70,153 research outputs found
Smooth Key-framing using the Image Plane
This paper demonstrates the use of image-space constraints for key frame interpolation. Interpolating in image-space results in sequences with predictable and controlable image trajectories and projected size for selected objects, particularly in cases where the desired center of rotation is not fixed or when the key frames contain perspective distortion changes. Additionally, we provide the user with direct image-space control over {\em how} the key frames are interpolated by allowing them to directly edit the object\u27s projected size and trajectory. Image-space key frame interpolation requires solving the inverse camera problem over a sequence of point constraints. This is a variation of the standard camera pose problem, with the additional constraint that the sequence be visually smooth. We use image-space camera interpolation to globally control the projection, and traditional camera interpolation locally to avoid smoothness problems. We compare and contrast three different constraint-solving systems in terms of accuracy, speed, and stability. The first approach was originally developed to solve this problem [Gleicher and Witken 1992]; we extend it to include internal camera parameter changes. The second approach uses a standard single-frame solver. The third approach is based on a novel camera formulation and we show that it is particularly suited to solving this problem
Multiparametric Continuous and Mixed-Integer Nonlinear Optimization with Parameters in General Locations
Convex programming has been a research topic for a long time, both theoretically and algorithmically. Frequently, these programs lack complete data or contain rapidly shifting data. In response, we consider solving parametric programs, which allow for fast evaluation of the optimal solutions once the data is known. It has been established that, when the objective and constraint functions are convex in both variables and parameters, the optimal solutions can be estimated via linear interpolation. Many applications of parametric optimization violate the necessary convexity assumption. However, the linear interpolation is still useful; as such, we extend this interpolation to more general parametric programs in which the objective and constraint functions are biconvex. The resulting algorithm can be applied to scalarized multiobjective problems, which are inherently parametric, or be used in a gradient dual ascent method. We also provide two termination conditions and perform a numerical study on synthetic parametric biconvex optimization problems to compare their effectiveness
Relativistic MHD and black hole excision: Formulation and initial tests
A new algorithm for solving the general relativistic MHD equations is
described in this paper. We design our scheme to incorporate black hole
excision with smooth boundaries, and to simplify solving the combined Einstein
and MHD equations with AMR. The fluid equations are solved using a finite
difference Convex ENO method. Excision is implemented using overlapping grids.
Elliptic and hyperbolic divergence cleaning techniques allow for maximum
flexibility in choosing coordinate systems, and we compare both methods for a
standard problem. Numerical results of standard test problems are presented in
two-dimensional flat space using excision, overlapping grids, and elliptic and
hyperbolic divergence cleaning.Comment: 22 pages, 8 figure
Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation
We apply multivariate Lagrange interpolation to synthesize polynomial
quantitative loop invariants for probabilistic programs. We reduce the
computation of an quantitative loop invariant to solving constraints over
program variables and unknown coefficients. Lagrange interpolation allows us to
find constraints with less unknown coefficients. Counterexample-guided
refinement furthermore generates linear constraints that pinpoint the desired
quantitative invariants. We evaluate our technique by several case studies with
polynomial quantitative loop invariants in the experiments
Domain-Type-Guided Refinement Selection Based on Sliced Path Prefixes
Abstraction is a successful technique in software verification, and
interpolation on infeasible error paths is a successful approach to
automatically detect the right level of abstraction in counterexample-guided
abstraction refinement. Because the interpolants have a significant influence
on the quality of the abstraction, and thus, the effectiveness of the
verification, an algorithm for deriving the best possible interpolants is
desirable. We present an analysis-independent technique that makes it possible
to extract several alternative sequences of interpolants from one given
infeasible error path, if there are several reasons for infeasibility in the
error path. We take as input the given infeasible error path and apply a
slicing technique to obtain a set of error paths that are more abstract than
the original error path but still infeasible, each for a different reason. The
(more abstract) constraints of the new paths can be passed to a standard
interpolation engine, in order to obtain a set of interpolant sequences, one
for each new path. The analysis can then choose from this set of interpolant
sequences and select the most appropriate, instead of being bound to the single
interpolant sequence that the interpolation engine would normally return. For
example, we can select based on domain types of variables in the interpolants,
prefer to avoid loop counters, or compare with templates for potential loop
invariants, and thus control what kind of information occurs in the abstraction
of the program. We implemented the new algorithm in the open-source
verification framework CPAchecker and show that our proof-technique-independent
approach yields a significant improvement of the effectiveness and efficiency
of the verification process.Comment: 10 pages, 5 figures, 1 table, 4 algorithm
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