Fingerprint Theorems for Curvature and Torsion Zero-Crossings

The {\em scale space image} of a signal f(x) is constructed by extracting the zero-crossings of the second derivative of a Gaussian of variable size a convolved with the signal, and recording them in the $x-\sigma $ map. Likewise, the {\em curvature scale space image} of a planar curve is computed by extracting the curvature zero-crossings of a parametric representation of the curve convolved with a Gaussian of variable size. The curvature level-crossings and torsion zero-crossings are used to compute the {\em curvature} and {\em torsion scale space images} of a space curve respectively. It has been shown [Yuille and Poggio 1983] that the scale space image of a signal determines that signal uniquely up to constant scaling and a harmonic function. This paper presents a generalization of the proof given in [Yuille and Poggio 1983]. It is shown that the curvature scale space image of a planar curve determines the curve uniquely, up to constant scaling and a rigid motion. Furthermore, it is shown that the torsion scale space of a space curve determines the function $\tau (u) \kappa ^{2} (u)$ modulus a scale factor where $\tau (u)$ and $\kappa (u)$ represent the torsion and curvature functions of the curve respectively. Our results show that a 1-D signal can be reconstructed using only one point from its scale space image. This is an improvement of the result obtained by Yuille and Poggio. The proofs are constructive and assume that the parametrizations of the curves can be represented by polynomials of finite order. The scale maps of planar and space curves have been proposed as representations for those curves [Mokhtarian and Mackworth 1986, Mokhtarian 1988]. The result that such maps determine the curves they are computed from uniquely, shows that they satisfy an important criterion for any shape representation technique.