Structure detection with second order riesz transforms

A frequently applied indicator of tubular structures is based on the eigenvalues of the Hessian matrix of the original image convolved with a Gaussian, whose standard derivation depends on the size of the tubes. Hence the tube size must either be known in advance or a whole scale of standard deviations has to be tested resulting in higher computational costs - a serious obstacle for data with varying tube thickness. In this paper, we propose to modify the structure indicator by replacing the derivatives of the Gaussian smoothed function by the Riesz transform. We show by various numerical examples that the resulting structure indicator is scale independent. Smoothing with a Gaussian is just necessary to cope with the noise in the image, but is not related to the size of the tubular structures. We apply the novel structure indicator for the fiber orientation analysis of fibrous materials and for the segmentation of leather. The latter one was a special challenging application since all scales are present in the microstructure of leather.