Evolution Properties Of Space Curves

Abstract

An evolved version I?, of a space curve l7 is obtained by convolving a parametric representation of r with a Gaussian function of variance 0'. The process of generating the ordered sequence of curves {I',\o>O} is referred to as the evolution of r. Evolved space curves arise when computing the Torsion and Curvature Scale Space representation of a space curve. A number of evolution properties of space curves are investigated in this paper. It is shown that the evolution of space curves is invariant under rotation, uniform scaling and translation of those curves. This is an essential propert for any reliable shape representation. It is also shown tiat properties such as connectedness and closedness of a space curve are preserved during evolution of the curve and that the center of mass of a s ace curve remains the same as the curve evolves. AnotRer result is that a space curve remains inside its convex hull during evolution. The two main theorems of the paper examine a s ace curve during its evolution just before and just after tie formation of a cusp point. It is shown that strong constraints on the shape of the curve in the neighborhood of the cusp point exist just before and just after the formation of that point. Final1 it is argued that each one of the results obtained in tiis paper is important and useful for practical applications. I. Introduction A multi-scale representation for one-dimensional functions and signals was first proposed by Stansfield [1980] and later developed by Witkin [1983]. The signal fp) is convolved with a Gaussian function as its variance 0 varies from a small to a large value. The aero-crossings of the second derivative of each convolved signal are extracted and marked in the z-o space. The result is the Scale Space Image of the signal. Mokhtarian and Mackworth [1986] generalized that concept to planar curves. A planar curve r is parametrized by arc length U and represented using its coordinate functions. An evolved version of I' is computed by convolving each of its coordinate functions with a Gaussian function of variance 0' and denoted by I',,. The process of generating the ordered sequence of curven {r,lo>O} is referred to as the evolution of r. The curvature of each Fa can be expressed in terms of the first and second derivatives of convolved versions of functions z( U) and y(u). It is therefore possible to extract the curvature zero-crossings of each I', as 5 varies from a small to a large value and mark them in the U-o space. The result is referred to as the Curvature Scale Space Image of the curve. Mokhtarian [1988b] generalized the above concept further to space curves. The parametrization of a space curve can be expressed as: I' = (z(u),y(u),z(u)). Curvature and torsion of an evolved space curve can be expressed in terms of the first three derivatives of convolved versions of functions z(u), y(~) and n(u). A scale space representation for space curves consists of the Torsion and Curvature Scale Space Images which contain the torsion zero-crossings and the curvature level-crossings maps of the curve respectively. Scale space representations for planar and space curves satisfy several useful criteria such as Eficiency, Invariance, Sensitivity, Uniqueness, Detail and Robustness [Mokhtarian 1988bI. These properties make them specially suitable for recognition of arbitrarily shaped objects. Mackworth and Mokhtarian [1988] have investigated a number of evolution properties of planar curves. This paper generaliies some of those properties and investigates other evolution properties of space curves. Lemma 1 shows that evolution of a space curve is invariant under rotation, uniform scaling and translation of the curve. Lemmas 2 and 3 show that connectedness and closedness of a space curve are preserved during its evolution. Lemma 4 shows that the center of mass of a closed space curve does not move as the curve evolves. Lemma 5 shows that a space curve remains inside its convex hull during evolution.