Overview - University of Kentuckyrwalker/research/tat_recovery_seminar.pdfOverview Introduction...

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model

Classical FBPApproach

CT Analogy

FBP Formulas

Time Reversal

Neumann SeriesExpansion

References

Appendix

Approaches to Reconstruction in

Thermoacoustic Tomography

Ryan Walker

March 31, 2011

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Overview

Thermoacoustic tomography (TAT) is an importantemerging medical imaging technology.

In a TAT scan, electromagnetic (EM) energy excites thepatient’s tissue to generate an ultrasonic (US) signal.

Fundamentally different modality from classicaltomographic methods like computed tomography (CT) ornuclear magnetic resonance imaging (NMR/MRI) in thatTAT reconstructs an energy absorptivity profile instead ofa density profile.

Different from ultrasonic reflectivity imaging in that EMenergy excites a US signal, rather than a US inputreflecting a US output. Also, ultrasonic imaging recoversmechanical properties of tissue rather than absorptivity.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Overview

Potential Advantages of TAT:

Does not use dangerous x-ray radiationHybrid modality: High resolution and high contrastimaging

Imaging absorptivity profiles ⇒ high contrast.

US signal data ⇒ high resolution.

Relatively stable inverse acoustic source problem

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Overview

Potential downsides and implementation challenges:

Both electromagnetic excitation pulses and acoustic signalattenuate rapidly in tissue ⇒ small field of view (currentlyup to 6 cm) for non-damaging energy inputs.Mathematical model is extremely ad hoc and may notdescribe the actual physics of the imaging experiment.Reconstruction methods for TAT are far more complicatedthan those for classical modalities. The elegance of CTreconstruction is not to be found here.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Outline of the talk

Our goal in this talk is to give a broad sketch ofthermoacoustic reconstruction methods and indicate someexciting new directions.

1 The thermoacoustic signal generation model

2 The CT analogy and the quest for filtered back-projectiontype formulas

3 A new approach: the Stefanov-Uhlmann solution

4 The implied numerical method: Time-reversal

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

A.G. Bell’s Photoacoustic Effect I

In 1880, A.G. Bell published the interesting article On the

Production and Reproduction of Sound by Light [1]. In thiswork, Bell proposed the photophone, a device which usespulsed light to transmit sound. In his investigation of thephotophone, Bell made the remarkable discovery that anintense, pulsed light source incident on a thin sheet of materialcan produce audible sound. This effect is called thephotoacoustic effect. Moreover, he recognized that differentmaterials produced different tones.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Thermoacoustic Scan I

Photoacoustic effect is actually far more general: pulsedelectromagnetic energy incident on an absorptive materialcauses rapid thermal expansions and contractions,propagating a pressure wave.

Thermoacoustic scan:1 Input energy: EM pulses are applied to a slice of the

patient.2 The excitation causes thermal expansion of the tissue and

this propagates a pressure wave through the tissue.3 Transducers placed on the tissue boundary record this

pressure data.4 From this data, we try to recover the energy absorptivity

profile for the slice.5 The energy absorptivity profile shows how absorbing of the

input energy the slice structures are.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Thermoacoustic Scan II

Resolution: The pressure data is ultrasonic so thatrecovered data has high resolution. Moreover, traveldistance is 1/2 that of traditional ultrasound imaging sothat signal attenuation is reduced by a factor of 1/2 ⇒with uniform excitation we get the same resolution ofultrasound at twice the depth.

Contrast mechanism: Tissue species vary significantly intheir absorptivity to a particular energy type. In particular:cancerous tissues exhibit preferential absorptivity and the

absorptivity can be several orders of magnitude different

from healthy tissue.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Schematic of a TAT Scan

TransducerSurface

EM energy input layer TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Energy Inputs

Several types of EM excitation energies have been investigated

Optical (photoacoustic tomography): High contrast butrapid attenuation of optical signal in tissue limits depthpenetration to only a few cm.

Microwave: Better penetration but poor diagnosticcontrast.

Radio Frequency: Good contrast and excellent penetration.RF deposition is non-uniform but can often be corrected.Current implementations mostly employ RF excitation.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

TAT Generation

The universally accepted model for pressure wave propagationin the TAT literature is the acoustic wave equation. In its mostsimplified form, the standard model is:

[

∂2

∂t2− c(x)2∆

]

p(x, t) = 0, x ∈ Ω, t ∈ (0,∞)

p(x, 0) = f (x), x ∈ Ωpt(x, 0) = 0, x ∈ Ω

where:

c(x) is the acoustic speed in the medium.

f (x) is a quantity closely related to the input energyabsorptivity of the tissue at location x.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Some comments on the model

Assumes uniform heating by EM source, relativehomogeneity in density of the tissue.

This very ad hoc model also assumes that human tissue isliquid!

The assumption gives an easy derivation and a tractableinverse problem, but does this reflect the actual physics oftissue? Derivation

Magnetic resonance elasticity tomography assumes exactlythe opposite: human tissue is an elastic solid. Qualityimages are also produced under this modality.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

TAT Data

Transducers are placed at the boundary of the slice regionwe want to image.

Assume that the transducer array is continuous and thatthe array forms a surface S ⊂ R

3.

Often, the detection surface S will be assumed to be acircle or sphere. Can always add a coupling layer to obtainthis geometry.

The recorded signal in TAT is then the pressure h(y, t),where y ∈ S , t ∈ [0,∞).

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

TAT Inverse Problem

The pressure in a TAT experiment satisfies

[

∂2

∂t2− c(x)2∆

]

p(x, t) = 0, x ∈ R3, t ∈ (0,∞)

p(x, 0) = f (x), x ∈ R3

pt(x, 0) = 0, x ∈ R3

p(y, t) = h(y, t) y ∈ S , t ∈ [0,∞)

Formally, these equations define a mapping

Λf := h.

Λ maps the absorptivity profile f to measured pressure data h.Abstractly, a reconstruction method tells us how to invert Λ.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

The Spherical Mean Transform I

Let us assume that sound speed in the tissue is constantc(x) = c . Then the Cauchy wave equation problem:

[

∂2

∂t2− c2∆

]

p(x, t) = 0, x ∈ R3, t ∈ (0,∞)

p(x, 0) = f (x), x ∈ R3

pt(x, 0) = 0, x ∈ R3

has a well-known solution by the Poisson-Kirchoff formulas:

p(x, t) = c∂

∂t

(

t

∫

|θ|=1f (x+ tθ) dS(θ)

)

.

TAT Recon-struction

Ryan Walker

Overview

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Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

The Spherical Mean Transform II

p(x, t) = c∂

∂t

(

t

∫

|θ|=1f (x+ tθ) dS(θ)

)

For S a sphere, the inverse TAT problem is essentiallyequivalent to determining f (x) from the data

RS f (y, t) =1

ωn

∫

|θ|=1f (y + tθ) dS(θ)

for all y ∈ S .

The map RS is called the spherical mean operator.

The collected data in a TAT scan is basically the averageof f over all spheres with centers on the detection surfaceS .

So solving the inverse problem amounts to inverting RS .

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Geometry of RS

TAT Recon-struction

Ryan Walker

Overview

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FBP Formulas

Time Reversal


References

Appendix

Example 1

x

y

Simple Phantom

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Experimental Setup

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Example 1

x

y

Simple Phantom

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450

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from

Det

ecto

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TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Example 2

xy

Another Simple Phantom

50 100 150 200 250 300 350 400 450 500

450

400

350

300

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100

50

0

Experimental Setup

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Example 2

x

yAnother Simple Phantom

50 100 150 200 250 300 350 400 450 500

450

400

350

300

250

200

150

100

50

0

Rad

ius

from

Det

ecto

r

Detector Angle

Signal

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TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Similarity to CT Problem

The methods of computed tomography (CT) havemotivated many of the attempts to solve the TAT inverseproblem, so it is worthwhile to review the filtered-backprojection approach to the Radon transform.

In CT, the task is to reconstruct a function f fromknowledge of its Radon transform. The Radon transformof f ∈ S(R2) is the map given by:

Rf (t, ω) =

∫

Lt,ω

f (x) dx =

∫ ∞

−∞f (tω + sω⊥) ds..

Here the parameter t and the unit vector ω uniquelydetermine the line Lt,ω that is perpendicular to ω and atdistance t from the origin.

The basic CT problem is: knowing the value of every lineintegral of a function f over the domain Ω ⊂ R

2, can werecover f ?

TAT Recon-struction

Ryan Walker

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CT Analogy

FBP Formulas

Time Reversal


References

Appendix

The Radon Transform

Rf (t, ω) =

∫

Lt,ω

f (x) dx =

∫ ∞

−∞f (tω + sω⊥) ds..

TAT Recon-struction

Ryan Walker

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Time Reversal


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Appendix

Back-Projection I

Considering the geometry of the problem, it seemsreasonable that to recover f at a point x from its Radondata, we should compute the average of Rf (ω, t) over allLω,t passing through x.

Define the back-projection operator R♯g for g(t, ω) by

R♯g(x) =1

2π

∫

S1

g(x · ω, ω) dω.

This is precisely the operator which averages g over everyline passing through x. It also happens to be the formaladjoint of R.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Back-Projection II

For functions f from a “good domain” space, recallingthat t = x · ω, compute:

R#Rf (x) =

∫

|ω|=1Rf (x · ω, ω) dS(ω)

=

∫

|ω|=1

∫ ∞

−∞f ((x · ω)ω + sω⊥) dsdS(ω)

Basically, a polar integral. Rewrite:

R#Rf (x) =

∫

R2

2f (y)

|y − x|dy =

2

|x|∗ f (x).

Not f (x) but a blurred version of f (x)!

TAT Recon-struction

Ryan Walker

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Time Reversal


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Appendix

Back-Projection III

Now define the operator (−∆)1/2 by

F((−∆)1/2f )(ξ) = |2πξ|F f (ξ)

then by Fourier transform properties:

f (x) = (−∆)1/2R♯Rf (x).

We note that the kernel K (x, y) = 1/|y − x| appearing in

R#Rf (x) =

∫

R2

2f (y)

|y − x|dy

suggested what operator should be applied to recover f .

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Back-Projection for CT Problem

A Phantom

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix


Radon Data:

TAT Recon-struction

Ryan Walker

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Model


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FBP Formulas

Time Reversal


References

Appendix


Back-Projection:

TAT Recon-struction

Ryan Walker

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FBP Formulas

Time Reversal


References

Appendix

Filtered Back-Projection for CT Problem

Filtered Back-Projection:

TAT Recon-struction

Ryan Walker

Overview

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FBP Formulas

Time Reversal


References

Appendix

Brief History of Spherical Mean Inversion I

Can we recover f from the data

RS f (y, t) =1

ωn

∫

|θ|=1f (y + tθ) dS(θ)

for y in the detection surface S?

(Norton 1980 [9]) First suggestion of spherical meaninversion for tomographic imaging modality.

Found a series reconstruction for the case of a circulardetection surface in 2D by exploiting rotational symmetry.Proof is a la Allan Cormack [3], expanding f and thecollected data h in spherical harmonic series andestablishing a relation on the coefficients.Impractical formulas which involve division by Besselfunctions with infinitely many zeros.

TAT Recon-struction

Ryan Walker

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FBP Formulas

Time Reversal


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Appendix

Brief History of Spherical Mean Inversion II

Also proved that the Radon transform is the limit of thespherical mean transform for the circular detection surfaceas the radius ρ → ∞.First instance of a much more general approach to theinversion: series expansion in a useful basis. Some verypromising recent approaches along this line have producedinversion formulas on an arbitrary domain Ω by expandingin the eigenfunction basis for the Dirichlet Laplacian −∆on Ω (Kunyansky 2007, [8]).

Formulas in various special cases found throughout the 80sand 90s.

Many researchers obtained very good approximateinversion with parametrices (e.g. Popov 2005 [10]),providing numerical evidence that there should be somekind of FBP analog for PAT reconstruction.

TAT Recon-struction

Ryan Walker

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FBP Formulas

Time Reversal


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Appendix

Brief History of Spherical Mean Inversion III

Finally, in 2004 Finch, Patch, and Rakesh [5] producedexact filtered back-projection formulas for the sphericaldetection surface in odd dimensions. Even dimensionalanalogs took several more years and were obtained byFinch, Haltmeier, and Rakesh in 2007 [4].

TAT Recon-struction

Ryan Walker

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FBP Formulas

Time Reversal


References

Appendix

Sketch of Finch et. all n = 3 FBP I

For the spherical detection geometry in R3 and f smooth

and compactly supported on B(0, ρ) with detectionsurface S = ∂B(0, ρ)

f (x) = c(−∆)(R ♯St

2RS)f (x).

As before, R ♯S is the formal adjoint of RS and is easy to

compute. In particular, for g(y, t) it is given by

RS♯g(x) =

1

4π

∫

|y|=ρ

g(y, |y − x|)

|x− y|dSy.

For g = RS f , this operator forms a weighted average of allthe integrals of f over spheres with centers on thedetection surface and which pass through x.

TAT Recon-struction

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Appendix

Sketch of Finch et. all n = 3 FBP II

TAT Recon-struction

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Appendix

Sketch of Finch et. all n = 3 FBP III

Compute R♯St

2RS f . A version of the coarea formulaapplies to expand all the spherical surface integrals intosolid integrals.

A few changes of variables can then be used to reduceR ♯t2Rf to

R♯St

2Rf (x, t) = c

∫

R3

f (y)

|x− y|dy .

Noting the Green’s function kernel, it follows that

f (x) = c(−∆)(R ♯S t

2R)f (x).

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

Disadvantages of FBP

FBP approaches are currently restricted to sphericaldetection geometries, are difficult to formulate, and lessintuitive than their counterparts in Radon theory.

No direct way to incorporate variable sound speed.

Only work when the support of f is completely enclosed bythe detection surface S . Interestingly, different exactinversion formulas lead to different incorrect results whenpart of the support of f is outside the detection surface(Hristova et al. [7]).

These considerations suggest that a different approach willprobably be needed to solve more general PAT inversionproblems.

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

The Wave Equation Reconsidered I

Recall that the pressure in a 3D TAT experiment satisfies:

[

∂2

∂t2− c(x)2∆

]

p(x, t) = 0, x ∈ R3, t ∈ (0,∞)

p(x, 0) = f (x), x ∈ R3

pt(x, 0) = 0, x ∈ R3

p(y, t) = h(y, t), y ∈ S , t ∈ [0,∞)

Now let Ω be a domain containing the support of f andwhose boundary is the detection surface S .

TAT Recon-struction

Ryan Walker

Overview

Introduction

Outline

Model


CT Analogy

FBP Formulas

Time Reversal


References

Appendix

The Wave Equation Reconsidered II

Recall the following version of Huygen’s principle: Forconstant sound speed in 3D, a wave originating from asource in Ω will propagate out of Ω in finite time.

Hence for c(x) = c , there is a time T so that p(x, t) = 0for t > T for all x ∈ Ω.

TAT Recon-struction

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Time Reversal


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Appendix

Rewinding the Wave Equation

For the case that c(x) = c , this suggests that we can solve

[

∂2

∂t2− c2∆

]

p(x, t) = 0, x ∈ Ω, t ∈ [0,T ]

p(x,T ) = 0, x ∈ Ωpt(x,T ) = 0, x ∈ Ωp(y, t) = h(y, t), y ∈ S , t ∈ [0,T ]

and rewind the solution to time t = 0 to recover

f (x) = p(x, 0).

TAT Recon-struction

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Time Reversal


References

Appendix

Time Reversal Method I

Of course, this only works for constant sound speed andodd dimension! In the more general case, there is no valueof T so that p(x,T ) ≡ 0 for t > T .

If however we have some a priori understanding of howp(x, t) decays then we can choose T very large so thatp(x, t) ≈ 0 for t > T . Then solving the problem

[

∂2

∂t2− c2∆

]

p(x, t) = 0, x ∈ Ω, t ∈ [0,T ]

p(x,T ) = 0, x ∈ Ωpt(x,T ) = 0, x ∈ Ωp(y, t) = h(y, t)φǫ(t), y ∈ S , t ∈ [0,T ]

we should see that

p(x, 0) ≈ f (x).

TAT Recon-struction

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Time Reversal


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Appendix

Time Reversal Method II

Here φǫ(t) is a smooth cutoff function which has φ(t) = 1for t ∈ (−∞,T − ǫ] and φ(t) = 0 for t ≥ T . We needthis to be sure that we do not accidentally introducesingularities into f by putting singularities in h.

This is the idea of the time reversal method for TATreconstruction which is carefully formulated in Hristova2009 [6]. We now outline this approach.

TAT Recon-struction

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FBP Formulas

Time Reversal


References

Appendix

Energy Decay Estimates for Wave Equation

Solutions I

To establish that time reversal gives a good approximation, wemust get decay estimates in Ω for p(x, t), the solution to

[

∂2

∂t2− c2(x)∆

]

p(x, t) = 0, x ∈ Rn, t ∈ [0,∞)

p(x, 0) = f (x), x ∈ Rn

pt(x, 0) = 0, x ∈ Rn

p(y, t) = h(y, t)φǫ(t), y ∈ S , t ∈ [0,∞]

For this, we need the sound speed to satisfy a non-trapping

condition, which basically ensures that singularities introducedin Ω at time t = 0 propagate out of the domain after a finiteamount of time T .

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Appendix


Solutions II

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Appendix


Solutions III

Assuming that such a condition holds ( More ) , we canmake the following estimate on the decay of p(x, t):

∣

∣

∣

∣

∣

∂k+|m|p(x, t)

∂kt ∂

mx

∣

∣

∣

∣

∣

≤ Cηk(t)‖f ‖L2 for x ∈ Ω, t > T0

where

ηk(t) =

t−n+1−k for even n

e−δt for odd n.

Thus after a sufficient amount of time has elapsed,solutions to the non-trapping wave equation decaypolynomially for even dimensions and exponentially forodd dimensions. The constant δ depends only on thesound speed.

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Appendix

Error Estimate I

Let u(x, t) be the solution to

[

∂2

∂t2− c2(x)∆

]

u(x, t) = 0, x ∈ Ω, t ∈ [0,T ]

u(x,T ) = p(x ,T ), x ∈ Ωut(x, 0) = pt(x ,T ), x ∈ Ωu(y, t) = h(y, t), y ∈ S , t ∈ [0,∞]

and v(x, t) be the solution to

[

∂2

∂t2− c2(x)∆

]

v(x, t) = 0, x ∈ Ω, t ∈ [0,T ]

v(x,T ) = 0, x ∈ Ωvt(x,T ) = 0, x ∈ Ωv(y, t) = h(y, t)φǫ(t), y ∈ S , t ∈ [0,T ]

TAT Recon-struction

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Appendix

Error Estimate II

Note that u is the solution to the exact pressure problemposed at t = T and v is the approximation obtained bysetting p = 0 for t = T . We rewind v to zero to get anapproximation for the initial absorptivity profile f .

Given the solution decay estimates, we can establish thatthere is a sufficiently large T so that for non-trappingsound speed in even dimensions

‖f − v(0)‖H1(Ω) ≤ C (ǫ)(T − ǫ)−n+1‖f ‖L2(Ω)

and for non-trapping sound speed in odd dimensions

‖f − v(0)‖H1(Ω) ≤ C (ǫ)e−δ(T−ǫ)‖f ‖L2(Ω).

TAT Recon-struction

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Appendix

Error Estimate III

The proof considers a a hyperbolic problem related to theerror e = u − v . With some technical but standardestimates from hyperbolic PDE theory and the decayestimates described above the result follows directly.

Simple numerical implementation: Discretize the PDE

[

∂2

∂t2− c2(x)∆

]

v(x, t) = 0, x ∈ Ω, t ∈ [0,T ]

v(x,T ) = 0, x ∈ Ωvt(x,T ) = 0, x ∈ Ωv(y, t) = h(y, t)φǫ(t), y ∈ S , t ∈ [0,T ]

and find v(x, 0) ≈ f (x). (Burgholzer, et al. 2007 [2] ;Hristova et al. 2008 [7]; Hristova 2009 [6])

Important difficulty: we do not have control over theparameter δ in the decay estimates.

TAT Recon-struction

Ryan Walker

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Time Reversal


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Appendix

Neumann Series Approach

Motivated by the time reversal method, Stefanov &Uhlmann (2009) [11] provide a surprisingly simple andcomplete TAT inversion theory.

Let Ω ⊂ Rn be a smooth bounded domain and let f be

supported in Ω; Assume c(x) is non-trapping.

Suppose we run the TAT experiment for only a finite timeT > diam(Ω). From the previous estimates, we know thepressure is approximated by the solution to the problem

[

∂2

∂t2− c2(x)∆

]

p(x, t) = 0, x ∈ Ω, t ∈ [0,T ]


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Appendix

Approximate Inverse

Here we will obtain a Neumann series expansion whichexactly recovers f .

Recall that Λf = h is the measured data we collect at thedetection surface S = ∂Ω.

Solve the backward wave problem and evaluate thesolution at t = 0 to recover f : define

A0h := p0(0, x) for x ∈ Ω.

We think of A0 as an approximate inverse for Λ.

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Appendix

Approximate Inverse

[

∂2

∂t2− c2(x)∆

]

p(x, t) = 0, x ∈ Ω, t ∈ [0,T ]


If T = ∞ then this is the exact equation for pressure andthe Cauchy data at t = ∞ is justified by finite energy.Recover f exactly by p(x, 0).

For finite T , this is approximate and error inapproximating f by p(x, 0) is O(e−δT ).

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Appendix

Neumann Series I

[

∂2

∂t2− c2(x)∆

]

p(x, t) = 0, x ∈ Ω, t ∈ [0,T ]


Problem is that there could be a jump discontinuity in p

at time T . This discontinuity propagates back in time andwill affect the value of p(0, x).

Define an approximate inverse A for Λ by

Ah = p(·, 0).

AΛf could be more singular than f !

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Appendix

Neumann Series II

Note that in the time reversal method, the fix was toreplace h with h · φǫ with φǫ a smooth cut-off function.

Here we will modify the approximate in such a way thatthe problem becomes nice and the operator A defined bythe problem is contractive.

For an approximate inverse A of Λ we define the erroroperator by

K = I − AΛ

Insight: Define A in such a way that K is compact andcontractive (‖K‖ < 1). Rearranging:

AΛf = (I − K )f

and so

f = (AΛ− I )−1f =

∞∑

n=0

KnAΛf .

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Appendix

Neumann Series III

The Neumann series converges since ‖K‖ < 1.

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Appendix

Sketch of the Proof I

To define A in the right way, we form the new backwardproblem

[

∂2

∂t2− c2(x)∆

]

v(x, t) = 0 in [0,T ] ×Ω

v(y, t) = h(y, t) for y ∈ ∂Ω, t ∈ [0,T ]v(x,T ) = φ(x) for x ∈ Ω∂tv(x,T ) = 0 for x ∈ Ω

where−∆φ(x) = 0, x ∈ Ω φ(y) = h(y,T ).

We then define A by

Ah := v(y, 0).

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Appendix

Sketch of the Proof II

[

∂2

∂t2− c2(x)∆

]

p(x, t) = 0 in (0,T ) × Ω

p(y, t) = h(y, t) for y ∈ ∂Ω, t ∈ [0,T ]p(x,T ) = 0 for x ∈ Ω∂tp(x,T ) = 0 for x ∈ Ω

[

∂2

∂t2− c2(x)∆

]

v(x, t) = 0 in (0,T ) × Ω

v(y, t) = h(y, t) for y ∈ ∂Ω, t ∈ [0,T ]v(x,T ) = φ(x) for x ∈ Ω∂tv(x,T ) = 0 for x ∈ Ω

−∆φ(x) = 0, x ∈ Ω φ(y) = h(y,T ).

We make a third problem:

[

∂2

∂t2− c2(x)∆

]

w(x, t) = 0 in (0,T ) × Ω

w(y, t) = 0 for y ∈ ∂Ω, t ∈ [0,T ]w(x,T ) = p(x,T )− φ(x) for x ∈ Ω∂tw(x,T ) = pt(x,T ) for x ∈ Ω

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Sketch of the Proof III

Note that on [0,T ] × Ω, p and v + w satisfy the sameproblem! So by uniqueness

p = v + w

and restricting to t = 0:

f (x) = Ah + w(x, 0) = AΛf (x) + w(x, 0).

SinceI = AΛ + K

we should haveKf (x) = w(x, 0).

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Appendix

Sketch of the Proof IV

Defining the Dirichlet norm

‖f ‖2HD:=

∫

Ω

(

|Df |2 + c(x)−2|f |2)

dx

it is not difficult to prove that for f 6= 0

‖Kf ‖HD (Ω) ≤ ‖f ‖HD(Ω)

A more tricky analysis with a version of Holmgren’s theorem( More ) is needed to show that this inequality is actually strict.The proof that K is compact will then follow by considering thevarious smooth mappings of f into data at time T defined bythe problems for p, v , and w and then expressing K as acomposition of these maps.

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Appendix

Sketch of the Proof V

For the compactness of K :

Since T > diam(Ω), all singularities starting in Ω have leftΩ by time T . In particular, the mappings

f 7→ p(·,T )− φ f 7→ pt(·,T )

are compact on HD(Ω).

The solution operator from t = T to t = 0 of the w

equation:

(u(·,T ) − φ, uT (·,T )) 7→ (w(·, 0),wt (·, 0))

is unitary in HD(Ω)⊕ L2(Ω).

The mapping K can be expressed as the composition ofthese two maps, the first compact and the secondbounded, so K is itself compact.

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Appendix

References I

Alexander Graham Bell.On the production and reproduction of sound by light.American Journal of Science, 20:305, Oct 1880.

Peter Burgholzer, Gebhard J. Matt, Markus Haltmeier, andGunther Paltauf.Exact and approximative imaging methods forphotoacoustic tomography using an arbitrary detectionsurface.Phys. Rev. E, 75(4):046706, Apr 2007.

A. M. Cormack.Representation of a function by its line integrals, with someradiological applications.Journal of Applied Physics, 34(9):2722–2727, 1963.

TAT Recon-struction

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Appendix

References II

D. Finch, M. Haltmeier, and Rakesh.Inversion of spherical means and the wave equation in evendimensions.ArXiv Mathematics e-prints, January 2007.

David Finch and Sarah K. Patch.Determining a function from its mean values over a familyof spheres.SIAM Journal on Mathematical Analysis, 35(5):1213–1240,2004.

Yulia Hristova.Time reversal in thermoacoustic tomography—an errorestimate.Inverse Problems, 25(5):055008, 2009.

TAT Recon-struction

Ryan Walker

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Appendix

References III

Yulia Hristova, Peter Kuchment, and Linh Nguyen.Reconstruction and time reversal in thermoacoustictomography in acoustically homogeneous andinhomogeneous media.Inverse Problems, 24(5):055006, 2008.

Leonid A Kunyansky.A series solution and a fast algorithm for the inversion ofthe spherical mean radon transform.Inverse Problems, 23(6):S11, 2007.

Stephen J. Norton.Reconstruction of a reflectivity field from line integrals overcircular paths.The Journal of the Acoustical Society of America,67(3):853–863, 1980.

TAT Recon-struction

Ryan Walker

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Appendix

References IV

D. A. Popov and D. V. Sushko.Image restoration in optical-acoustic tomography.Problems of Information Transmission, 40:254–278, 2004.10.1023/B:PRIT.0000044261.87490.05.

Plamen Stefanov and Gunther Uhlmann.Thermoacoustic tomography with variable sound speed.Inverse Problems, 25(7):075011, 2009.

Andrew C. Tam.Applications of photoacoustic sensing techniques.Rev. Mod. Phys., 58(2):381–431, Apr 1986.

TAT Recon-struction

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Appendix

References V

Minghua Xu and L.V. Wang.Time-domain reconstruction for thermoacoustictomography in a spherical geometry.Medical Imaging, IEEE Transactions on, 21(7):814 –822,2002.

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Appendix

TAT Generation Derivation I

Assume patient is inhomogeneous in her absorption of EMenergy but relatively homogeneous in density.

The photoacoustic effect is then governed by an acousticwave equation. The standard reference for a carefulderivation is [12]. Following, [13],

For a short EM pulse duration, the change in temperatureis proportional to the applied thermal energy. Let theheating function H(x, t) be thermal energy per unit timeand T (x, t) be the temperature:

ρCp∂T

∂t= H(x, t)

where ρ is density and Cp is specific heat.

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Appendix

TAT Generation Derivation II

Acoustic pressure p(x, t) is the deviation from the ambientpressure caused by the propagation of a sound wave. Letu(x, t) be the acoustic displacement of the slice.

Pressure gradient drives displacement, so Newton’s lawgive

ρ∂2

∂t2u = −∇p.

Temperature rise drives displacement through thermalexpansion:

∇ · u = −p

ρc(x)2+ βT (x , t).

Here c(x) is the speed of sound in the medium and β is anisobaric volume expansion constant.

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Appendix

TAT Generation Derivation III

Differentiating twice in t:

∇ · utt = −ptt

ρc(x)2+ βTtt(x , t).

and substituting ρutt = −∇p and ρCp∂T∂t = H(x, t)

∆p −ptt

c(x)2= −

β

Cp

Ht

Now assume that H can be separated into a product ofthe spatial absorptivity function and a temporalillumination function:

H(x, t) = f (x)I (t).

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Appendix

TAT Generation Derivation IV

In PAT models, the heating pulse is idealized so thatI (t) = δ(t).

The model is:

(∆− c(x)−2∂tt)p(x, t) = f (x)δ′(t)

where p(x, t) = 0 when t < 0.

Go back

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Appendix

Non-trapping Condition I

Consider the Hamiltonian system in R2nx ,ξ of pairs (x , ξ) of

n-dimensional vectors and with H = c(x)2

2 |ξ|2:

x ′t =∂H∂ξ = c2(x)ξ

ξ′t = −∂H∂x = 1

2∇(c2(x))|ξ|2

x |t=0 = x0, ξ|t=0 = ξ0

The solution of this system are called bicharateristics and theirprojections onto R

nx are called rays. We call c(x) non-trapping

if all rays with ξ0 6= 0 tend to +∞ when t → ∞. Go back

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Appendix

Holmgren’s Theorem I

Holmgren’s uniqueness theorem is a uniqueness theorem forCauchy problems for linear PDEs with analytic coefficients anddata prescribed on a noncharacteristic surface. In this context,the theorem allows us to extend a solution satisfyingu(x, t) = 0 on a narrow cylinder to cone. Go back

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