Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels

Abstract

The reconstruction of images is an important operation in many applications. From sampling theory, it is well known that the sine-function is the ideal interpolation kernel which, however, cannot be used in practice. In order to be able to obtain an acceptable reconstruction, both in terms of computational speed and mathematical precision, it is required to design a kernel that is of finite extent and resembles the sinc-function as much as possible. In this paper, the applicability of the sine-approximating symmetrical piecewise nth-order polynomial kernels is investigated in satisfying these requirements. After the presentation of the general concept, kernels of first, third, fifth and seventh order are derived. An objective, quantitative evaluation of the reconstruction capabilities of these kernels is obtained by analyzing the spatial and spectral behavior using different measures, and by using them to translate, rotate, and magnify a number of real-life test images. From the experiments, it is concluded that while the improvement of cubic convolution over linear interpolation is significant, the use of higher order polynomials only yields marginal improvement.

Quantitative evaluation of convolution-based methods for medical image interpolation

“…These aliasing errors may have influenced the results and conclusions. For example, in some cases the cubic convolution kernel resulting from the flatness constraintreferred to as the modified cubic spline-performed statistically significantly worse than linear interpolation, while it is known from many other studies [24,33,38,40,41,61] (including the present one) that the former kernel is generally superior. Second, the evaluation does not assess the performance of entire kernels, but only of a few distinct function values of these kernels.…”

“…A frequently used alternative approach to study the performance of interpolation kernels for the purpose of applying geometrical transformations, is to apply these transfor-mations to a number of test-images, followed by the inverse transformation so as to bring the images back in their original position [6,13,28,33,38]. Ideally, the forward-backward transformed images should be identical to their respective originals, so that a quantitative performance measure can be based on the grey-value differences between the images.…”

“…, m, where the superscript (l) denotes the lth-order derivative. It can be shown [33] that, given any odd degree n 3, the maximum allowable value for l that will not result in an over-constrained problem is n−2, in which case the total number of equations to be solved is (n+1)m−1. This implies Table 1.…”

“…In order to obtain a unique value for α, one additional constraint needs to be imposed. In this paper, we used the following constraints: (i) the slope constraint [44], which implies that the slope of the kernel is constrained in such a way that it equals the slope of the sinc function at x = 1; (ii) the continuity constraint [54], which implies that the kernel is constrained in such a way that its (n − 1)th-order derivative is continuous at x = 1; (iii) the flatness constraint [33,40], which implies that the frequency spectrumH(f ) of the kernel is required to be flat at f = 0. It can be shown that the latter constraint yields the mathematically most precise interpolant, in the sense that the Taylor series expansion agrees in as many terms as possible with the original signal [24,31].…”

“…The same holds for the studies of Schreiner et al [52] and Chuang [5], in which interpolation techniques were compared for the purpose of generating maximum intensity projections (MIPs) from MRA data, and surface rendering, respectively. Finally, we mention our recent study [33], in which we analyzed the effects of several piecewise polynomial interpolation kernels on the geometrical transformation of images. However, no medical images were included in that study.…”