Application and assessment of multiscale bending energy for morphometric characterization of neural cells

The estimation of the curvature of experimentally obtained curves is an important issue in many applications of image analysis including biophysics, biology, particle physics, and high energy physics. However, the accurate calculation of the curvature of digital contours has proven to be a difficult endeavor, mainly because of the noise and distortions that are always present in sampled signals. Errors ranging from 1% to 1000% have been reported with respect to the application of standard techniques in the estimation of the curvature of circular contours [M. Worring and A. W. M. Smeulders, CVGIP: Im. Understanding, 58, 366 (1993)]. This article explains how diagrams of multiscale bending energy can be easily obtained from curvegrams and used as a robust general feature for morphometric characterization of neural cells. The bending energy is an interesting global feature for shape characterization that expresses the amount of energy needed to transform the specific shape under analysis into its lowest energy state (i.e., a circle). The curvegram, which can be accurately obtained by using digital signal processing techniques (more specifically through the Fourier transform and its inverse, as described in this work), provides multiscale representation of the curvature of digital contours. The estimation of the bending energy from the curvegram is introduced and exemplified with respect to a series of neural cells. The masked high curvature effect is reported and its implications to shape analysis are discussed. It is also discussed and illustrated that, by normalizing the multiscale bending energy with respect to a standard circle of unitary perimeter, this feature becomes an effective means for expressing shape complexity in a way that is invariant to rotation, translation, and scaling, and that is robust to noise and other artifacts implied by image acquisition.