Comparing nonlinear texture measures for quantifying trabecular bone structures using surrogates

We generalize the methods of constrained randomization in order to assess different nonlinear texture measures for the quantitative characterisation of trabecular bone structures as seen in high resolution MR images of the distal radius for patients with and without osteoporotic bone fractures. We demonstrate that it is feasible to produce surrogates which preserve texture measures sensitive to higher-order correlations. Specifically, we preserve for two-dimensional images the three Minkowski functionals (MF) which can be interpreted as the surface, the perimeter and the Euler-Characteristic of an excursion set. The surrogates preserving the MF's are generated by using simulated annealing techniques, where the constraints are specified in terms of a cost function which has a global minimum when the constraints are fulfilled. The cost function has to be minimized among all permutations of the image pixels. The surrogates and the original data are quantified by estimating their local scaling properties by means of the calculation of the spectrum of weighted scaling indices (WSI). It is shown that a significant discrimination between original and surrogate data is made possible by comparing the probability distributions of the weighted scaling indices. This proves that the two nonlinear texture measures (MF an WSI) are complementary since they are sensitive to different morphological aspects of the trabecular bone structures. It turns out that the generalized method of constrained randomization is a vital tool for assessing the quality of texture measures in terms of sensitivity to images structures and discrimination power.