Geometry of Shape Manifolds

1
Geometry of Shape Manifolds

• Constraints define a manifold embedded in q0 L2
• Move along manifold by moving in tangent space
  and projecting back to manifold
• Tangent space is infinite dimensional, but normal
  space is characterized by three constraints
  defined in f1

2
Tangents and Normals

• The derivative of f1 in the direction of f at ?
  is
• Implies df1 is surjective
• If f is orthogonal to 1, sin q, cos q, then
  df10 in the direction of f and hence f is in the
  tangent space

3
Projections

• Want to find the closest element in C1 to an
  arbitrary q ? q0 L2
• Basic idea move orthogonal to level sets so
  projections under f form a straight line in R3
• For a point b ? R3, we define the level set as
• Let b1(p,0,0). Then its level set is the
  preshape space C1

4
Approximate Projections

• If points are close to C1, then one can use a
  faster method
• Let dq be the normal vector at q for which
  f(qdq)b1. Can do first order approximation to
  compute this
• Approximate Jacobian as

5
Iterative algorithm

• Define the residual (error) vector as
• Then
• where
• Iteratively update q dq? q until the error goes
  to zero
• Call this projection operator P

6
Example Projections
Fig. 1 Projections of arbitrary curves into C1
7
Geodesics

• Definition For a manifold embedded in Euclidean
  space, a geodesic is a constant speed curve whose
  acceleration vector is always perpendicular to
  the manifold
• Define the metric between two shapes as the
  distance along the manifold between the shapes
  with respect to the L2 inner product
• Nice features
• Defined for all closed curves
• Interpolants are closed curves
• Finds geodesics in a local sense, not necessarily
  global

8
Paths from initial conditions

• Assume we have a q in C1 and an f in the tangent
  space
• Approximate geodesic along manifold by moving to
  qfDt and projecting that back onto the manifold
  (Dt is step size)
• So q(tDt) P(q(t)f(t)Dt)

9
Transporting the tangent vector

• Now f(t) is not in the tangent space of q(tDt)
• Two conditions for a geodesic
• The acceleration vector must be perpendicular to
  the manifold simply project f into the next
  tangent space
• The curve must move at constant speed
  renormalize so f(t1)f(t)
• hk is the orthonormal basis of the normal space

10
Geodesics on shape spaces

• S1 is a quotient space of C1 under actions of S1
  by isometries, so finding geodesics in S1
  equivalent to finding geodesics in C1 which are
  orthogonal to S1 orbits
• S1 acting by isometries implies that if a
  geodesic in preshape space is orthogonal to one
  S1 orbit, its orthogonal to all S1 orbits which
  it meets
• So now normal space has one additional component
  spanned by
• The algorithm is the same as detailed earlier
  except with an expanded normal space

11
Geodesics between shapes

• We know how to generate geodesic paths given q
  and f
• Now we want to construct a geodesic path from q1
  to q2
• So we need to find all f that lead from q1 to an
  S1 orbit of q2 in unit time, and then choose the
  one that leads to the shortest path
• Let Y define the geodesic flow, with ?(q1,0,f)q1
  as the initial condition
• We then want Y(q1,1,f)q2

12
Finding the geodesic

• Define an error functional which measures how
  close we are to the target at t1
• Choose the geodesic as the flow Y which has the
  smallest initial velocity f
• i.e., min f s.t. Hf0
• Hard because infinite dimensional search

13
Fourier decomposition

• f ? L2, so it has a Fourier decomposition
• Approximate f with its first m1 cosine
  components and its first m sine components
• Let a be the vector containing all of the Fourier
  coefficients
• Now optimization problem is min a s.t. Ha0

14
Geodesic paths
Fig. 2 Geodesic paths between two shapes