Displacement (geometry) | EPFL Graph Search

In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final position of the point trajectory. A displacement may be identified with the translation that maps the initial position to the final position. A displacement may be also described as a relative position (resulting from the motion), that is, as the final position xf of a point relative to its initial position xi. The corresponding displacement vector can be defined as the difference between the final and initial positions: s = x_\textrm{f} - x_\textrm{i} = \Delta{x} In considering motions of objects over time, the instantaneous velocity of the object is the rate of change of the displacement as a function of time. The instantaneous speed, then, is distinct from v

Looping probabilities of elastic chains: A path integral approach

Ludovica Cotta-Ramusino, John Maddocks

We consider an elastic chain at thermodynamic equilibrium with a heat bath, and derive an approximation to the probability density function, or pdf, governing the relative location and orientation of the two ends of the chain. Our motivation is to exploit continuum mechanics models for the computation of DNA looping probabilities, but here we focus on explaining the novel analytical aspects in the derivation of our approximation formula. Accordingly, and for simplicity, the current presentation is limited to the illustrative case of planar configurations. A path integral formalism is adopted, and, in the standard way, the first approximation to the looping pdf is obtained from a minimal energy configuration satisfying prescribed end conditions. Then we compute an additional factor in the pdf which encompasses the contributions of quadratic fluctuations about the minimum energy configuration along with a simultaneous evaluation of the partition function. The original aspects of our analysis are twofold. First, the quadratic Lagrangian describing the fluctuations has cross-terms that are linear in first derivatives. This, seemingly small, deviation from the structure of standard path integral examples complicates the necessary analysis significantly. Nevertheless, after a nonlinear change of variable of Riccati type, we show that the correction factor to the pdf can still be evaluated in terms of the solution to an initial value problem for the linear system of Jacobi ordinary differential equations associated with the second variation. The second novel aspect of our analysis is that we show that the Hamiltonian form of these linear Jacobi equations still provides the appropriate correction term in the inextensible, unshearable limit that is commonly adopted in polymer physics models of, e. g. DNA. Prior analyses of the inextensible case have had to introduce nonlinear and nonlocal integral constraints to express conditions on the relative displacement of the end points. Our approximation formula for the looping pdf is of quite general applicability as, in contrast to most prior approaches, no assumption is made of either uniformity of the elastic chain, nor of a straight intrinsic shape. If the chain is uniform the Jacobi system evaluated at certain minimum energy configurations has constant coefficients. In such cases our approximate pdf can be evaluated in an entirely explicit, closed form. We illustrate our analysis with a planar example of this type and compute an approximate probability of cyclization, i.e., of forming a closed loop, from a uniform elastic chain whose intrinsic shape is an open circular arc.

2010

A discrete choice pedestrian behavior model for pedestrian detection in visual tracking systems

Gianluca Antonini, Michel Bierlaire, Jean-Philippe Thiran

Different approaches to the moving object detection in multi-object tracking systems use dynamic-based models. In this paper we propose the use of a discrete choice model (DCM) of pedestrian behavior and its application to the problem of the target detection in the particular case of pedestrian tracking. We analyze real scenarios assuming to have a calibrated monocular camera, allowing a unique correspondence between the image plan and the top view reconstruction of the scene. In our approach we first initialize a large number of hypothetical moving points on the top view plan and we track their corresponding projections on the image plan by means of a simple correlation method. The resulting displacement vectors are then re-projected on the top view and pre-filtered using distance and angular thresholds. The pre-filtered trajectories are the inputs for the discrete choice behavioral filter used to decide whether the pre-filtered targets are real pedestrians or not.

IEEE2004

Pedestrian head detection using automatic scale selection for feature detection and statistical edge curvature analysis

Gianluca Antonini, Jean-Philippe Thiran

In this report we focus on pedestrian head detection and tracking in video sequences. The task is not trivial in real and complex scenarios where the deformation induced by the perspective field requires a multi-scale analy- sis. Multi-scale shape models for the human head are considered to identify the correct size of the region of interest. Anisotropic diffusion is used as a pre-processing step and edge detection is performed using an automatic scale selection process. A non parametric statistical description is given for the edge curvature and detection is performed by means of goodness-of-fit tests. The head detector is used as a validation tool in a correlation-based tracker. The local maxima of the correlation matrix are analyzed. Tracking is performed associating the displacement vector of the target with that local maximum which maximizes the goodness-of-fit with the distribution of the edge curvature of the head.

2004

Looping probabilities of elastic chains: A path integral approach

Ludovica Cotta-Ramusino, John Maddocks

2010