Nonuniform Generalized Sampling

Milana Gataric

Cambridge Centre for AnalysisFaculty of MathematicsUniversity of Cambridge

Cambridge, 11 November 2015

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Problem formulation

Assume f ∈ L2(D) is an unknown function, where

L2(D) =f ∈ L2(Rd ) : supp(f ) ⊆ D

, D ⊆ Rd compact,

and we are given finite data f (ω) : ω ∈ Ω

,

wheref (ω) =

∫Df (x)e−i2πω·x dx , ω ∈ Rd ,

andΩ ⊆ Rd

is a set of nonuniformly-spaced sampling points.

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Problem formulation

The main problem

Under what conditions is it possible to recover f ∈ L2(D) stably & accurately,from the finite Fourier data f (ω) : ω ∈ Ω, in a desired finite-dimensionalreconstruction space

T ⊆ L2(D) ?

How can this be achieved via numerical algorithm?

Applications

MRI, CT, geophysical imaging, electron microscopy, helium atom scattering...

Figure : Different sampling sets Ω

Jittered Radial Spiral

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Main contributions

(i) Guarantees for such reconstruction in terms of conditions on samplingdensity and sampling bandwidth (any d).

Sampling density corresponds to 1/the maximal gap of the sampling points.Sampling bandwidth measures how far the sampling points are stretched in Rd .

(ii) Estimate of the sampling bandwidth K when T consists of wavelets ordifferent types of polynomials (d = 1).

(iii) Stable numerical algorithm to achieve such reconstruction (any d), withefficient implementation when T consists of boundary-correctedDaubechies wavelets (d = 1, 2).

(iv) Understanding of the reduction in the sampling density if set ofmeasurements includes derivatives of f (any d).

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Fourier frames

In order to tackle the main problem, we extract a subproblem...

Subproblem

Under what conditions does an arbitrary set of sampling points Ω give rise to aFourier frame for L2(D)? That is, find conditions that ensure existence ofA,B > 0 such that

A‖f ‖2 ≤∑ω∈Ω

|f (ω)|2 ≤ B‖f ‖2, f ∈ L2(D). (?)

If (?) holds, Ω is called a set of sampling.

If (?) is replaced by

A‖f ‖2 ≤∑ω∈Ω

µω|f (ω)|2 ≤ B‖f ‖2, f ∈ L2(D), (??)

for some weights µω > 0, we have a weighted Fourier frame for L2(D) and Ω iscalled a weighted set of sampling.

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Fourier frames

In order to tackle the main problem, we extract a subproblem...

Subproblem

Under what conditions does an arbitrary set of sampling points Ω give rise to aFourier frame for L2(D)? That is, find conditions that ensure existence ofA,B > 0 such that

A‖f ‖2 ≤∑ω∈Ω

|f (ω)|2 ≤ B‖f ‖2, f ∈ L2(D). (?)

If (?) holds, Ω is called a set of sampling.

If (?) is replaced by

A‖f ‖2 ≤∑ω∈Ω

µω|f (ω)|2 ≤ B‖f ‖2, f ∈ L2(D), (??)

for some weights µω > 0, we have a weighted Fourier frame for L2(D) and Ω iscalled a weighted set of sampling.

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Definition of weights

Typicallyµω = vol(V ∗ω ),

where V ∗ω is the Voronoi region at ω ∈ Ω with respect to the |·|∗-norm, given by

V ∗ω =y ∈ Rd : ∀λ ∈ Ω, λ 6= ω, |ω − y |∗ ≤ |λ− y |∗

.

Figure : Voronoi regions with respect to the Euclidean norm

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The need for weights

number of radial lines = 19, “maximum gap” < 1/4

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The need for weights

increasing the sampling bandwidth...

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The need for weights

increasing the sampling bandwidth...

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The need for weights

increasing the sampling bandwidth...

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The need for weights

increasing the sampling bandwidth...

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The need for weights

increasing the sampling bandwidth...

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The need for weights

number of radial lines = 19, “maximum gap” > 1/4

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The need for weights

number of radial lines = 35, “maximum gap” < 1/4

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Existing results for classical (unweighted) Fourier frames

[Beurling, 1966]: If D be the unit Euclidean ball in Rd and Ω is separated

∃η > 0 ∀ω, λ ∈ Ω (ω 6= λ⇒ |ω − λ| ≥ η) ,

with densityδ := sup

y∈Rdinfω∈Ω|ω − y | < 1/4,

then Ω is a set of sampling. If δ ≥ 1/4, this does not have to hold.

[Benedetto & Wu, 2000]: Extension to D compact, convex and symmetric.

Pros: Sharp density condition δ < 1/4.

Cons: Requires separation of the sampling points.

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Existing results for weighted Fourier frames

[Gröchenig, 1992]: If D = [−1, 1]d and Ω has density

δ = supy∈Rd

infω∈Ω|ω − y | < log 2

2πd,

then Ω is a weighted set of sampling, with weights µω defined as volumes ofVoronoi regions. The frame bounds satisfy

√A ≥ 2− e2πδd > 0,

√B ≤ e2πδd < 2.

Pros: No separation condition. Explicit estimates for the frame bounds.

Cons: Sampling density increases with the dimension d and not sharp.

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Result 1

Question

Can Ω cluster arbitrarily while having the sharp density condition δ(Ω) < 1/4?

Theorem (Adcock, G. and Hansen (2014))

If D ⊆ Rd is compact, convex and symmetric and Ω has density

δD := supy∈Rd

infω∈Ω|ω − y |D < 1/4,

where D is the polar set of D, then Ω a weighted set of sampling, withweights defined as volumes of Voronoi regions.

Pros: No separation condition. Sharp density condition.

Cons: No explicit estimates on the frame bounds.

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Result 2

Question

Can we estimate frame bounds for improved, dimension independent densities?

Theorem (Adcock, G. and Hansen (2014))

If D ⊆ Rd is compact and Ω has density

δ∗ := supy∈Rd

infω∈Ω|ω − y |∗ <

log 22πmDc∗

,

where mD := supx∈D |x | and c∗ is such that |·| ≤ c∗ |·|∗, then Ω is a weightedset of sampling, with weights defined as volumes of Voronoi regions. The framebounds satisfy

√A ≥ 2− e2πmDδ > 0,

√B ≤ e2πmDδ < 2.

If |·|∗ is Euclidean then c∗ = 1.If D is the unit cube then mD =

√d : improvement for factor

√d .

If D is the unit ball then mD = 1 : dimensionless density δ < log 2/(2π) ≈ 0.11.

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Function recovery from nonuniform Fourier measurements

Now that we know conditions that ensure Fourier frames, what about theproblem of stable and accurate function recovery?

Back to the main problem...

Main problem

Given finite Fourier data f (ω) : ω ∈ Ω

,

and given a desired finite-dimensional reconstruction space

T ⊆ L2(D)

under what conditions is it possible to recover unknown function f ∈ L2(D)stably and accurately in T? How can this be achieved via numerical algorithm?

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Different approaches for function recovery from nonuniform samples

Common approaches:

Direct discretization of the inverse Fourier integral aka gridding

Iterative reconstruction techniques for recovery of pixels

ACT algorithm for recovery of bandlimited functions

Inversion of the frame operator

. . .

Our approach:

Nonuniform Generalized Sampling (NUGS): based on a general approachof sampling and reconstruction in abstract Hilbert spaces, known asgeneralized sampling introduced by B. Adcock and A. Hansen in 2012.

1 Recovery in any reconstruction space is possible.

2 Stable and accurate recovery is guaranteed.

3 Fast algorithm for wavelets.

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Nonuniform Generalized Sampling (NUGS)

Let Ω = ωnNn=1 ⊆ ZK , where ZK ⊆ Rd is such that f |ZK → f , K →∞.

Let T ⊆ L2(D), dim(T) = M <∞.

Definition (NUGS reconstruction)

Given Fourier data f (ωn)Nn=1, define the NUGS reconstruction G(f ) ∈ T as

G(f ) := argming∈T

N∑n=1

µn

∣∣∣f (ωn)− g(ωn)∣∣∣2 .

Challenge: Find conditions on δ(Ω), K and M to ensure stability and accuracy.In particular, provide a positive constant C = C(Ω,T)∞ such that

∀f ∈ L2(D), ‖f − G(f )‖ ≤ C‖f − PTf ‖, (1)

where PT is orthogonal projection on T, and also, such that

κ(G) = supf∈L2(D)

f 6=0

‖G(f )‖√∑Nn=1 µn|f (ωn)|2

≤ C . (2)

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NUGS guarantees

Theorem (Adcock, G. and Hansen (2014))

(1) If D ⊆ Rd is compact and Ω ⊆ ZK has density

δ(Ω) <log(2)

2πmD,

and sampling bandwidth K large enough so that

RK (T) ≤ ε <√

(2− e2πmDδ)e2πmDδ,

then G(f ) ∈ T exists uniquely and

C ≤ e2πmDδ

√1− ε2 + 1− e2πmDδ

.

Pros: Explicit bound. Residual RK (T) can be estimated for different T.

Cons: Density not sharp.

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NUGS guarantees (2)

Theorem (Adcock, G. and Hansen (2014)– Cont.)

(2) If D ⊆ Rd is compact, convex and symmetric, and Ω ⊆ ZK has density

δD(Ω) < 1/4

and sampling bandwidth K large enough so that

RK (Ω,T) ≤ ε <√A,

then G(f ) ∈ T exists uniquely and

C ≤√

BA− ε2 ,

where A,B > 0 are the frame bounds arising from ωnn∈N.

Pros: Sharp density condition.

Cons: Bound not explicit. Residual RK (Ω,T) complicated to estimate.

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NUGS guarantees (summary)

Summary

Stable and accurate recovery in any T is possible subject to

exactly the same densities δ that guarantee weighted Fourier frames, and

sufficiently large sampling bandwidth K .

Question

How large does K need to be?

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NUGS guarantees (summary)

Summary

Stable and accurate recovery in any T is possible subject to

exactly the same densities δ that guarantee weighted Fourier frames, and

sufficiently large sampling bandwidth K .

Question

How large does K need to be?

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Scaling of the sampling bandwidth K with the number of coefficients M

Theorem (Adcock, G. and Hansen (2014))

In univariate case d = 1, for any ε > 0,

RK (T) = supf∈T‖f ‖=1

‖f ‖R\(−K ,K) < ε,

provided that

K ≥ c(ε)M, if T = M compactly supported wavelets,

K ≥ c(ε)M, if T = trig. pol. up to degree M,

K ≥ c(ε)M2, if T = alg. pol. up to degree M,

K ≥ c(ε)M2/η, if T = splines of degree M & min knot spacing η,

K ≥ c(ε)M2/η, if T = piece. pol. up to degree M & min spacing η.

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Remark

So far, we have dealt with recovery of f ∈ L2(D) from f (x) : x ∈ X, but wecould have equivalently considered recovery of f from f (x) : x ∈ X via

G(f ) := argming∈T

∑x∈X

µx

∣∣∣f (x)− g(x)∣∣∣2 .

That is, we have seen under what conditions we can recover a bandlimitedfunction h from its direct samples h(x) : x ∈ X.

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Nonuniform sampling of bandlimited functions with derivatives

Let f ∈ B(Ω) be an unknown function, where

B(Ω) =f ∈ L2(Rd ) : supp(f ) ⊆ Ω

, Ω ⊆ Rd compact,

and let X be a set of nonuniformly-spaced sampling points.

Question

Suppose one has access to the samplesDαf (x) : x ∈ X , α = (α1, . . . , αd ), |α|1 ≤ k

.

How does this extra information improve sampling and recovery of f ∈ B(Ω)?

Motivation

Seismology: Recently-developed sensors can record functions values and spatialgradient information.

Uniform sampling with derivatives: [Jagerman & Fogel, 1956], [Linden & Abramson,1960], [Papoulis, 1977]

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Nonuniform sampling of bandlimited functions with derivatives

The problem

Under what conditions does X form a weighted set of sampling with derivativesfor B(Ω)? That is, under what conditions there exist A,B > 0 such that

A‖f ‖2 ≤∑x∈X

∑|α|1≤k

µx,α |Dαf (x)|2 ≤ B‖f ‖2, f ∈ B(Ω). (F)

Our contributions (Adcock, G. and Hansen (2015))

Ifδ∗(X ) <

C(k, d)

2πmΩc∗,

then (F) holds. In particular,

C(k, d) ∼ k + 1e

, k →∞.

In addition, we can explicitly estimate the corresponding frame bounds.

Previous result in 1D: [Razafinjatovo, 1995]

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Open problems and current investigations

1 What is sufficient and necessary sampling bandwidth K for stable recoveryin different reconstruction spaces in dimensions d > 1?

2 What if we nonuniformly sample f ∗ hl , l = 1, . . . , L, hl ∈ L2(D)? Howdoes the sufficient sampling density scale with L?

We have tackled this problem when hl (x) = (−i2πx)l , l = 1, . . . , L, whichcorresponds to sampling the derivatives of f .

3 What if we want to reconstruct a non-compactly supported andnon-bandlimited function f in shift-invariant space from

Dαf (x) : x ∈ X , |α|1 ≤ k

?

Thank you for your attention.

More details in (joint work with B. Adcock and A. Hansen):On stable reconstructions from nonuniform Fourier measurements, SIAMJ. Imaging Sci., DOI:10.1137/130943431, 2014

Weighted frames of exponentials and stable recovery of multidimensionalfunctions from nonuniform Fourier samples, Appl. Comput. Harmon. Anal.,DOI:10.1016/j.acha.2015.09.006, 2015

Recovering piecewise smooth functions from nonuniform Fourier measurements,Proc. ICOSAHOM, 2014

Density theorems for nonuniform sampling of bandlimited functions usingderivatives or bunched measurements, Submitted, 2015

Fast recovery of wavelet coefficients in (joint work with C. Poon)A practical guide to the recovery of wavelet coefficients from Fouriermeasurements, + MATLAB code, To be published in SIAM J. Sci. Comput.,2015

http://milanagataric.wix.comm.gataric@maths.cam.ac.uk