Digital Curvature Estimation: An Operator Theoretic Approach

• This thesis is divided into two parts. The first part is devoted to the curvature estimation of piecewise smooth curves using variation diminishing splines. The variation diminishing property combined with the ability to reconstruct linear functions leads to a convexity preserving approximation that is crucial if additional sign changes in the curvature estimation have to be avoided. To this end, we will first establish the foundations of variation diminishing transforms and introduce the Bernstein and the Schoenberg operator on the space of continuous functions and its generalization to the Lp-spaces. In order to be able to detect C2-singularities in piecewise smooth curves, we establish lower estimates for the approximation error in terms of the second order modulus of smoothness for Schoenberg’s variation diminishing operator. Afterwards, we consider smooth curve approximations using only finitely many samples of the curve, where the approximation, its first, and its second derivative converge uniformly to its corresponding part ofThis thesis is divided into two parts. The first part is devoted to the curvature estimation of piecewise smooth curves using variation diminishing splines. The variation diminishing property combined with the ability to reconstruct linear functions leads to a convexity preserving approximation that is crucial if additional sign changes in the curvature estimation have to be avoided. To this end, we will first establish the foundations of variation diminishing transforms and introduce the Bernstein and the Schoenberg operator on the space of continuous functions and its generalization to the Lp-spaces. In order to be able to detect C2-singularities in piecewise smooth curves, we establish lower estimates for the approximation error in terms of the second order modulus of smoothness for Schoenberg’s variation diminishing operator. Afterwards, we consider smooth curve approximations using only finitely many samples of the curve, where the approximation, its first, and its second derivative converge uniformly to its corresponding part of the curve to be approximated. In this case, we can show that the estimated curvature converges uniformly to the real curvature if the number of samples goes to infinity. Based on the lower estimates that relates the decay rate of the approximation error with smoothness we propose a multi-scale algorithm to estimate the curvature and to detect C2-singularities. We numerically evaluate our algorithm and compare it to others to show that our algorithm achieves competitive accuracy while our curvature estimations are significantly faster to compute. The second part deals with generalizations of the established lower estimates for the Schoenberg operator. We will show that such estimates can be obtained for linear operators on a general Banach function space with smooth range provided that the iterates of the operator converge uniformly and a semi-norm defined on the range of the operator annihilates the fixed points of the operator. To this end, we will prove by spectral properties that the iterates of every positive finite-rank operator converge uniformly. As highlight of this thesis, we show a constructive way using a Gramian matrix where the dual fixed points operate on the fixed points of an operator to derive the limit of the iterates for an arbitrary quasi-compact operator defined on a general Banach space.