We present our work, "A parametric analysis of discrete Hamiltonian functional maps".
The scope of this paper, as the title suggests,
is to perform an in-depth theoretical study
of Hamiltonian functional maps, which we introduce
here for the first time. This formalism arises
naturally when relating the eigenfunctions
of two distinct discrete differential operators:
the former being the standard Laplace-Beltrami
operator and the latter being the Hamiltonian
operator, its localized version.
Here we show an eigenfunction of the Laplacian
on the left and an eigenfunction of the Hamiltonian
on the right, which is locally supported.
Hamiltonian functional maps allow to build
a correspondence between these two eigenbases
and let us pass easily from one to the other.
Our goal is to characterize formally this
phenomenon, performing a parametric analysis
which enables us to understand how the eigenfunction
on the left transforms to the one on the right,
both from a global and local point of view.
In geometry processing the Laplace Beltrami
operator is ubiquitous.
Its spectral quantities, solution
to the eigenvalue equation, capture intrinsic
geometric information of manifold surfaces.
When dealing with bounded domains, boundary
conditions should be imposed on its eigenfunctions:
In this work we impose Dirichlet conditions
which constrain the eigenfunctions to vanish
on the border of the domain.
The Hamiltonian operator can be obtained from
the Laplacian by adding a potential term,
which is a real valued function defined on
the surface.
In this work we focus on a simple step potential,
composed by an orthogonal projection operator
P and real valued parameter tau.
These two set the potential to zero on a certain
region (in white) and to a finite value tau
on its complement (in red). The value of tau
corresponds to the height of the step and
directly identifies with the potential energy.
When tau tends to infinity we recover the
well-studied model in physics of the infinite
potential barrier.
It is well known in quantum mechanics that
in the continuous case, Hamiltonian eigenfunctions
with eigenvalues less than the potential energy,
vanish on the high potential area, enforcing
localization. We can observe this phenomenon
by looking at the eigenfunctions with lowest
energy.
When tau tends to infinity, Hamiltonian eigenfunctions
will vanish on the boundary of the potential
region. This behavior is equivalent to the
one enforced by Dirichlet boundary conditions
on Laplacian eigenfunctions, justifying our
previous choice.
The same equivalence holds with isometrically
deformed surfaces, due to invariance properties
of the laplacian eigenfunctions.
Our aim is to study the dynamic process depending
on tau, which evolves the laplacian basis
to the Hamiltonian's.
To achieve this, we employ the functional
map formalism. A functional map is a linear
operator between the functional spaces defined
on manifolds. It allows to tackle correspondence
problems by mapping functions between surfaces
instead of considering points.
In our case the Hamiltonian functional map
gives the correspondence between the full
Laplacian eigenbases on the top row and the
Hamiltonian basis on the bottom row, on the
same domain. But what do these Hamiltonian
functional maps look like?
In this picture we show the full discrete
representation of Hamiltonian functional maps
evolving with the parameter tau. We increase
the potential energy from zero to infinity
and observe how the maps are changing.
At the start, when tau = 0, we have an identity
because the Hamiltonian is equal to the Laplacian.
When we start to increment tau smoothly, the
diagonal separates into two components, converging
to two slanted diagonals. The goal of the
paper is to characterize mathematically the
structural evolution of these maps, which
we observed experimentally.
The visualization of this phenomenon, although
interesting, does not provide much information
of the underlying process. We could gather
more knowledge by analyzing the parametric
evolution of the eigenvalues instead.
In this slide we show how eigenvalues evolve
while increasing tau. On the x axis of the
plot we have the values of tau, and each curve
shows the i-th indexed eigenvalue, ordered
increasingly. Here we show the first 20. The
evolution of the eigenvalue curves displays
some interesting regularities: they increase
monotonically and often seem to interact by
repelling one another. We notice that eigenvalue
curves stabilize after tau has reached a certain
value.
On the right we show two eigenfunctions with
different indices, the tenth and the twentieth.
We notice that the lower indexed eigenfunction
on the left converges faster, in contrast
to the higher indexed eigenfunction on the
right. This is related to the fact that their
eigenvalue curves stabilize at different values
of tau. It is clear that before convergence
the eigenfunctions transform in a complex
way. We will discuss in detail the nature
of these transformations in our analysis.
Our theoretical analysis will investigate
the dynamics of these eigenvalue curves and
that of the associated eigenfunctions in the
discrete setting. We have divided the spectral
analysis in two main steps.
The first is an asymptotic analysis that will
tell us how the spectral quantities behave
when tau tends to infinity.
The second analysis is local in nature and
explains the interactions between eigenfunctions
when eigenvalue curves come close to each
other.
Local interactions then will be quantified
according to a metric called: veering index.
Finally we combine asymptotic and local information
to give a mathematical description of the
Hamiltonian maps.
In order to perform our analysis we will use
perturbation theory, a branch of quantum mechanics
that studies perturbation of differential
operators. The tools that we require are specified
by
three equations which describe the rate of
change of spectral quantities, with respect
to the parameter tau. We need first and second
order eigenvalue derivatives and the first
derivative of the eigenfunctions. We notice
that each of the three equations contains
the inner product between pairs of eigenfunctions,
weighted by the mass matrix A and the projection
matrix P of the Hamiltonian.
This quantity is called modal coupling and
measures the intensity of a spectral interaction.
With the equations presented in the last slides
we have proven the first theorem in our paper,
which characterizes the behavior of eigenvalues
and eigenfunctions when tau tends to infinity.
We indicate with k_0 the r vertices on the
zero potential region
and with k_1 the n - r vertices on the high
potential region.
The thesis can be divided into two parts:
the former is related to the smallest r eigenvalues.
We proved that their limit is finite and bounded
by the lowest and greatest initial Laplacian
eigenvalues, mu_1 and mu_n; the limit of the
derivative of these eigenvalues is zero and
the components on the high potential value
of the lowest eigenfunction is zero.
More interestingly, the other n-r eigenvalues
will tend to infinity, their derivative tends
to 1 and the associated eigenfunctions are
equal to zero on the low potential support.
Let's look at what this result means in practice.
First let's concentrate on the eigenvalues
behavior, visualizing a simplified real example.
We consider a tetrahedron and put the high
potential on two vertices.
We see that two eigenvalues converge to finite
values with derivative equal to zero while
the other two diverge, with derivative equal
to one.
The most important aspect of the theorem however
relates to eigenfunctions.
Here we consider a Laplacian eigenfunction
of a given index, and observe that, when tau
is large enough, it will be decomposed in
two components:
one on the low potential region, which corresponds
to the body of the elephant on top,
while the other on the high potential region
on the head.
This shows that the continuous domain quantum
mechanical results are not valid in the discrete
settings anymore: there exist eigenvalues
with value less than tau, whose eigenfunctions
do not localize on the region of low potential
but instead on the complementary high potential
region.
Now let us look at local interactions between
spectral quantities. Principally we want to
study the behavior of two eigenvalue curves
when they get close enough.
To do this, we use the second derivative equation
presented previously.
We see that the denominators in each sum are
high for indices different from j and i. Thus,
the contributions in the sum given by eigenvalues
different from i and j are negligible.
We can perform the following approximation
considering only the contributions of lambda_i
and lambda_j.
We notice that these expressions have the
same absolute value and inverse sign. If the
curves come very close, the denominator component
gets smaller and the second derivatives increment,
meaning that they bend and repel each other.
This interaction is called a veering interaction.
Two curves may cross only if the modal coupling
in the numerator is zero, because in that
case the second derivative would be null.
Now let us discuss how the eigenfunctions
associated to two veering eigenvalues interact.
Using the perturbative equations it is not
difficult to see that when eigenvalue curves
veer, their associated eigenvectors rotate
on a plane. We can observe this phenomenon
in the left figure. When eigenvalues begin
to veer, their eigenfunctions in blue change
with the derivatives in red. At the end of
the veering interaction as we can see, the
transformed eigenfunctions have swapped their
eigenspaces. In an ideal veering case when
the second eigenvalue derivative becomes an
equality, a perfect exchange takes place.
In general however only approximate rotations
occur: the interacting eigenvalues will be
associated with linear combinations of the
initial eigenfunctions.
To spot where eigenvalue curves veer we can
use a specific metric denoted as veering index,
introduced in the structural mechanics literature.
This metric takes into account two factors:
the cross sensitivity quotient, that measures
the intensity of the modal coupling and
the modal dependency factor, that measures
the degree of approximation made on the second
derivative.
We show an example of the veering index corresponding
to two eigenvalues interacting on the shape
of a bunny. The veering index peaks nearly
at one, its maximum, indicating an approximate
exchange of eigenfunctions. This metric can
be exploited to track automatically eigenfunctions
along perturbations of tau, following the
path with stronger interactions.
Exploiting the asymptotic and local analysis
performed, we are now able to present our
main theoretical result, namely the parametric
characterization of the Hamiltonian functional
map structure.
Recalling that eigenvalue curves have only
horizontal and oblique asymptotes we make
the fundamental assumption that all veering
interactions are perfect, that is the veering
index peaks always at 1.
This case is ideal but simplifies the analysis
enough to make it tractable. With this assumption
in mind let us first look at the following
toy example: on top we see the interactions
of 4 eigenvalue curves. The asymptotes are
drawn in black, while the underlying eigenvalue
curves are illustrated by the dashed blue
lines. Let us focus on the first asymptote
in red. The starting eigenfunction is phi_1
and is exchanged twice on the eigenvalue curves
snapping to the red asymptote. Underneath
we observe the corresponding functional map
that changes accordingly.
In a certain sense an eigenfunction flows
along an asymptote
We can now illustrate the general case.
We start by depicting the map as a matrix
C and consider the first Laplacian eigenfunctions
that have horizontal asymptotes. Since there
are no crossings they map in themselves, and
the corresponding entries in the map will
stay diagonal.
Now we consider the last Laplacian eigenfunctions
having oblique asymptotes. As for the former
case no intersections occur between them so
the corresponding entries in the map will
be diagonal as well.
At this point we take into account the first
eigenfunction with an oblique asymptote, depicted
in blue in the central section.
This eigenfunction must map in the lowest
indexed Hamiltonian eigenfunction on the high
potential area psi_r+1. Its entry therefore
will be distributed on the right half of the
map.
Symmetrically we look at the last eigenfunction
with an horizontal asymptote, depicted in
red in the central section.
This eigenfunction maps in the highest indexed
Hamiltonian eigenfunction on the low potential
area psi_r. Its entry will be distributed
on the left half of the map.
Based on the arrangement of oblique asymptotes,
some Laplacian eigenfunctions will map on
higher indexed Hamiltonian values on the high
potential area (the C2 component), and their
entries will slide to the right.
Conversely, given the arrangement of horizontal
asymptotes, Laplacian eigenfunctions will
map on lower indexed Hamiltonian values on
the low potential area (the C1 component),
with their entries sliding to the left.
Each row of C1 and C2 contains a column match
but not both, since an initial diagonal entry
slides on the left or on the right depending
on the asymptote.
We recall that the structure that we have
presented is an ideal construction. However
we can generalize to the real case by the
following decomposition of the Hamiltonian.
The Ld matrix is the diagonal of the Laplacian
while Lo the off-diagonal part. Since Ld +
tau P is diagonal, the eigenvalue curves coincide
with the asymptotes. It follows that its spectral
dynamics is ideal as described precedingly.
Applying Weyl's inequality we can bound the
real eigenvalues to the ideal ones using the
lowest and greatest eigenvalues of P_o. This
perturbation gives rise to non ideal veering
interactions, which cause off-diagonal dispersion
in the functional map, as we see in the corresponding
figures.
To characterize the practical usefulness of
Hamiltonian functional maps, we present an
application to solve the partial shape matching
problem.
Given a source shape M and a target shape
N, which we assume to be a nearly isometric
deformation of a subset of M,
our aim is to find a correspondence between
the two shapes.
We do this by exploiting the Hamiltonian functional
maps formalism in a fully spectral and automatic
pipeline.
We split the correspondence problem into two
subproblems, often seen interchangeable in
the past literature, but which we find complementary:
the former is to find an Hamiltonian functional
map C between the Hamiltonian eigenbasis on
M and the Laplacian eigenbasis on N;
while, the latter is a region localization
problem, where we seek the region v on M which
corresponds to an isometric deformation of
N.
Our approach is iterative:
First, we estimate a map C between the two
eigenbases, by exploiting a set of arbitrary
descriptors in the spectral domain in addition
with a set of regularizers on C to better
attain a local minimum.
Then we guess the region v corresponding to
N localized on M, by transferring an indicator
function in the spectral domain, exploiting
the current estimate of the map.
We can optimize for v starting with this guess,
by aligning the spectra of the Hamiltonian
and Laplacian, using a method previously explored
in the literature.
As a last step, we recompute the Hamiltonian
eigenbasis according to the new potential
region, obtained as the complement of the
current estimate for the region v on M.
We can repeat these steps, until we converge
to a satisfactory solution.
Our pipeline, leveraging on the relationship
between the Laplacian and Hamiltonian eigenbasis,
places itself at the state of the art from
a quantitative perspective.
Our approach offers qualitatively better results
than previous ones, encouraging solutions
which correspond to contiguous regions, and,
therefore, attaining a better performance
in the case of symmetric cases, which are
notoriously more difficult to handle.
In this work we performed a parametric analysis
of the Hamiltonian functional map formalism.
In future developments other than finding
more applications which could benefit from
the theory developed, our analysis could be
in principle extended to handle more complex
eigensystems, such as Hamiltonians with potentials
depending on more than one parameter.
We show two preliminary examples of this:
the former is a step potential Hamiltonian
where we study the variation of the potential
region.
The latter takes a more general perspective
considering geometric deformation flows, where
one must also take into account the change
of domain across time. Here we show experimental
evidence of how spectral quantities change
under the action of conformalized mean curvature
flow.
