Evaluation of Directed Secant Methods in 2D

The secant method is one of the first methods to compute anisotropy from textured 2D images or 3D volumes. While computer science developed more accurate methods to compute anisotropy, the secant method keeps being applied in the analysis of bone micro-structure. Over the last decades, different improvements of the method have been suggested, in particular smoothing of the input signal, applying a three-pixel wide secant line and computation in gray scale. We evaluate these modifications and suggest instead of the commonly applied single pixel secant and nearest neighbor interpolation a two-pixel wide secant and bi-cubic interpolation. We provide pseudo-code of the tested 2D algorithms and compare them on precision and accuracy of the shape and size based on the root-mean square error, and median and inter-quantile range of repeated experiments to compute the mean intercept density and degree of anisotropy. Our results showed large improvements in precision and accuracy when incorporating smoothing, two-pixel wide secants and bi-cubic interpolation, achieving on 2D slices of CT scans of human bone a reduction of up to 91% for size accuracy errors and 57% for shape errors. Our methods are similarly applicable in 3D, either in binary or grayscale. We strongly suggest the consideration of our modifications for applications that require the secant method, in particular for the micro-structural analysis of human bone with CT.