New Closed Formula for the Univariate Hermite Interpolating Polynomial of Total Degree and its Application in Medical Image Slice Interpolation

This work investigates the usefulness of univariate Hermite interpolation of the total degree (HTD) for a biomedical signal processing task: slice interpolation in a variety of medical imaging modalities. The HTD is an algebraically demanding interpolation method that utilizes information of the values of the signal to be interpolated at distinct support positions, as well as the values of its derivatives up to a maximum available order. First a novel closed form solution for the univariate Hermite interpolating polynomial is presented for the general case of arbitrarily spaced support points and its computational and algebraic complexity is compared to that of the classical expression of the Hermite interpolating polynomial. Then, an implementation is proposed for the case of equidistant support positions with computational complexity comparable to any convolution-based interpolation method. We assess the proposed implementation of HTD interpolation with equidistant support points in the task of slice interpolation, which is usually treated as a one-dimensional problem. We performed a large number of interpolation experiments for 220 Magnetic Resonance Imaging (MRI) datasets and 50 Computer Tomography (CT) datasets and compared the proposed HTD implementation to several other well established interpolation techniques. In our experiments, we approximated the signal derivatives using finite differences, however the proposed HTD can accommodate any type of derivative calculation. Results show that the HTD interpolation outperforms the other interpolation methods under comparison, in terms of root mean square error (RMSE), in every one of the interpolation experiments, resulting in higher accuracy interpolated images. Finally, the behavior of the HTD with respect to its controlling parameters is explored and its computational complexity is determined.