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C. Schmitt, H.-B. Neuner:
"Knot estimation on B-Spline curves";
Österreichische Zeitschrift für Vermessung und Geoinformation (VGI), begutachteter Beitrag (invited), 103 (2015), 2+3; 188 - 197.

Surface-based metrology, like terrestrial laser scanner, needs new surface-based evaluation methods. These approximation methods are one of the main challenges making the information of 3D point clouds suitable and taking benefits from the redundancy. Freeform curves and surfaces are promising approximation methods to create parameterized curves and surfaces for further evaluation steps, like shape information for structural analysis of built objects. In the past it was shown that the freeform shapes significantly improve the approximation quality, compared to approximations of geometric primitives.

As contactless surveying metrology the terrestrial laser scanner was deployed scanning a concrete-freeform dome. One profile of the panoramic scan was extracted for our application.
Focusing on B-spline curves which are able to capture the local behavior of the profile are investigated in this study. Typically, the only parameter set treated as unknowns are the control points of the B-Spline. Their location is determined by least squares adjustment. The second parameter set, the knots, which are part of the basis functions derived from the Bernstein polynomials, are placed at stable locations. The approach with fixed knots leads to a linear system, but it intuitively restricts the B-Spline curve in its flexibility. Nevertheless the residuals of the approximation may still contain systematic effects. However estimating the control points and the locations of the knots at the same time succeeds in full flexibility of
B-Splines and optimizes the approximation. The accrued system of equations is highly non-linear. To enhance the convergent behavior, constraints and adequate starting values are necessary. The constraints are derived from Schoenberg-Whitney theorems. The starting values for the knots are chosen with the bottom up method, beginning by the minimum number of knots and adding one knot at each iteration step at particular curve sections until the convergent criterion is reached. The decision to insert a knot and at which location, is based on the analysis of the residuals in each section, referred to as the knot span. The starting values of the control points are computed from the linear system.

Freiformkurven können zur Approximation von Punktwolken von terrestrischen Laserscannern genutzt werden.
Im Speziellen werden in dieser Untersuchung B-Spline Kurven eingesetzt, die je nach Parameterwahl lokale Gegebenheiten in einer globalen Approximation darstellen können. Typischerweise werden bei einer Approximation von B-Splines die Kontrollpunkte in einem linearen Modell geschätzt. Die Knoten sind ein weiterer Parametersatz, mithilfe derer die Basisfunktionen erstellt werden. Die gemeinsame Schätzung der Knoten mit den Kontrollpunkten ergibt ein hochgradig nichtlineares Gleichungssystem. Die volle Flexibilität zur lokalen Anpassung wird erst durch die Schätzung beider Parametergruppen erreicht.
Zur Stützung des nichtlinearen Gleichungssystems werden Bedingungsgleichungen und verbesserte Näherungswerte eingeführt. Diese Näherungswerte für die Knoten werden mit einer neuen Methode ermittelt. Diese basiert auf den Residuen der linearen Schätzung der Kontrollpunkte, die in Teilbereichen, sogenannter Spans, analysiert werden. Begonnen wird die Approximation mit der Minimalkonfiguration, den Bézier-Kurven, innerhalb derer die Knoten festgelegt sind.

freeform curve, B-Spline curve, TLS profile approximation, free knots