Curvature of n-dimensional space curves in grey-value images

Abstract

Abstract-Local curvature represents an important shape parameter of space curves which are well described by differential geometry. We have developed an estimator for local curvature of space curves embedded in -dimensional ( -D) grey-value images. There is neither a segmentation of the curve needed nor a parametric model assumed. Our estimator works on the orientation field of the space curve. This orientation field and a description of local structure is obtained by the gradient structure tensor. The orientation field has discontinuities; walking around a closed contour yields two such discontinuities in orientation. This field is mapped via the Knutsson mapping to a continuous representation; from a -D vector to a symmetric 2 -D tensor field. The curvature of a space curve, a coordinate invariant property, is computed in this tensor field representation. An extensive evaluation shows that our curvature estimation is unbiased even in the presence of noise, independent of the scale of the object and furthermore the relative error stays small.

A systematic approach to nD orientation representation

“…For G; the smoothing and an eigenvalue analysis of G is now standard. In general, statements are possible about the properties laid down in the requirements, the norm of a variational vector kdMðxÞk and the norm of a mapped vector kMðxÞk: The uniform stretch property can be utilized to compute different properties than local structure by applying other filters than blurring filters [16,17].…”

A methodology for analyzing curvature in the developing brain from preterm to adult

“…Other methods exist for constructing surfaces that do not rely on explicit segmentation, as proposed recently by Rieger (Rieger and van Vliet, 2002;. These methods are based on orientation fields obtained by using gradient structure tensors at different scales.…”

Three-dimensional gesture comparison using curvature analysis of position and orientation

“…As described by Reiger et al [6], the curvature of a curve can completely describe the shape of this curve in 2D space. If one add a second parameter, the torsion, one can describe a curve in 3D space.…”