Image interpolation techniques often are required in medical imaging for image generation (e.g., discrete back projection for inverse Radon transform) and processing such as compression or resampling. Since the ideal interpolation function spatially is unlimited, several interpolation kernels of finite size have been introduced. This paper compares 1) truncated and windowed sinc; 2) nearest neighbor; 3) linear; 4) quadratic; 5) cubic B-spline; 6) cubic; g) Lagrange; and 7) Gaussian interpolation and approximation techniques with kernel sizes from 1 x 1 up to 8 x 8. The comparison is done by: 1) spatial and Fourier analyses; 2) computational complexity as well as runtime evaluations; and 3) qualitative and quantitative interpolation error determinations for particular interpolation tasks which were taken from common situations in medical image processing. For local and Fourier analyses, a standardized notation is introduced and fundamental properties of interpolators are derived. Successful methods should be direct current (DC)-constant and interpolators rather than DC-inconstant or approximators. Each method's parameters are tuned with respect to those properties. This results in three novel kernels, which are introduced in this paper and proven to be within the best choices for medical image interpolation: the 6 x 6 Blackman-Harris windowed sinc interpolator, and the C2-continuous cubic kernels with N = 6 and N = 8 supporting points. For quantitative error evaluations, a set of 50 direct digital X rays was used. They have been selected arbitrarily from clinical routine. In general, large kernel sizes were found to be superior to small interpolation masks. Except for truncated sinc interpolators, all kernels with N = 6 or larger sizes perform significantly better than N = 2 or N = 3 point methods (p << 0.005). However, the differences within the group of large-sized kernels were not significant. Summarizing the results, the cubic 6 x 6 interpolator with continuous second derivatives, as defined in (24), can be recommended for most common interpolation tasks. It appears to be the fastest six-point kernel to implement computationally. It provides eminent local and Fourier properties, is easy to implement, and has only small errors. The same characteristics apply to B-spline interpolation, but the 6 x 6 cubic avoids the intrinsic border effects produced by the B-spline technique. However, the goal of this study was not to determine an overall best method, but to present a comprehensive catalogue of methods in a uniform terminology, to define general properties and requirements of local techniques, and to enable the reader to select that method which is optimal for his specific application in medical imaging.

This paper analyzes B-spline interpolation techniques of degree 2, 4, and 5 with respect to all criteria that have been applied to evaluate various interpolation schemes in a recently published survey on image interpolation in medical imaging (Lehmann et al., 1999). It is shown that high-degree B-spline interpolation has superior Fourier properties, smallest interpolation error, and reasonable computing times. Therefore, high-degree B-splines are preferable interpolators for numerous applications in medical image processing, particularly if high precision is required. If no aliasing occurs, this result neither depends on the geometric transform applied for the tests nor the actual content of images.

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Iris Schubert, yang telah mendukung dan mengawasi proyek ini. Proyek GTZ ProBangkit, yang terdiri dari Roto Priyono, dan Pimpinan Tim Proyek Manfred Poppe, yang mendukung proyek ini melalui kerja samanya yang sangat baik, khususnya selama tahap akhir penyiapan panduan. Tak lupa ucapan terima kasih kami untuk Lini Wollenberg dan Godwin Limberg dari proyek CIFOR-BMZ yang telah mengkaji ulang draf awal panduan ini. Bundesministerium für Wirtschaftliche Zusammenarbeit und Entwicklung (BMZ), Jerman yang menyediakan dana untuk penelitian dan pengembangan publikasi ini.

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C e n t e r f o r I n t e r n a t i o n a l F o r e s t r y R e s e a r c h

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