Nat. Center for Res. Resources, Nat. Inst. of Health, Bethesda, MD.
IEEE Trans Image Process. 1995;4(3):247-58. doi: 10.1109/83.366474.
The purpose of this paper is to derive optimal spline algorithms for the enlargement or reduction of digital images by arbitrary (noninteger) scaling factors. In our formulation, the original and rescaled signals are each represented by an interpolating polynomial spline of degree n with step size one and Delta, respectively. The change of scale is achieved by determining the spline with step size Delta that provides the closest approximation of the original signal in the L(2)-norm. We show that this approximation can be computed in three steps: (i) a digital prefilter that provides the B-spline coefficients of the input signal, (ii) a resampling using an expansion formula with a modified sampling kernel that depends explicitly on Delta, and (iii) a digital postfilter that maps the result back into the signal domain. We provide explicit formulas for n=0, 1, and 3 and propose solutions for the efficient implementation of these algorithms. We consider image processing examples and show that the present method compares favorably with standard interpolation techniques. Finally, we discuss some properties of this approach and its connection with the classical technique of bandlimiting a signal, which provides the asymptotic limit of our algorithm as the order of the spline tends to infinity.
本文旨在推导用于任意（非整数）比例因子放大或缩小数字图像的最优样条算法。在我们的公式中，原始信号和重采样信号分别由阶数为 n 的具有步长 1 和 Delta 的插值样条表示。通过确定具有步长 Delta 的样条来实现比例变化，该样条在 L(2)-范数中提供对原始信号的最接近逼近。我们表明，这种逼近可以通过三个步骤来计算：（i）数字预滤波器，提供输入信号的 B 样条系数，（ii）使用扩展公式进行重采样，该公式具有显式依赖于 Delta 的修改采样核，以及（iii）数字后滤波器，将结果映射回信号域。我们为 n=0、1 和 3 提供了显式公式，并提出了这些算法的有效实现解决方案。我们考虑图像处理示例，并表明与标准插值技术相比，本方法具有优势。最后，我们讨论了该方法的一些性质及其与信号带限的经典技术的关系，这为我们的算法提供了样条阶数趋于无穷大时的渐近极限。
