Approach for the smoothing of three-dimensional reconstructions of the human spine using dual Kriging interpolation

In numerous situations, 3-D reconstructions of the spine are represented as curves in space, with the vertebral centroids as control points. Interpolation functions such as splines, polynomials or Fourier series have been used to minimise measurement errors and to perform specific calculations. A more general approach, dual Kriging, is presented which incorporates in a single formulation several methods, such as piece-wise linear interpolation, splines and least square functions as a limit case. To minimise user interaction and to control the different Kriging parameters, a computer program is developed allowing efficient use of these interpolation techniques in a clinical environment. Given different drift and covariance functions, the program determines the most suitable Kriging model for specific spine geometries and controls the amount of smoothing performed on raw data. Validation of the technique is with analytical 3-D curves, where random noise is added to represent reconstruction errors. A maximum absolute mean difference of 1.85 +/- 0.50 mm is found between the analytical and noisy curves smoothed with the Kriging technique for 200 points. Results obtained on actual 3-D reconstructions of scoliotic patients are very promising.