Mobile sampling of bandlimited fields
Karlheinz Gröchenig∗ , José Luis Romero∗ , Jayakrishnan Unnikrishnan† and Martin Vetterli‡
∗
Faculty of Mathematics, University of Vienna,
Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
† General Electric Global Research,
1 Research Circle, Niskayuna, NY 12309, USA
‡ Audiovisual Communications Laboratory,
School of Computer and Communication Sciences,
Ecole Polytechnique Fédérale de Lausanne (EPFL),
EPFL-IC-LCAV, Station 14, CH-1015 Lausanne, Switzerland
Abstract—We investigate a new method for the acquisition of
bandlimited functions, which we call mobile sampling. A field is
sampled by a mobile sensor that moves along a continuous path.
In this context, it is possible to increase the spatial sampling rate
along the sensor’s path with marginal additional cost. Hence we
assume that the sensors acquire the field values on its path at an
arbitrarily high resolution. While in classical sampling theory the
relevant performance metric is the average number of samples
per unit area, in mobile sampling the main acquisition cost comes
from the total distance that needs to be traveled by the moving
sensor. We consider a performance metric tailored to mobile
acquisition and study the mobile sampling problem with respect
to this metric. The introduction of this metric completely changes
the mathematical nature of the sampling problem. We present
in a brief and simplified form recent results that show to which
extent the classical Nyquist sampling theory admits a parallel in
the context of mobile acquisition. We also provide a brief account
of the main technical tools.
I. T HE MOBILE SAMPLING PROBLEM
A. Sampling on curves
Let us use the following normalization for the Fourier
transform
Z
fb(ω) =
f (x)e−2πihω,xi dx,
ω ∈ Rd .
Rd
For a compact set Ω ⊆ Rd , let
n
o
BΩ := f ∈ L2 (Rd ) : supp(fˆ) ⊆ Ω .
We consider the following sampling problem.
Given a family
of rectifiable curves P ≡ pi : R → Rd , we want to recover
a d-dimensional bandlimited field f ∈ BΩ from the samples
{f (pi (t)) : t ∈ R, i ∈ I}.
B. Performance metrics
In traditional sampling theory a bandlimited signal is measured by evaluation on a discrete set Λ ⊆ Rd . The set
Λ is a stable sampling set for BΩ if there exist constants
A, B ∈ (0, +∞) such that the following frame inequalities
hold:
X
Akf k2 ≤
|f (λ)|2 ≤ Bkf k2 , for all f ∈ BΩ
(1)
λ∈Λ
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
If Λ is a sampling set for BΩ , every function f ∈ BΩ can
be stably reconstructed from its samples {f (λ) : λ ∈ Λ}. The
usual performance metric is Beurling’s lower density [2], [3]:
D− (Λ) := lim inf inf
a→∞ x∈Rd
#(Λ ∩ Bad (x))
,
|Bad |
(2)
where Bad (x) denotes the d-dimensional Euclidean ball. In
practical terms D− (Λ) measures the average number of sensors per unit area (or volume) that are used to acquire f .
Perfect acquisition is possible only if D− (Λ) > |Ω| [8],
[9], [4], [12]. Moreover, in several concrete situations this
condition is also sufficient. For example, this is the case if Λ
is a lattice and Ω is the fundamental domain of the so-called
dual lattice.
In contrast, in mobile sampling, the relevant performance
metric is the average length covered by a collection of travelling sensors, instead of the average number of samples. To
formalize this observation, in [17], [18],
some of us introduced
the notion of path-density. If P ≡ pi : R → Rd , i ∈ I is a
set of (rectifiable) trajectories, we let MP (a, x) denote the
total length of the curves pi within the ball Bad (x). The upper
and lower path-densities of P are defined as
MP (a, x)
,
a→∞ x∈Rd
|Bad |
MP (a, x)
,
`+ (P ) := lim sup sup
|Bad |
a→∞ x∈Rd
`− (P ) := lim inf inf
(3)
(4)
see Figure 1. These metrics are directly relevant in applications
like environmental monitoring using moving sensors [14] and
for the design of trajectories for Magnetic Resonance Imaging
(MRI) [1], where the path density can be used as a proxy for
the total scanning time per unit area in space.
C. Admissible trajectories
A trajectory set P ≡ pi : R → Rd , i ∈ I is called a stable
Nyquist trajectory set for BΩ - denoted P ∈ NyqΩ - if P
satisfies the following conditions:
(i) [Nyquist] There exists a set of points Λ on the
trajectories in P , Λ ⊂ {pi (t) : t ∈ R, i ∈ I},
such that Λ forms a set of stable sampling for
BΩ .
(a) The Beurling density of a
lattice.
(b) The path density of a trajectory.
Fig. 1. Beurling and path densities.
(a) Classical pointwise sampling: (b) Sampling on parallel lines:
Minimum sampling density ∝ Minimum path density ∝ Volume
Vol(Ω).
of minimum section through the
center of Ω.
Fig. 3. Sampling limits for a convex symmetric spectrum Ω.
A. The unconstrained optimization problem is ill-posed
In [6] we have shown that the full optimization problem (5)
has a trivial solution. More precisely, we have the following
result.
Proposition 2.1 ([6]): Let Ω ⊆ R2 be a compact set. For
every > 0 there exists a stable Nyquist trajectory set P for
the spectrum Ω, such that `+ (P ) < . Thus,
inf
P ∈NyqΩ
Fig. 2. An illustration of a curve (shown in dark) that satisfies the regularity
condition (ii) in the definition of admissible trajectory set.
(ii) [Non-degeneracy] There exists a function δ :
R+ → R+ such that δ(a) = o(ad ) with
the following property: For every x ∈ Rd
and every a 1, there is a rectifiable curve
α : [0, 1] → Rd , (depending on x and a),
such that (i) `(α) = MP (a, x) + δ(a), and
(ii) P ∩ Bad (x) ⊂ α([0, 1]), i.e., the curve α
contains the portion of the trajectory set P that
is located within Bad (x).
`+ (P ) = 0.
Proposition 2.1 says that for every compact set Ω it is
possible to design a stable Nyquist trajectory set for BΩ
with arbitrarily small path density. This answers negatively
a question posed in [18].
We show below that the optimization problem in (5) becomes well-posed when we optimize over smaller classes of
trajectories or when we add additional constraints.
B. Parallel lines
The regularity condition in item (ii) is motivated by the
model of mobile sensors. It states that for all x ∈ Rd a
single sensor moving along a rectifiable curve can cover the
portions of the trajectories in the ball Bad (x) with total length
MP (a, x)+o(ad ). Thus, although there may be a several paths
in P , a single sensor can cover the portions of P inside Bad (x),
without significantly affecting the total distance traveled per
unit area. (See Figure 2 for an illustration.)
We now restrict our attention to trajectories consisting of
parallel lines. In this case, the optimization problem in (5)
can be given a complete solution. Given a compact spectrum
Ω and q ∈ Rd \ {0} we denote by Pq the class of all trajectory
sets that are admissible for Ω and also consist of lines parallel
to q. Also, we denote P := ∪q6=0 Pq .
Theorem 2.2 ([6]): Let Ω ⊂ Rd be a centered symmetric
convex body. Then
inf `− (P ) : P ∈ Pq = |Ω ∩ q ⊥ |,
q ∈ Rd \ {0}. (6)
Here, |Ω ∩ q ⊥ | denotes the d − 1 dimensional measure of the
section of Ω through the hyperplane perpendicular to q. In
particular,
inf `− (P ) : P ∈ P = min |Ω ∩ q ⊥ |.
(7)
q∈Rd \{0}
II. T HE OPTIMIZATION PROBLEM
Given a compact set (spectrum) Ω and a class of admissible
trajectories C we consider the following optimization problem:
Minimize `− (P ), subject to P ∈ C.
(5)
Hence, according to Theorem 2.2, when sampling a bandlimited function with (convex symmetric) spectrum Ω along
parallel lines, the optimal direction is the one that minimizes
the area of the cross section of Ω through the corresponding
hyperplane.
C. Optimization with a constraint on the condition number
We now return to the problem of optimizing the pathdensity over all admissible sampling trajectories that allow
for stable reconstruction. According to Proposition 2.1, this
problem has a trivial solution: there are sampling trajectories
with arbitrarily low path-density. However, a close inspection
into the examples provided in [6] shows that the condition
number provided by these trajectories is also arbitrarily high.
To formalize these observations, given a spectrum Ω and
numbers A, B > 0, let us denote by NyqA,B
the class of
Ω
all stable Nyquist trajectory sets that contain a sampling set
for BΩ with sampling bounds A, B (cf. (1)). The following
result from [6] shows that the optimization of the path-density
within the class NyqA,B
is a meaningful problem.
Ω
Theorem 2.3: Let Ω ⊆ Rd be a compact set with smooth
boundary. Then
inf
P ∈NyqA,B
Ω
`− (P ) ≥ Cd
A|Ω|
Bσ(∂Ω)
P ∈NyqA,B
Ω
`− (P ) ≥
π2
The results presented help identify trajectory sets consisting
of parallel lines that possess minimal path density and permit
stable reconstruction of bandlimited fields from measurements
taken along these trajectories. We also have shown that the
problem of minimizing the path density is ill-posed if we allow
arbitrary trajectory sets that admit stable reconstruction. As a
positive result it was shown that the problem is well-posed if
we restrict the trajectory sets to contain a stable sampling set
with given stability parameters.
This work opens up several possible research directions.
One important pending question is whether we can exactly
determine the minimum of the path-density among all trajectories that provide a certain stability margin. So far, we have
only derived non-trivial bounds. Another interesting question
concerns the optimization of the path-density among trajectory
sets consisting of arbitrary lines (not necessarily parallel).
d−1
,
where Cd is a constant that depends only on d and σ(∂Ω)
denotes the perimeter of Ω, i.e., the d − 1 measure of ∂Ω.
For concrete spectra, all the constants in Theorem 2.3 can
be made explicit. For example, we have shown in [6] that for
Ω = [−1/2, 1/2]2 ,
inf
IV. C ONCLUSIONS
A
√ .
2B
ACKNOWLEDGMENT
K. Gröchenig was partially supported by National Research
Network S106 SISE and by the project P 26273-N25 of
the Austrian Science Fund (FWF). J. L. Romero gratefully
acknowledges support from an individual Marie Curie fellowship, within the 7th. European Community Framework program, under grant PIIF-GA-2012-327063. J. Unnikrishnan and
M. Vetterli were supported by ERC Advanced Investigators
Grant: Sparse Sampling: Theory, Algorithms and Applications
SPARSAM no. 247006.
III. B RIEF COMMENTS ON THE TECHNICAL TOOLS
R EFERENCES
In [6], we study the problem of sampling along parallel lines
by reducing it to a collection of sampling problems in smaller
dimensions. The challenge lies in carrying out this reduction
while maintaining a sampling rate that is close to critical. Here
the geometry of the spectrum Ω comes into play. In order to
efficiently reduce the dimension of the sampling problem, we
resort to the theory of universal sampling as established by
Olevsky and Ulanovsky [13] and by Matei and Meyer [10],
together with tools from convex geometry like the the BrunnMinkowski inequality [5].
In order to prove that the unconstrained optimization of the
path-density is an ill-posed problem, we leverage the fact that
it is possible to construct sampling sets containing arbitrarily
large gaps. Indeed, when the sampling spectrum is an interval,
the success of acquisition is (almost) determined by Beurling’s
lower density (cf. (2)) which depends only on the asymptotic
distribution of points in large cubes.
For the optimization problem with constraints on the condition number, we exploit the key fact that the size of the largest
hole of a sampling set is determined by the condition number
in the corresponding frame reconstruction. This was shown by
Iosevich and Pedersen in a very general setting [7]. Moreover,
for concrete examples we derive explicit and effective bounds
on maximal spectral gaps. These are instrumental to give lower
bounds on the path-density.
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