# The ICICS/CS Reading Room

## UBC CS TR-88-09 Summary

- No on-line copy of this technical report is available.

- Fingerprint Theorems for Curvature and Torsion Zero-Crossings, April 1988 Farzin Mokhtarian
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.
\n 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.
\n 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.
\n 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.

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