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## Natural Metrics and Least-Committed Priors for Articulated Tracking

### Citations

3854 | A new approach to linear filtering and prediction problems - Kalman - 1960 |

3313 | Numerical Optimization
- Nocedal, Wright
- 1999
(Show Context)
Citation Context ... in the animation and robotics literature. As this is an important tool in much applied research, much work has gone into finding good solvers; we apply a projected steepest descent with line-search (=-=Nocedal and Wright, 1999-=-), as empirical results have shown it to be both fast and stable (EngellNørreg̊ard and Erleben, 2011). The search is started in θt−1, which practically ensures that a good optimum is found as the nume... |

1492 |
Lévy processes and infinitely divisible distributions, Cambridge
- Sato
- 1999
(Show Context)
Citation Context ...ion model in angle space has some rather unintuitive properties, which cannot be avoided by scaling the coordinates. Formally, Euclidean Brownian motion, also known as the Wiener process, is defined (=-=Sato, 1999-=-) as a stochastic process Wt on Rd having independent increments, such that for any partitioning, n ≥ 1 and 0 ≤ t0 < t1 < . . . < tn, Wt0 ,Wt1 −Wt0 , . . . ,Wtn −Wtn−1 are independent random variables... |

1449 |
Numerical solution of stochastic differential equations, volume 23 of Applications of Mathematics (New York
- Kloeden, Platen
- 1992
(Show Context)
Citation Context ...d a Brownian motion model that respects the manifold metric. We now set out to simulate this model using the SDE in eq. 10. While there exists literature on both simulating SDE’s in Euclidean spaces (=-=Kloeden and Platen, 1992-=-) and solving ODE’s on manifolds (Hairer et al., 2004), to the best of our knowledge, no general solvers for manifold-valued SDE’s have been described in the literature. The most basic scheme for simu... |

1226 | Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation
- Belkin, Niyogi
- 2003
(Show Context)
Citation Context ...ladh et al. (2000) learned a lowdimensional linear subspace using Principal Component Analysis and used a linear motion model in this subspace. Sminchisescu and Jepson (2004) use Laplacian Eigenmaps (=-=Belkin and Niyogi, 2003-=-) to learn a nonlinear motion manifold. Similarly, Lu et al. (2008) use a Laplacian Eigenmaps Latent Variable Model (Carreira-Perpinan and Lu, 2007) to learn a manifold. All three learning schemes can... |

877 |
Stochastic Differential Equations: An Introduction with Applications, 6th ed
- Øksendal
- 2003
(Show Context)
Citation Context ... to scale the individual joint angles to encode that some joints move more than others. This corresponds to introducing a covariance matrix in eq. 4. Formally, this makes the model an Itô diffusion (=-=Øksendal, 2000-=-), but we will simply treat it as a Brownian motion in the scaled coordinate system. However, as we shall see, the Brownian motion model in angle space has some rather unintuitive properties, which ca... |

417 |
Directional Statistics
- Mardia, Jupp
(Show Context)
Citation Context ...ong as they are physically possible. 3.4. Relations to Directional Statistics A large part of the work on manifold-valued statistics has been done on spheres; this is known as directional statistics (=-=Mardia and Jupp, 1999-=-). Here easy-touse Brownian motion models are available in the Von Mises distribution. In sequential analysis, this has found uses in such different areas as multi-target air plane tracking (Miller et... |

383 | Stochastic Tracking of 3D Human Figures Using 2D Image Motion,” - Sidenbladh, Black, et al. - 2000 |

219 |
Spatial control of arm movements
- MORASSO
- 1981
(Show Context)
Citation Context ...esponding to the length of the spatial curves that joint positions follow during the movement. Interestingly, this natural metric is well in tune with how humans plan, think about and discuss motion (=-=Morasso, 1981-=-; Abend et al., 1982). Using our spatial representation, we define a Brownian motion model on the Riemannian representation manifold that reflects the metric. The Brownian motion model is expressed as... |

211 | Style-based inverse kinematics. - GROCHOW, MARTIN, et al. - 2004 |

201 | 3D people tracking with Gaussian process dynamical models”, CVPR, - Urtasun, Fleet, et al. - 2006 |

181 | Principal Geodesic Analysis for the study of nonlinear statistics of shape.
- Fletcher, Lu, et al.
- 2004
(Show Context)
Citation Context ...odels are often crafted by learning manifolds to which the motion is confined. An obvious next step is, thus, to learn a submanifold of the kinematic manifoldM using e.g. Principal geodesic analysis (=-=Fletcher et al., 2004-=-) or Geodesic PCA (Huckemann et al., 2010). This can then be used to restrict the tracking system. In this paper, we have focused exclusively on models of human motion. The Brownian motion model is, h... |

175 |
Vision-based human motion analysis: An overview.
- Poppe
- 2007
(Show Context)
Citation Context ...uations, Numerical Solutions to SDEs 1. Introduction This paper is concerned with least-committed priors for probabilistic articulated tracking, i.e. estimation of human poses in sequences of images (=-=Poppe, 2007-=-). When treating such problems, a maximum a posteriori estimate is typically found by solving an optimisation problem, and the optimisation is then guided by a prior model for predicting future motion... |

158 | Gaussian process dynamic models for human motion. - Wang, Fleet, et al. - 2008 |

122 | Priors for people tracking from small training sets. - Urtasun, Fleet, et al. - 2005 |

114 | An overview of existing methods and recent advances in sequential Monte Carlo. - Cappe, Godsill, et al. - 2007 |

89 | Stochastic analysis on manifolds - Hsu - 2002 |

89 | Generative Modeling for Continuous Non-Linearly Embedded Visual Inference - Sminchisescu, Jepson - 2004 |

79 |
Human arm trajectory formation
- Abend, Bizzi, et al.
- 1982
(Show Context)
Citation Context ...e length of the spatial curves that joint positions follow during the movement. Interestingly, this natural metric is well in tune with how humans plan, think about and discuss motion (Morasso, 1981; =-=Abend et al., 1982-=-). Using our spatial representation, we define a Brownian motion model on the Riemannian representation manifold that reflects the metric. The Brownian motion model is expressed as a manifold-valued s... |

48 |
Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations
- Hairer, Lubich, et al.
- 2002
(Show Context)
Citation Context ...c. We now set out to simulate this model using the SDE in eq. 10. While there exists literature on both simulating SDE’s in Euclidean spaces (Kloeden and Platen, 1992) and solving ODE’s on manifolds (=-=Hairer et al., 2004-=-), to the best of our knowledge, no general solvers for manifold-valued SDE’s have been described in the literature. The most basic scheme for simulating Stratonovich SDE’s in Euclidean domains is the... |

47 | Hierarchical Implicit Surface Joint Limits to Constrain Video-Based Motion Capture - Herda, Urtasun, et al. - 2004 |

34 | Conditional-mean estimation via jump-diffusion processes in multiple target tracking/recognition.
- Miller, Srivasta, et al.
- 1995
(Show Context)
Citation Context ...upp, 1999). Here easy-touse Brownian motion models are available in the Von Mises distribution. In sequential analysis, this has found uses in such different areas as multi-target air plane tracking (=-=Miller et al., 1995-=-) and white matter tracking in Diffusion Tensor MRI (Zhang et al., 2007). Except for the special case of the kinematic skeleton consisting of only one bone, the kinematic manifold is not spherical and... |

33 | People tracking with the Laplacian eigenmaps latent variable model - Lu, Perpinan, et al. - 2007 |

30 |
The Laplacian Eigenmaps latent variable model.
- Carreira-Perpinan, Lu
- 2007
(Show Context)
Citation Context ...ce. Sminchisescu and Jepson (2004) use Laplacian Eigenmaps (Belkin and Niyogi, 2003) to learn a nonlinear motion manifold. Similarly, Lu et al. (2008) use a Laplacian Eigenmaps Latent Variable Model (=-=Carreira-Perpinan and Lu, 2007-=-) to learn a manifold. All three learning schemes can be phrased in terms of pair-wise distances between training data, where the metric is the joint angle distance discussed in sec. 2.4. The above ap... |

29 | Physics-based person tracking using the anthropomorphic walker,” IJCV - Brubaker, Fleet, et al. - 2010 |

27 | Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric Lie group actions. Statist. Sinica 20 1–100. MR2640651
- HUCKEMANN, HOTZ, et al.
- 2010
(Show Context)
Citation Context ...folds to which the motion is confined. An obvious next step is, thus, to learn a submanifold of the kinematic manifoldM using e.g. Principal geodesic analysis (Fletcher et al., 2004) or Geodesic PCA (=-=Huckemann et al., 2010-=-). This can then be used to restrict the tracking system. In this paper, we have focused exclusively on models of human motion. The Brownian motion model is, however, applicable to many other domains.... |

20 | M.: Tracking people interacting with objects - Kjellstrom, Kragic, et al. - 2010 |

14 | Scene Constraintsaided Tracking of Human Body - Yamamoto, Yagishita - 2000 |

7 | K.S.: Stick it! articulated tracking using spatial rigid object priors.
- Hauberg, Pedersen
- 2010
(Show Context)
Citation Context ... the paper, we will introduce a novel method for simulating the manifold valued SDE’s numerically and use this for predicting human motion in an articulated tracking system. In (Hauberg et al., 2010; =-=Hauberg and Pedersen, 2011b-=-), we introduced the kinematic manifold and showed that it is suitable for modelling interactions with the environment. In these papers, a somewhat ad hoc predictive model was defined where motion was... |

6 |
K.S.: Predicting articulated human motion from spatial processes
- Hauberg, Pedersen
- 2011
(Show Context)
Citation Context ...́ et al., 2007). For the predictive model, p(θt+1|θt), we will compare different models in the following sections. We describe the likelihood system next; this likelihood was previously described in (=-=Hauberg and Pedersen, 2011a-=-). We use a small baseline consumer stereo camera1 for acquiring data. At each time instance we, thus, get a set of three dimensional points Zt = {z(1)t , . . . , z(K)t } that are mostly scattered aro... |

2 | 2011. A projected backtracking line-search for constrained interactive inverse kinematics - Engell-Nørreg̊ard, Erleben |