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Modelling character motions on infinite-dimensional manifolds

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Abstract

In this article, we will formulate a mathematical framework that allows us to treat character animations as points on infinite-dimensional Hilbert manifolds. Constructing geodesic paths between animations on those manifolds allows us to derive a distance function to measure similarities of different motions. This approach is derived from the field of geometric shape analysis, where such formalisms have been used to facilitate object recognition tasks. Analogously to the idea of shape spaces, we construct motion spaces consisting of equivalence classes of animations under reparametrizations. Especially cyclic motions can be represented elegantly in this framework. We demonstrate the suitability of this approach in multiple applications in the field of computer animation. First, we show how visual artefacts in cyclic animations can be removed by applying a computationally efficient manifold projection method. We next highlight how geodesic paths can be used to calculate interpolations between various animations in a computationally stable way. Finally, we show how the same mathematical framework can be used to perform cluster analysis on large motion capture databases, which can be used for or as part of motion retrieval problems.

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Notes

  1. Such as arclength parametrization, open/closed curves etc.

  2. Cyclic animations are periodic, i.e., they can be repeated continuously, such as for example a walking motion.

  3. The root bone can also have rotational and translational transformations with respect to a global coordinate system.

  4. i.e., translation w.r.t to its parent.

  5. By a bone’s degrees of freedom we mean, more precisely, the corresponding joint’s degrees of freedom.

  6. Also known as elastic metric.

  7. They are parametrized by weights balancing the influences of bending vs. stretching forces.

  8. See Fig. 2

    Fig. 2
    figure 2

    Reparametrization of a curve by composition with a diffeomorphism on the unit circle \(\mathbb {S}^1\). Both curves evolve clockwise. Note the changed starting point (the diamond) and increased density of sampling points close to the start

    Full size image

  9. We could also construct a similar space over the space of open curves \(\mathcal {C}^o\), where the start point remains fixed but the sampling rate along the curve varies.

  10. See also the attached videos.

    Fig. 3
    figure 3

    Optimization of the periodicity of a running animation. For simplicity, only half the character is shown. The top figure shows the original animation, played two times, with a noticeable gap as the animation repeats. The bottom figure shows a more cyclic animation derived using the projection defined in Sect. 4.1 and Algorithm 1

    Full size image
    Fig. 4
    figure 4

    Optimization of the periodicity of a jumping animation. For simplicity, only half the character is shown. The top figure shows the original animation, played two times. There is again a noticeable gap between repetitions, where the animation ends with the character leaning backwards and then immediately starts with a forward leaning pose. The bottom figure shows a more cyclic animation derived using the projection defined in Sect. 4.1 and Algorithm 1

    Full size image

  11. The Palais-metric; see, e.g., [17].

  12. Corresponding to general and cyclic animations, respectively.

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Acknowledgments

The author would like to thank Elena Celledoni and Markus Grasmair for valuable discussions and feedback. This research was supported in part by the GeNuIn Applications project grant from the Research Council of Norway. The data used in this project was obtained from mocap.cs.cmu.edu. The database was created with funding from NSF EIA-0196217.

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  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 , Trondheim, Norway

    Markus Eslitzbichler

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Correspondence to Markus Eslitzbichler.

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Eslitzbichler, M. Modelling character motions on infinite-dimensional manifolds. Vis Comput 31, 1179–1190 (2015). https://doi.org/10.1007/s00371-014-1001-y

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