Random warping functions for curve alignment
Karthik Bharath
University of Nottingham
Curve alignment on
[0, 1]
f1 and f2
f1 and f2 ◦ γ
warping function
Warping or Reparameterization: speed of traversal along curve
Interplay between two dierent notions of shape
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Domain and range are the same.
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Increasing.
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Orientation should be preserved.
General setting for pairwise registration problem
I
Registration/alignment of two functions or parametric curves
fi : D → Rk , i = 1, 2, where D is a compact set, can be achieved by
computing an `optimal' warping function that is:
arg min d(f1 (t), f2 (γ(t))),
γ∈Γ
General setting for pairwise registration problem
I
Registration/alignment of two functions or parametric curves
fi : D → Rk , i = 1, 2, where D is a compact set, can be achieved by
computing an `optimal' warping function that is:
arg min d(f1 (t), f2 (γ(t))),
γ∈Γ
or
arg max likelihood on {f1 (t) − f2 (γ(t))},
γ∈Γ
General setting for pairwise registration problem
I
Registration/alignment of two functions or parametric curves
fi : D → Rk , i = 1, 2, where D is a compact set, can be achieved by
computing an `optimal' warping function that is:
arg min d(f1 (t), f2 (γ(t))),
γ∈Γ
or
arg max likelihood on {f1 (t) − f2 (γ(t))},
γ∈Γ
or is based on the posterior from a model such as
likelihood on {f1 (t) − f2 (γ(t))} and prior distribution on classΓ,
General setting for pairwise registration problem
I
Registration/alignment of two functions or parametric curves
fi : D → Rk , i = 1, 2, where D is a compact set, can be achieved by
computing an `optimal' warping function that is:
arg min d(f1 (t), f2 (γ(t))),
γ∈Γ
or
arg max likelihood on {f1 (t) − f2 (γ(t))},
γ∈Γ
or is based on the posterior from a model such as
likelihood on {f1 (t) − f2 (γ(t))} and prior distribution on classΓ,
where Γ := {γ : D → D} is usually a group of dieomorphisms or
homeomorphisms that preserve orientation and are monotone-increasing.
In this talk...
0
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𝑡" 𝑡#
1
If on f1 a landmark is present at t1 ∈ [0, 1] and correspondingly at t2 on f2 ,
then there is a warping constraint: γ(t2 ) = t1 .(Non-dierentiability?
Jump?)
In this talk...
0
I
𝑡" 𝑡#
1
If on f1 a landmark is present at t1 ∈ [0, 1] and correspondingly at t2 on f2 ,
then there is a warping constraint: γ(t2 ) = t1 .(Non-dierentiability?
Jump?)
I
Construct probability distribution on class of warping functions Γ that can
handle such constraints.
Desiderata of the distribution
Pθ
Pθ(b−a)
Stochastic process with sample paths that are:
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are non-decreasing (subordinator);
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possess similar localized behavior (self-similar);
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possess some sort of Markov property;
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are continuous/dierentiable?
Desiderata of the distribution
Pθ
Pθ(b−a)
Stochastic process with sample paths that are:
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are non-decreasing (subordinator);
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possess similar localized behavior (self-similar);
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possess some sort of Markov property;
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are continuous/dierentiable?
A case for the Dirichlet process.
ECG data example: Constructed Dirichlet process as prior on
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Γ
Data: Cycles fi : [0, 1] → R+ , i = 1, 2. Square Root Velocity (SRV)
transform q := sign(f 0 ) |f 0 |.
p
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Model: γ ∼ Dirichlet process centered at γ(t) = t ;
{q1 − (q2 , γ)}|γ ∼ zero-mean Gaussian process with known covariance.
ECG data example: Dirichlet process as prior on
Γ
(green is posterior mean)
Construction of the distribution
In practice: dene a distribution on the increments, increase sampling. (Bayesian
registration of functions and curves.
W. Cheng, I. L. Dryden and X. Huang. Bayesian Analysis,
)
11, 447-475, 2015
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Since
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Suitably transform the coordinates pi such that as n → ∞, we obtain a
Pn
i=1
pi = 1, consider a distribution on the simplex ∆n .
sensible limiting distribution on ∆∞ .
`Uniform' distribution on warping functions
Randomness in:
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increments pi (jumps of the process);
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partition 0 = t0 < t1 < · · · < tn < tn+1 = 1 (times of jumps).
`Uniform' distribution on
Γ
centered at
γ(t) = t
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Let Γ := {γ : [0, 1] → [0, 1], right continuous, non-decreasing}.
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Partition: 0 = t0 < t1 < · · · < tn < tn+1 = 1, let pi = γ(ti ) − γ(ti−1 ) of
xed length 1/n (discrete uniform).
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(p1 , . . . , pn ) uniform on the simplex ∆n .
`Uniform' distribution on
Γ
centered at
γ(t) = t
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Let Γ := {γ : [0, 1] → [0, 1], right continuous, non-decreasing}.
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Partition: 0 = t0 < t1 < · · · < tn < tn+1 = 1, let pi = γ(ti ) − γ(ti−1 ) of
xed length 1/n (discrete uniform).
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(p1 , . . . , pn ) uniform on the simplex ∆n .
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The simplex can be coordinatized by yi = p1 + · · · + pi , with
0 = y0 ≤ y1 ≤ y2 ≤ · · · ≤ yn ≤ 1.
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−1
For θ > 0, let νn,θ (y ) = P(Gamma( θn , 1) > y ); let ξi := νn,θ
(yi ) .
`Uniform' distribution on
Γ
centered at
γ(t) = t
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Let Γ := {γ : [0, 1] → [0, 1], right continuous, non-decreasing}.
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Partition: 0 = t0 < t1 < · · · < tn < tn+1 = 1, let pi = γ(ti ) − γ(ti−1 ) of
xed length 1/n (discrete uniform).
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(p1 , . . . , pn ) uniform on the simplex ∆n .
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The simplex can be coordinatized by yi = p1 + · · · + pi , with
0 = y0 ≤ y1 ≤ y2 ≤ · · · ≤ yn ≤ 1.
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−1
For θ > 0, let νn,θ (y ) = P(Gamma( θn , 1) > y ); let ξi := νn,θ
(yi ) .
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On [0, 1] × R+ dene the point process/random measure
Nn :=
n
X
i=1
δ( i ,ξi ) .
n
`Uniform' distribution on
Γ
centered at
γ(t) = t
Theorem
I As
n → ∞ , Nn
converges weakly (vague) to a Poisson point process
with intensity measure
θe −y
dt
×
dy .
|{z}
y
uniform partition | {z }
jump rate
N
`Uniform' distribution on
Γ
centered at
γ(t) = t
Theorem
I As
n → ∞ , Nn
converges weakly (vague) to a Poisson point process
N
with intensity measure
θe −y
dt
×
dy .
|{z}
y
uniform partition | {z }
jump rate
I If
N = {(ηx , ηy ) ∈ [0, 1] × R+ },
Gamma process, and
Dθ (t) :=
then for
t ∈ [0, 1], G(t) :=
ηx ≤t
G(θt)
is a Dirichlet process with
G(θ)
E Dθ (t) = t.
|
{z
}
Centred at identity warping
.
P
ηy ,
is a
Comments on theorem and non-uniform distribution
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To centre the process at non-identity warping, partition using a
non-uniform distribution on [0, 1].
Comments on theorem and non-uniform distribution
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To centre the process at non-identity warping, partition using a
non-uniform distribution on [0, 1].
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Proof relies on limit theory for spacings of uniform random variables, since:
d
pi = Ui:n − Ui−1:n ,
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d
and npi −→
Exp(1).
Can consider the Xn (t) =
Pdnte
i=1
npi as a random function in D([0, 1]), but
tightness is hard to prove.
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Convergence of point process is proved in total variation.
`Non-uniform' distribution on
Γ
Suppose the vector of increments X1 , . . . , Xn are from a distribution F with
density f on [0, 1].
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Then, pi can be viewed as spacings:
pi = Xi:n −Xi−1:n = F −1 (Ui:n )−F −1 (Ui−1:n ) =
d
d
1
f (F −1 (αi,n ))
(Ui:n −Ui−1:n ).
`Non-uniform' distribution on
Γ
Suppose the vector of increments X1 , . . . , Xn are from a distribution F with
density f on [0, 1].
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Then, pi can be viewed as spacings:
pi = Xi:n −Xi−1:n = F −1 (Ui:n )−F −1 (Ui−1:n ) =
d
d
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1
f (F −1 (αi,n ))
With some smoothness conditions on f we can ensure that
#
f F −1 (i/n)
max
− 1 = Op (1).
i
f (F −1 (αi,n ))
"
(Ui:n −Ui−1:n ).
Γ
`Non-uniform' distribution on
Suppose the vector of increments X1 , . . . , Xn are from a distribution F with
density f on [0, 1].
I
Then, pi can be viewed as spacings:
pi = Xi:n −Xi−1:n = F −1 (Ui:n )−F −1 (Ui−1:n ) =
d
d
I
1
f (F −1 (αi,n ))
(Ui:n −Ui−1:n ).
With some smoothness conditions on f we can ensure that
#
f F −1 (i/n)
max
− 1 = Op (1).
i
f (F −1 (αi,n ))
"
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With pi∗ = f F −1 (i/n) pi , again:
Theorem
As
n → ∞, Nn
converges weakly to the same Poisson point process
N.
Properties of
Pθ
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EPθ (γ(t)) = t (Regularization).
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as θ → 0, Pθ converges to uniform distribution on the set
{I[t,1] : 0 ≤ t ≤ 1}.
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as θ → ∞, Pθ converges to point mass at identity map.
Properties of
Pθ
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EPθ (γ(t)) = t (Regularization).
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as θ → 0, Pθ converges to uniform distribution on the set
{I[t,1] : 0 ≤ t ≤ 1}.
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as θ → ∞, Pθ converges to point mass at identity map.
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Pθ has full support on Γ for every θ (several topologies coincide).
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Pθ is compatible with composition (quasi-invariance).
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Restricting Pθ to [a, b] ⊂ [0, 1] just re-scales the parameter to θ|b − a|
(self-similarity), and is independent of [0, 1] − [a, b] except at a and b
(`Markov').
Behaviour of large jumps
The map t 7→ γ(t) is increasing, but only in jumps with jump locations dense
in [0, 1]. (How big are the jumps? How often do they occur?)
Behaviour of large jumps
The map t 7→ γ(t) is increasing, but only in jumps with jump locations dense
in [0, 1]. (How big are the jumps? How often do they occur?) Set
ζi = nf (F −1 (i/n))pi − log n and dene the point process Nn :=
Proposition
Nn
converges weakly to
N
with mean measure
dt × θ
R
e −y dy .
Pn
k=1
δ{ k ,ζk }
n
Dirichlet process for warpings on
S1
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Consider the quotient group S1 = R/Z = [0, 1]/ ∼ with addition.
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Map [0, 1] to S1 by considering φ : R → S1 with φ(t) = t(mod1).
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n
o
Γ̃ := γ̃ : R → R, increasing and satisfying γ̃(t + 1) = γ̃(t) + 1, t ∈ R .
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This induces an orientation preserving and increasing map on S1 :
γ(t) := φ(γ̃)(t) = γ̃(t)(mod1).
Dirichlet process for warpings on
S1
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Consider the quotient group S1 = R/Z = [0, 1]/ ∼ with addition.
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Map [0, 1] to S1 by considering φ : R → S1 with φ(t) = t(mod1).
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n
o
Γ̃ := γ̃ : R → R, increasing and satisfying γ̃(t + 1) = γ̃(t) + 1, t ∈ R .
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This induces an orientation preserving and increasing map on S1 :
γ(t) := φ(γ̃)(t) = γ̃(t)(mod1).
As n → ∞, an almost identical
measure (rotation invariant) can be
constructed for γ based on maximal
gaps and point processes on the circle.