Results

**1 - 3**of**3**### Optimal Spline Approximation via 0-Minimization

"... Splines are part of the standard toolbox for the approximation of functions and curves in Rd. Still, the problem of finding the spline that best approximates an input function or curve is ill-posed, since in general this yields a “spline ” with an infinite number of segments. The problem can be regu ..."

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Splines are part of the standard toolbox for the approximation of functions and curves in Rd. Still, the problem of finding the spline that best approximates an input function or curve is ill-posed, since in general this yields a “spline ” with an infinite number of segments. The problem can be regularized by adding a penalty term for the number of spline segments. We show how this idea can be formulated as an 0-regularized quadratic problem. This gives us a notion of optimal approximating splines that depend on one parameter, which weights the approximation error against the number of segments. We detail this concept for different types of splines including B-splines and composite Bézier curves. Based on the latest development in the field of sparse approximation, we devise a solver for the resulting minimization problems and show applications to spline approximation of planar and space curves and to spline conversion of motion capture data.

### Real-Time Nonlinear Shape Interpolation

"... We introduce a scheme for real-time nonlinear interpolation of a set of shapes. The scheme exploits the structure of the shape interpolation prob-lem, in particular, the fact that the set of all possible interpolated shapes is a low-dimensional object in a high-dimensional shape space. The interpola ..."

Abstract
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We introduce a scheme for real-time nonlinear interpolation of a set of shapes. The scheme exploits the structure of the shape interpolation prob-lem, in particular, the fact that the set of all possible interpolated shapes is a low-dimensional object in a high-dimensional shape space. The interpolated shapes are defined as the minimizers of a nonlinear objective functional on the shape space. Our approach is to construct a reduced optimization prob-lem that approximates its unreduced counterpart and can be solved in mil-liseconds. To achieve this, we restrict the optimization to a low-dimensional subspace that is specifically designed for the shape interpolation problem. The construction of the subspace is based on two components: a formula for the calculation of derivatives of the interpolated shapes and a Krylov-type sequence that combines the derivatives and the Hessian of the objective functional. To make the computational cost for solving the reduced opti-mization problem independent of the resolution of the example shapes, we combine the dimensional reduction with schemes for the efficient approxi-mation of the reduced nonlinear objective functional and its gradient. In our experiments, we obtain rates of 20-100 interpolated shapes per second even for the largest examples which have 500k vertices per example shape.