poses manifold angles have inherent periodicity, which is not compatible with vector space structure. marisasa mae

http://image.diku.dk/hauberg/sommeretal10.pdf

https://www.researchgate.net/post/What_is_the_idea_behind_the_definition_of_an_exponential_map_on_a_manifold2

https://www.researchgate.net/profile/Stefan_Sommer

marissa mae


Just a remark concerning the motivation and the terminology: If your manifold happens to be a Lie-group, or more specifically, a matrix group M such that its elements A can be represented in the form exp(Bt), then the Lie-algebra, that is the tangent space of M at the identity I, is the vector space of such matrices B. The "exponential map at I" in the sense of differential geometry is, in this case, the matrix exponential map. Just to consider the simple case: O(2)=S^1: In this case the tangent space of O(2) at I is the set of skew-symmetric matrices \begin{pmatrix} 0 & -s \\ s &0 \end{pmatrix}, which can be identified with the imaginary numbers \{ ix | x\in\R\}. The matrix exponential
T_I O(2) \to O(2) can be written as ix \mapsto e^{ix}, which is exactly the map that corresponds to "following the circle with speed x for time 1."


4.3 Human Poses In this experiment, we consider human poses obtained using tracking software. A consumer stereo camera4 is placed in front of a test person, and the tracking software described in [10] is invoked in order to track the pose of the persons upper body. The recorded poses are represented by the human body end-effectors; the end-points of each bone of the skeleton. The placement of each end-effector is given spatial coordinates so that an entire pose with k end-effectors can be considered a point in R 3k . To simplify the representation, only the end-effectors of a subset of the skeleton are included, and, when two bones meet at a joint, their end-points are considered one end-effector. Figure 5 shows a human pose with 11 end-effectors marked by thick dots. −1 −1.5 0 −0.5 1 0.5 2 1.5 2.5 −0.5 0 0.5 0 0.5 1 1.5 2 2.5 Fig. 5. Camera output superimposed with tracking result (left) and a tracked pose with 11 end-effectors marked by thick dots (right). The fact that bones do not change length in short time spans gives rise to a constraint for each bone; the distance between the pair of end-effectors must be constant. We incorporate this into a pose model with b bones by restricting the allowed poses to the preimage F −1 (0) of the map F : R 3k → R b given by F i (x) = kei1 − ei2 k 2 − l 2 i , (12) where ei1 and ei2 denote the spatial coordinates of the end-effectors and li the constant length of the ith bone. In this way, the set of allowed poses constitute a 3k − b-dimensional implicitly represented manifold. We record 26 poses using the tracking setup, and, amongst those, we make 20 random choices of 8 poses and perform linearized PGA and exact PGA. For each experiment, τSvˆ , ˜τSvˆ , ρ, and σ are computed and plotted in Figure 6. The

Computer Vision -- ECCV 2010: 11th European Conference on ...

https://books.google.com/books?isbn=3642155677

Kostas Daniilidis, ‎Petros Maragos, ‎Nikos Paragios - 2010 - ‎Computers

Vectors in the tangent space are often mapped back to the manifold using the exponential map, Expp, which maps straight lines trough the origin of TpM to ...

Manifolds and Manifold Valued Statistics The interest in manifolds as modeling tools arises from the non-linearity apparent in a variety of problems. We will in the following exemplify this by considering the pose of a human skeleton captured by e.g. a tracking system or motion capture equipment. Consider the position of a moving hand while the elbow and the rest of the body stay fixed. The hand cannot move freely as the length of the lower arm restricts it movement. Linear vector space structure is not present; if we multiply the position of the hand by a scalar, the length of the arm would in general change in order to accommodate the new hand position. Even switching to an angular representation of the pose of the elbow joint will not help; angles


Manifolds and Manifold Valued Statistics The interest in manifolds as modeling tools arises from the non-linearity apparent in a variety of problems. We will in the following exemplify this by considering the pose of a human skeleton captured by e.g. a tracking system or motion capture equipment. Consider the position of a moving hand while the elbow and the rest of the body stay fixed. The hand cannot move freely as the length of the lower arm restricts it movement. Linear vector space structure is not present; if we multiply the position of the hand by a scalar, the length of the arm would in general change in order to accommodate the new hand position. Even switching to an angular representation of the pose of the elbow joint will not help; angles have inherent periodicity, which is not compatible with vector space structure. Though the space of possible hand positions is not linear, it has the structure of a manifold since it possesses the property that it locally can be approximated 4 Sommer, Lauze, Hauberg, Nielsen by a vector space. Furthermore, we can, in a natural way, equip it with a Riemannian metric [14], which allows us to make precise notions of length of curves on the space and intrinsic acceleration. This in turns defines the Riemannian manifold equivalent of straight lines: geodesics. The length of geodesics connecting points defines a distance metric on the manifold.

What is the idea behind the definition of an exponential map on a manifold?

Given a Riemannian Manifold, the exponential map at a point is a function acting on a vector in the tangent space at that point defined using a constant speed geodesic originating at that point with initial velocity as the tangent vector on which it is acting. What is this definition actually serving? What is the motivation behind the definition?

TOPICS

• Riemannian Geometry
• Differential Geometry

Sep 26, 2013

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POPULAR ANSWERS

• 
  Rogier Brussee · Hogeschool Utrecht

  It models traveling with fixed velocity (in size and direction) for a unit period of time along great circles on earth (i.e as the bird flies). Equivalently, it models traveling in a fixed direction, for a certain period of time along a great circles. The exponential map is of great use in Riemannian geometry because it relates in a bijective way a neighborhood of a point to tangent data in this point. For example it gives a map of the earth centred in Amsterdam that describes Bangalore as flying (with 1000 km/hour) for 6 hours in south eastern direction, London as flying 15 minutes south west west, Rome one hour south south east, New York 6 hours west etc. This description breaks down for New Zealand because it is antipodal to Amsterdam: Wellington is both 20 hours south east and 20 hours north west.

   Sep 26, 2013

ALL ANSWERS (6)

• 
  Rogier Brussee · Hogeschool Utrecht

  It models traveling with fixed velocity (in size and direction) for a unit period of time along great circles on earth (i.e as the bird flies). Equivalently, it models traveling in a fixed direction, for a certain period of time along a great circles. The exponential map is of great use in Riemannian geometry because it relates in a bijective way a neighborhood of a point to tangent data in this point. For example it gives a map of the earth centred in Amsterdam that describes Bangalore as flying (with 1000 km/hour) for 6 hours in south eastern direction, London as flying 15 minutes south west west, Rome one hour south south east, New York 6 hours west etc. This description breaks down for New Zealand because it is antipodal to Amsterdam: Wellington is both 20 hours south east and 20 hours north west.

   Sep 26, 2013
• 
  Robert J. Low · Coventry University

  To expand on Rogier's answer, the exponential map lets you use the tangent space at a point as coordinates in a neighbourhood of the point in a nice way. In particular, curves passing through the point which are straight lines in this coordinate system are geodesics. (Which in the case he describes are the great circles.) It also gives you a helpful handle on some geometric quantities, for example the sectional curvature.

   Sep 27, 2013
• 
  Stefan Born · Technische Universität Berlin

  Just a remark concerning the motivation and the terminology: If your manifold happens to be a Lie-group, or more specifically, a matrix group M such that its elements A can be represented in the form exp(Bt), then the Lie-algebra, that is the tangent space of M at the identity I, is the vector space of such matrices B. The "exponential map at I" in the sense of differential geometry is, in this case, the matrix exponential map. Just to consider the simple case: O(2)=S^1: In this case the tangent space of O(2) at I is the set of skew-symmetric matrices \begin{pmatrix} 0 & -s \\ s &0 \end{pmatrix}, which can be identified with the imaginary numbers \{ ix | x\in\R\}. The matrix exponential
  T_I O(2) \to O(2) can be written as ix \mapsto e^{ix}, which is exactly the map that corresponds to "following the circle with speed x for time 1."

   Oct 17, 2013
• 
  Richard L. Bishop · University of Illinois, Urbana-Champaign

  A particularly simple example, a special case of the Lie group, has been used "forever": the usual real or complex exponential function from the field to the nonzero elements. For the matrix Lie groups the usual exponential function series or differential equation is satisfied and is the Riemannian metric exponential for the nicest invariant metrics.

   Oct 23, 2013
• 
  Maung Min-Oo · McMaster University

  As Professor Bishop pointed out the name is a relic from matrix exponentiation used in Lie group theory but let me point out some basic facts from Riemannian geometry.
  The exponential map defines a special coordinate system near a point (called normal coordinates) and if you expand the metric tensor in those coordinates the first order terms (Christoffel symbols) vanish and the second order terms give you the curvature tensor on the nose. Besides, the second variation formula for geodesics (Jacobi fields) controls the derivative of the exponential map and that can be used to prove comparison theorems in Riemannian geometry. In fact, understanding the way geodesics are "focusing" or "expanding" is also important in General Relativity.