The effect of NPS calculation method on power-law coefficient estimation accuracy in breast texture modeling

Abstract

In breast X-ray imaging, breast texture has been characterized by a radial noise power spectrum (NPS) that has an inverse power-law shape with exponent β. The technique to estimate the radial power-law coefficient β is typically based on averaging 2-dimensional noise power spectra (NPS), calculated from partly overlapping image regions each weighted by a suitable window function. The linear regression applied over a selected frequency range to the logarithm of the 1- dimensional NPS as a function of the logarithm of the radial frequencies, gives β. For each step in this process, several alternative techniques have been proposed. This paper investigates the effect of image region of interest (ROI) size, image data windowing and alternative ways to determine radial frequency in terms of bias, variance and root mean square error (RMSE) in the estimated β. The effects of these three factors were analytically derived and evaluated using synthetic images with known β varying from 1 to 4 to cover the range of textures encountered in 2D and 3D breast X-ray imaging. Our results indicate that the RMSE in estimated β is smallest when the ROIs are multiplied with an appropriate window function and either no radial averaging or radial averaging with small frequency bins is applied. The ROI size yielding the smallest RMSE depends on several factors and needs to be validated with numerical simulations. In clinical practice however, there might be a need to compromise in the choice of the ROI size to balance between the RMSE magnitudes inherent to the applied β estimation technique and encompass the breast texture range so as to obtain an accurate shape of the NPS. When using 2.56 cm x 2.56 cm ROI sizes, applying a 2D Hann window and no radial frequency averaging, the RMSE in the estimated β ranges from 0.04 to 0.1 for true β values equal to 1 and 4. While many subtleties in real images were not modeled to simplify the mathematics in deriving our results, this work is illustrative in demonstrating the limits of commonly used algorithm steps to estimate accurate β values.