Real Time Waveguide Parameter Estimation Using Sparse Multimode Disperse Radon Transform

Osteoporosis and associated fragility fractures are still a societal problem. Several quantitative ultrasound approaches have been proposed to overcome limitations of the current gold standard DXA. Bi Directional Axial Transmission (BDAT) is based on the measurement of waves guided by the cortical bone shell. Cortical thickness (Ct.Th) and porosity (Ct.Po) estimates correspond to the maxima of the objective function Proj(Ct.Th, Ct.Po), initially defined as the projection of a tested model in the singular vector basis (method 1). Each model matrix has the same dimension, i.e., Nf=124 x Nk=256, 512, 1024 or 2048 pixels, of an ultrasonic guided wave spectrum experimental image Norm(f, k). The total number of models is equal to Nth, =38 x Npo=25, i.e., the number of cortical thickness and porosity taken into account, ranging respectively from 0.8 to 4.5 mm and 1 to 25%. Finally, each pixel of the alternative objective function (NthxNpo pixels) corresponds to the pixel-wise image multiplication between one model and the experimental guided wave spectrum image (method 2) or a sparse matrix multiplication between experimental and model reshaped vectors (method 3). The three methods were tested on data obtained on 400 measurements. It was observed that methods 2 and 3 provided the same Ct.Th Ct.Po values while differences with method 1 decreased with Nk. Acceptable differences, i.e., lower than the typical measurement resolution (0.2 mm for Ct.Th and 1% for Ct.Po) were achieved for Nk=2048. Using Matlab on a standard desktop, this calculation took 20, 4 and 0.3 s, for the methods 1 to 3, respectively. Method 3 calculation was achieved in 5 ms using C++. This last value opens perspective toward guiding interface improvement using real time objective function.