Consequently, reconstruction methods better than linear interpolation became an attractive filed of research. Hou and Andrews [19] investigated, at the end of the seventies, the applicability of Bsplines for image processing tasks like interpolation, smoothing, filtering, enlargement, and reduction. They concentrate on how to choose an optimal and yet easy to implement basis function, the socalled Bspline, and use it for interpolation and data smoothing. Only a visual comparison of Bspline interpolation to other interpolation methods is given by the authors, whereby they compare it only to nearest neighbor and linear interpolation. For testing purposes they performed magnification and minification on digital pictures and come to the result that Bspline interpolation is superior.
Keys [25] defines the
cubic interpolation kernel as piecewise cubic polynomials on the
intervals [2,1],[1,0],[0,1]
and [1,2] (see Eq. 4.7). Outside the
interval [2,2] the kernel is zero, which means that the number of
sample points that contribute to one reconstructed value reduce to
four. Since an interpolation kernel must be symmetric this leads to
eight unknown coefficients. Further, the kernel must be one at
position zero and zero at all other integer positions and it must be
continuous and have continuous derivatives. These constraints lead to
the oneparametric family of cardinal splines:
This cubic convolution kernel then is compared to nearest neighbor and linear interpolation both by comparing their frequency responses and by visual comparison of interpolated images. Again, conclusions are that cubic convolution kernels are more accurate than nearest neighbor or linear interpolation. However, Keys notes that they are not as accurate as cubic splines but can be performed much more efficiently.
Park and Schowengerdt [44] point out that the
parameter a of the cardinal cubic splines has a physical meaning
since it is the slope of the filter at x=1. They further analyze the
cardinal splines in frequency domain and come to the same result as
Keys in spatial domain that
is superior which is not
the standard value commonly referenced in the literature (that would
be a = 1.0 which duplicates the slope of the ideal interpolation
function (i.e., the sinc function) at x = 1.0 so that simpleminded
researchers obviously concluded that would be best for the cardinal
splines too). Also, they rewrite
Equation 2.1 as
h(x)=h_{0}(x)+ah_{1}(x)  (2.2) 
(2.3) 
(2.4) 
The introduction of the Bsplines and cardinal splines entailed some work of researchers comparing different reconstruction methods. Parker et al. [45] compared five different interpolation filters, namely nearest neighbor and linear interpolation, cubic Bsplines and cardinal splines (which they call highresolution cubic splines) with a=1 and a=0.5. The comparison is done in frequency domain as well as by visual inspection of resampled images. Nearest neighbor, being the cheapest method with just one contributing sample point, has a very poor stop zone response. Linear interpolation has a slightly better overall response but extends over two sample points. The Bspline and cardinal splines are even more expensive with four contributing sample points. However, they have better response in both the pass and stop band what is just to be expected from the increased filter length. They conclude that the choice between the length four interpolating functions depends upon the task. The cubic Bspline provides the most smoothing whereas the cardinal cubic spline with a=1 provides the best highfrequency response along with some high frequency enhancement and the cardinal cubic spline with a=0.5 has both a flat lowfrequency response and good stopband performance.
Schreiber and Troxel [53] conducted a series of experiments using ``perceptual rather than mathematical criteria''. They compared eight different interpolation methods, nearest neighbor (which they call sampleandhold) and linear interpolation, a truncated sinc filter, Bsplines, two types of sharpened cubic splines (which are not explained in more detail), a truncated Gaussian and a truncated sharpened Gaussian filter. Sharpening a Gaussian means that two negative Gaussian pulses are added to one central Gaussian pulse, one earlier and one later. They uses two test pictures, which were enlarged and shown to ten ``subjects'', whereas ``most of the subjects were affiliated with our laboratory and were considered experienced, if not expert, observers''. They conclude that the result show the clear superiority of the sharpened Gaussian to all the others with respect to image quality. However, the sharpened Gaussian is not a very popular reconstruction filter, so this conclusion is at least questionable.
Mitchell and Netravali [39] eventually introduced
a quite popular family of cubic splines, the BCsplines. A cubic
spline filter is in its general form given by
(2.5) 
(2.6) 
They note that a reconstruction filter now has two tasks
Two new filters were proposed, the first with and suppresses postaliasing but is unnecessarily blurring, the second with and turns out to be a satisfactory compromise between ringing, blurring, and anisotropy. Furthermore, a simple subjective test was carried out which allowed to divide the B,C parameter space into ringing, blurring, anisotropy, and satisfactory regions.This parameter space is depicted in Fig. 2.1 (image taken from the paper [39]).

At this point two excellent papers have to be mentioned, both by Jim Blinn published in Jim Blinn's corner of Computer Graphics & Applications and called ``What We Need Around Here Is More Aliasing'' [4] and ``Return of the Jaggy'' [3]. In these papers Jim Blinn first explains in a very comprehensive way the effects of sampling and reconstruction in frequency domain. He then explains the box filter (nearest neighbor interpolation), the tent filter (linear interpolation), Gaussian, and ideal (the sinc) filter.He notes that the sinc filter is impracticable because of its infinite extend in spatial domain and therefore some approximation must be used which results in the conclusion that an achievable filter never is a perfect lowpass filter.
Ken Turkowski [63] published a comparative study of some reconstruction filters. These include box, tent, and Gaussian filter. Further, he notes that the sinc function should be multiplied by an appropriate windowing function, where he states that the Lanczos window is most appropriate in a way that it ``produced the best quality with an acceptable amount of ringing.'' [62]. This was found out in an earlier study comparing the sinc function windowed with several windows like Hann, Hamming, or raised cosine window.