Progressive and iterative approximation for least squares B-spline curve and surface fitting

Optimal Spline Approximation via ℓ0‐Minimization

“…These problems can cause bad performance, an example is the large error for the approximation of the Bahrain-curve. • For the case of B-splines, our 0 problem is addressing the free-knot optimization problem, so it is natural to com- Figure 11: A curve approximated by a cubic spline using our algorithm and using a refinement technique from [DL14], such that both results produce similar 2 error to the input curve. From left to right: input curve, cubic B-spline from our algorithm with 65 nodes, cubic B-spline from the refinement method in [DL14] with 102 nodes.…”

“…For B-splines, this means that a knot vector is fixed and the remaining parameters are optimized. Latest solvers for this problem can compute highly accurate solutions within few milliseconds even for a large number of points [DL14]. However, the approximation can be greatly improved by treating the knots as free variables, see [dB73,Bur74,Jup78] for some early work in this direction.…”

“…• The knot vector is iteratively extended using heuristics, such as integrated discrete curvature [LXZG05], largest accumulated L 2 error on a segment [DL14], or others [YWS04,LM88,Vas96]. These methods often require initial knot vectors, user-defined error bounds and other parameters.…”

Constructing B-spline solids from tetrahedral meshes for isogeometric analysis

“…Some iterative fitting formats were presented recently whose limits approximate the given data points, such as extended PIA (Lin and Zhang, 2011), and least square PIA (Deng and Lin, 2014). In Lin and Zhang (2013), an efficient iterative fitting algorithm using T-spline was proposed for fitting large data set.…”

Reconstructing B-patch surface from triangular mesh

“…In general, the standard surface reconstructing methods interpolate the smooth surfaces by solving linear equation systems and least square problems [1] [11]. The surfaces that are generated by these methods may very well lie close to data points, but they may not be very smooth.…”