Summary

We have introduced several new families of interpolation functions that are optimized for best performance. The family of Schoenberg's polynomial B-splines gives one prominent example, but it is, by far, not the only one!

Introduction

Image interpolation plays a crucial role in a variety of medical imaging tasks. Typical examples include tomographic reconstruction, re-slicing, resampling, compensation of geometrical distortions, image registration, and feature extraction (contours, differential geometry).

Main Contribution

We have introduced an approximation-theoretic framework for the quantitative assessment and comparison of interpolation algorithms. Thanks to this methodology, we have been able to design new basis functions that are optimized for best performance. In particular, we have specified a new family of interpolation functions with maximum order and minimum support (MOMS). We have performed a systematic comparison with other techniques; our experimental results clearly demonstrate that the MOMS functions (B-splines and O-MOMS, in particular) provide the best tradeoff between accuracy and computational cost. The superiority of our spline methods was also confirmed by two research groups in Germany (Lehmann et al ., IEEE Transactions on Medical Imaging, 2001) and in The Netherland (Meijering et al ., Medical Image Analysis, 2001).

This can be checked visually on this JAVA demo .

More recently, we have derived an explicit formula that generates every possible piecewise-polynomial kernel of given degree, support, regularity, and order of approximation (design parameters). This formula contains a set of coefficients that can be chosen freely and do not interfere with the four main design parameters; it is thus easy to tune the design to achieve any additional constraints that the designer may care for.