ratrictd to cubic polynomial basis finctions, but can generate filters of arbitrary smoothness and accu-
REF In our framework, we can interactively design complex filters in the frequency domain guided by the effects we see in the data.
Design of accurate and smooth filters for function and derivative reconstruction
VVS '98
Computer Science