Stochastic Systems Group
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
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Tangents and Normals
• The derivative of f1 in the direction of f at q is:
• Implies df1 is surjective
• If f is orthogonal to {1, sin q, cos q}, then df1=0 in the
direction of f and hence f is in the tangent space
Stochastic Systems Group
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
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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(q+dq)=b1. Can do first order
approximation to compute this
• Approximate Jacobian as:
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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
Stochastic Systems Group
Example Projections
Fig. 1: Projections of arbitrary curves into C1
Stochastic Systems Group
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
Stochastic Systems Group
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 q+fDt and projecting that back
onto the manifold (Dt is step size)
• So q(t+Dt) = P(q(t)+f(t)Dt)
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Transporting the tangent vector
• Now f(t) is not in the tangent space of q(t+Dt)
• 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(t+1)||=||f(t)||
• hk is the orthonormal basis of the normal space
Stochastic Systems Group
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, it’s 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
Stochastic Systems Group
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 Y(q1,0,f)=q1
as the initial condition
• We then want Y(q1,1,f)=q2
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Finding the geodesic
• Define an error functional which measures
how close we are to the target at t=1:
• Choose the geodesic as the flow Y which
has the smallest initial velocity ||f||
• i.e., min ||f|| s.t. H[f]=0
• Hard because infinite dimensional search
Stochastic Systems Group
Fourier decomposition
• f  L2, so it has a Fourier decomposition
• Approximate f with its first m+1 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. H[a]=0
Stochastic Systems Group
Geodesic paths
Fig. 2: Geodesic paths between two shapes