Scale selection for anisotropic scale-space: application to volumetric tumor characterization

A unified approach for treating the scale selection problem in the anisotropic scale-space is proposed. The anisotropic scale-space is a generalization of the classical isotropic Gaussian scale-space by considering the Gaussian kernel with a fully parameterized analysis scale (bandwidth) matrix. The "maximum-over-scales" and the "most-stable-over-scales" criteria are constructed by employing the "L-normalized scale-space derivatives", i.e., response-normalized derivatives in the anisotropic scale-space. This extension allows us to directly analyze the anisotropic (ellipsoidal) shape of local structures. The main conclusions are (i) the norm of the γ- and L-normalized anisotropic scale-space derivatives with a constant γ =1/2 are maximized regardless of the signal\´s dimension iff the analysis scale matrix is equal to the signal\´s covariance and (ii) the most-stable-over-scales criterion with the isotropic scale-space outperforms the maximum-over-scales criterion in the presence of noise. Experiments with 1D and 2D synthetic data confirm the above findings. 3D implementations of the most-stable-over-scales methods are applied to the problem of estimating anisotropic spreads of pulmonary tumors shown in high-resolution computed-tomography (HRCT) images. Comparison of the first- and second-order methods shows the advantage of exploiting the second-order information.