mathematical methods of computed tomography

Lecture 1. Introduction: Computed tomographyLectures 2–3. CAT scan and Radon/X-ray transform

L. 4, Emission tomographyHybrid methods

Mathematical Methods of Computed Tomography

Peter KuchmentDedicated to the memory of

Leon Ehrenpreis and Iosif Shneiberg,friends, great mathematicians and human beings

NSF-CBMS conference, UT Arlington, May 28– June 2, 2012

Peter Kuchment Dedicated to the memory of Leon Ehrenpreis and Iosif Shneiberg, friends, great mathematicians and human beingsMathematical Methods of Computed Tomography



A brief outline of the lectures

1 (L1) Introduction: meaning and history of X-ray tomographyand Radon transform.

2 (L2–3) X-ray CT and X-ray/Radon transform

3 (L4) Emission tomography

4 (L5) Microlocal analysis in tomography

5 (L6-7) MRI, EIT, OT, etc.

6 (L7-8) Thermo-/photoacoustic tomography

7 (L9) Ultrasound modulated EIT and OT

8 (L10) Miscellanea: John’s equation, 3D CT, κ-operator,MRE, SAR, etc.




Thanks to:

M. Agranovsky, V. Aguilar, G. Ambartsoumian, H. Ammari, S. Arridge,G. Bal, C. Berenstein, G. Beylkin, J. Boman, L. Borcea, A. Bukhgeim,P. Burgholzer, M. Cheney, A. Cormack, M. de Hoop, L. Ehrenpreis,A. Faridani, D. Finch, I. Gelfand, S. Gindikin, E. Grinberg, M. Haltmeier,S. Helgason, D. Isaacson, V. Isakov, H. Kang, A. Katsevich,L. Kunyansky, M. Lassas, S. Lissianoi, A. Louis, S. Lvin, D. Ma,A. Markoe, J. McLaughlin, M. Mogilevsky, F. Natterer, L. Nguyen,L. Paivarinta, V. Palamodov, G. Papanicolaou, V. Papanicolaou,I. Ponomarev, Rakesh, E. T. Quinto, B. Rubin, O. Scherzer, J. Schotland,L. Shepp, I. Shneiberg, D. Solomon, P. Stefanov, G. Uhlmann, L. Wang,Y. Xu ... and others whom I might have forgotten.

Thanks to L. Kunyansky and Y. Hristova for help with pictures.




Mathematical imaging

Image processing (often done in engineering)

Image understanding (AI)

Image reconstruction




Lecture 1. Introduction: Computed tomography

Tomography is an inverse problem

Direct problem: given input f and operator A, find the outputg = Af

Inverse problem: given (a variety of) input(s) f and output(s)g = Af , find the operator A

A typical kind of inverse problems is the recovery of coefficientsof a differential equations on a domain from some informationabout its solutions at the domain’s boundary.Another example of an inverse problem: the famous Mark Kac’sproblem “Can one hear the shape of the drum?”




Tomography

Tomography: from Greek slice (τoµoσ) and to write (γραψετε).It attempts to find the internal structure of a non-transparentobject by sending some signals (waves, radiation) through it.Electromagnetic waves of various frequencies (radio andmicrowaves, visual light, X-rays, γ-rays) and acoustic waves arecommon.In Computed Tomography, the image is not obtained directlyfrom the measurements (like in the usual X-ray pictures), butrather is the result of an intricate mathematical reconstructionfrom the measured data.




Tomography

Has existed since 1950s and is still alive and kicking (morethan before)

Applied in medicine (diagnostics and treatment), biology,geophysics, archeology, astronomy, material science, industry,oceanography, atmospheric sciences, homeland security ...

Inexhaustible source of wonderful hard mathematicalproblems. Involves Differential Equations, Numerical Analysis,Integral, Differential, and Algebraic Geometry, SeveralComplex Variables, Probability/Statistics, Number Theory,Discrete Mathematics, ... IS JUST FUN!




Examples of tomographic reconstructions

In medicine





Industry





Geophysics





Archeology





Homeland security




History of (mathematics of) CT

• 1895 Rontgen discovers X-rays. Receives Nobel Prize in 1901.• 1905-1906 Lorenz solves “Radon transform” inversion in 3D.• 1917 Radon publishes his 2D reconstruction.• 1925 Ehrenfest solves the n-dimensional problem.• 1936 Cramer & Wold reconstructed a probability distributionfrom its marginal distributions.• 1936 Eddington recovers the distribution of star velocities fromtheir radial components.• 1956 Bracewell solves inverse problem of radio astronomy.• 1958 Korenblyum (Ukraine) develops the 1st X-ray medicalscanner.• 1963 Cormack (South Africa, USA) implements tomographicreconstructions for an X-ray scanner.• 1969 Hounsfield builds a medical X-ray scanner (with Ambrose)in 1972.• 1979 Hounsfield and Cormack receive Nobel Prize in medicine.Peter Kuchment Dedicated to the memory of Leon Ehrenpreis and Iosif Shneiberg, friends, great mathematicians and human beingsMathematical Methods of Computed Tomography



Some main players




Types of tomography

Transmission: the radiation transverses the body and isdetected emerging “on the other side.” Example: standardclinical X-ray CT.

Reflection: the radiation bounces back and is detected whereit was emitted. Examples: some instances of the ultrasoundand geophysics imaging.

Emission: the radiation is emitted inside the body and isdetected emerging outside. Examples: clinical SPECT (singlephoton emission tomography) and PET (positron emissiontomography), plasma diagnostics, nuclear reactors testing,detection of illicit nuclear materials.




Some common types of CT

X-ray CT is the most commonly used version.

SPECT (Single Photon Emission Tomography)

PET (Positron Emission Tomography)

MRI (Magnetic Resonance Imaging, based upon the NuclearMagnetic Resonance effect)

Ultrasound Tomography

Optical Tomography

Electrical impedance Tomography

And MANY more:

Thermoacoustic, photoacoustic, ultrasound modulated optical,acousto-electric, magneto-acoustic, elastography, electrontomography, radar and sonar, Internet tomography, discretetomography, ....




Features to look for

Contrast: variation in tissues’ response to radiation.

Resolution: size of distinguishable details.

Uniqueness of determining the unknown quantity.

Inversion methods (formulas, algorithms).

Stability of inversion. Ill-posed problems!

Incomplete data effects.

Range conditions.




PDE classification in CT

Most imaging methods reduce mathematically to determiningcoefficients of a PDE from boundary data. The features wediscussed in the previous section are closely related to the type ofthe PDE involved. The main types arising are:

Transport equation (X-ray CT, SPECT, PET)

Elliptic equations (OT, EIT)

Wave equation (TAT/PAT, SAR, Ultrasound imaging)

Lately, new breeds of tomographic techniques have beenarising, the so-called hybrid (or coupled physics) methods,where one can relay upon some additional internalinformation.




Lectures 2–3. CAT scan and Radon/X-ray transform

X-ray tomography = CAT scan

CAT=Computer Assisted Tomography




Projection X-ray imaging




CAT scan (= X ray tomography) procedure

The X-ray tomography (=CAT scan) procedure:

I0, I1 - initial and terminal intensities.




Scanning geometries

Cone-beam geometry

parallel beam




Beer’s law

I (x) – intensity of X-ray beam traveling along line L at point x .

Beer’s law: dI = −f (x)I (x)dx ,

where f (x) – attenuation coefficient at x

dI (x)dx = −f (x)I (x)⇒ d ln I (x)

dx = −f (x)⇒ I1 = e−

∫L

f (x)dx

I0

I.e., measurements provide∫L

f (x)dx for any line L

The density plot of f (x) is the tomogram.




Transport equation

u(x , ω) density of particles at x moving in the direction ω.

ω · ∇xu(x , ω) + f (x)u(x , ω) =

∫σ(s, ω′, ω)u(x , ω′)dω′ + s(x).

f (x) - attenuation coefficient, σ -scattering coefficient, s(x)-sources density.In absence of scattering and sources, one gets Beer’s law.




Radon transform

Given a function f (x) Radon transform produces the values of∫L f (x)dx along all lines L:

Rf (L) :=

∫L

f (x)dx

In coordinates:




Radon transform in coordinates

Rf (ω, s) :=

∫x ·ω=s

f (x)dx , (ω, s) ∈ C := S1 × R




Sinograms




Symmetries of the Radon transform

Evenness Rf (ω, s) = Rf (−ω,−s)since x · ω = s ⇔ x · (−ω) = −sIdentify (ω, s) ∼ (−ω,−s)g := Rf is defined on the Mobius strip (S × R)/ ∼Shift invariance.Let (Taf )(x) := f (x + a), (tpg)(ω, s) := g(ω, s + p)

R(Taf )(ω, s) = ta·ωRf = Rf (ω, s + a · ω)

Exercise: Prove this.Corollary: Fourier transform methods might be useful




Symmetries

Rotation invariance.A – rotation in 2D

R (f (Ax)) (ω, s) = Rf (Aω, s)

Exercise: Prove this.Corollary: Fourier series expansions might be useful.




Dilation invariance and Mellin transform

Blackboard presentation




Relations with the Fourier transform

f (x) on R2

f (ξ) =1

2π

∫R2

f (x)e−ix ·ξdx

f (x) =1

2π

∫R2

f (ξ)e ix ·ξdξ

g(s) on R

g(σ) =1√2π

∫R

g(s)e−isσds g(s) =1√2π

∫R

g(σ)e isσdσ




Projection slice formula

Ridge function Q(x) = q(x · ω)

Coupling with ridge functions∫R2

f (x)Q(x)dx =∫R

Rf (ω, s)q(s)ds.

For Q(x) = e iξ·x = e iσω·x get∫R2

f (x)e−iσω·xdx =∫R

Rf (ω, s)e−iσsds

Projection-slice (Fourier-slice) formula

f (σω) = 1√2π

g(ω, σ)




Uniqueness

Uniqueness

Any compactly supported function f is uniquely determined by Rf .Indeed, Rf (ω, s) determines the Fourier transform of f , and thus fitself.




Dual Radon transform

R : f (x) 7→ Rf (ω, s) - Radon,R# : g(ω, s) 7→ R#g(x) dual Radon transform, such that∫

Rf (ω, s)g(ω, s)dωds =

∫f (x)R#g(x)dx




Backprojection

Exercise: Prove the formula:

R#g(x) =

∫S

g(ω, x · ω)dω

The lines with normal coordinates (ω, x · ω) are passing through x :

This explains the name backprojection operator.




Radon transform over d dimensional planes. X-raytransform.

In Rn (n > 2), one can take d-dimensional Radon transform forany 1 ≤ d < n:

f (x) 7→ Rd f (H) =

∫H

f (x)dx , where H − d-dimensional subspace

E.g., in R3:

d = 2 - Radon transform∫x ·ω=s f (x)dx

d = 1 - X-ray transform∫∞−∞ f (x + tω)dt




Fourier inversion

Projection-slice formula gives an inversion procedure:

f (x) ⇒ Rf (ω, s) ⇒ Rf (ω, σ) =√

2πf (σω)inverse FT︷︸︸︷⇒ f (x)




Fourier inversion in polar coordinates

Let’s elaborate on Fourier inversion, denoting g := Rf :

f (x) = 12π

∫e iξ·x f (ξ)dξ = 1

2π

∫|ω|=1

∫∞0 e iσω·x f (σω)|σ|dσdω

= 14π

∫|ω|=1

∞∫−∞· · · = 1

4π

∫|ω|=1

dω

1√2π

∞∫−∞

e iσω·x g(ω, σ)|σ|dσ

︸︷︷︸

H dgds

(ω,x ·ω)

= 14πR#

(H d

ds g)

(x) = 14πR#H d

ds g(x).




Filtered backprojection (FBP) inversion

FBP inversion

f = 14πR#H d

ds Rf , where H dds - filtration, R# -backprojection

Exercise

Compute R#Rf (i.e., filtration omitted) and see why it produces ablurred version of f .




Hilbert transform

H - Hilbert transform

Hu(t) =1

πp.v .

∞∫−∞

u(s)

t − sds.




An example of a Matlab FBP inversion

Phantom (left) and its reconstruction from 128 projections and128 detectors. USE SIMPLE, BUT REVEALING PHANTOMS!




Variations on the filtration theme

Riesz potentials

(Iαf )(ξ) := |ξ|−α, α < n (spacial dimension)

I−1 := Λ :=√−∆ - Calderon operator.

f =1

4πI−αR#Iα−1Rf , α < 2

f =1

4πI−αR#Iα−n+1Rf , α < n




Fourier series inversion (Cormack’s formulas)

Polar coordinates r , ω (i.e., x = rω) on the plane,ω = (cosφ, sinφ). Standard coordinates (ω, s) in Radon domain.If f (x) - the original function, g(ω, s) = Rf -its Radon image,expand into Fourier series w.r.t. φ:

f (r , ω) =∑l

fl(r)e ilφ

g(ω, s) =∑l

gl(r)e ilφ

Any good formulas for {fl} ⇔ {gl}?




Fourier series continued - Cormack’s formulas

Rotational invariance ⇒ gl depends on fl only.

Exercise: Straightforward calculation:

gl(s) = 2

∞∫s

(1− s2

r2

)−1/2

T|l |

(s

r

)fl(r)dr ,

where Tl(x) = cos(l arccos(x)) - Chebyshev polynomial of 1stkind.

fl(r) = − 1

π

∞∫r

(s2 − r2

)−1/2T|l |

(s

r

)g ′l (s)ds




A word of caution - left inversions

Blackboard again




Range conditions

Q: is a given g(ω, s) Radon transform of some f ? (appropriatefunction classes assumed)A: Let g = Rf .i) g(ω, s) = g(−ω,−s)ii) consider kth moment Gk(ω) :=

∫R skg(ω, s)ds - function on

the unit sphere.

Gk(ω) =

∞∫−∞

ds

∫x ·ω=s

f (x)dx =

∫R2

(x · ω)k f (x)dx

homogeneous polynomial of degree k in ω




Range conditions

Range conditions

Evenness. g(ω, s) = g(−ω,−s)

Moment conditions. For any k = 0, 1, 2, . . . , Gk(ω) extendsto a homogeneous polynomial of degree k of ω ∈ R2.




Range revisited in Fourier domain

Consider 1D FT g(ω, σ) of g = Rf . Get 2D FT of f along radiallines:

f (σω) = cg(ω, σ) Notice the role of evenness!

Function h(σω) := g(ω, σ) is smooth everywhere, possibly exceptthe origin.

Exercise

Prove that smoothness of h at zero implies that dk gdsk (ω, 0)

extends to a homogeneous polynomial of degree k in ω.

Prove that this is equivalent to the moment conditions on g .




Well and Ill posed problems

A problem of finding g from f is well posed (in Hadamard’s sense),if it has unique solution and small variations in f lead to smallvariations in g .Otherwise the problem is ill-posed.This will be specified in various settings throughout the workshop.The characteristic property of inverse problems istheir usual ill-posedness.




Stability of reconstruction

Stability:

Small variations in data lead to small changes in the result.

SVD reduces any linear problem to solving Ax = y with a largediagonal matrix A:

a1 0 0 . . . 00 a2 0 . . . 00 0 a3 0 . . .. . . . . . . . . . . . . . .

x1

x2

x3

. . .

=

y1

y2

y3

. . .

When ak → 0 fast - instability.




Stability of Radon inversion

Stability estimate for Radon transform in 2D

D - disk in R2, f ∈ Hs0(D), C1‖Rf ‖Hs+1/2 ≤ ‖f ‖Hs ≤ C2‖Rf ‖Hs+1/2

Conclusion:

X-ray transform smoothes functions by 1/2 derivatives. Same withthe dual. Thus, two of them add a derivative, which must beremoved during inversion (remember the filter H d

ds ?).

More generally, d-plane transform adds d/2 derivatives, thus inFBP d derivatives need to be removed by filtration.




L. 4, Emission tomography

Other transforms of Radon type arising in tomography:

S - surface in R3 or curve in R2. Spherical mean operator

f (x) 7→ MS f (p, r) :=

∫|θ|=1

f (p + rθ)dθ, p ∈ S , r ∈ R+

arises in thermo- and photo-acoustic tomography.

In R3 - integrals over cones with vertices on a surface S .Arises in Compton camera imaging.

Integrals over horocycles and geodesics in the hyperbolic plane- arise in electrical impedance imaging.

Generalizations to Riemannian manifolds, to tensor fieldsrather than functions, etc.

Weighted Radon transform.Non-uniqueness example.




SPECT (Single Photon Emission Computed Tomography)

A patient is given a pharmaceutical labeled with a radionuclide,which emits γ-photons. The goal – recover distribution of thesources f (x).

µ(x) - attenuation, f (x) - sources distribution.




Attenuated Radon transform

Total count at detector is attenuated Radon transform of f :

(Tµf )(L) :=

∫L

f (x)e−

∫Lx

µ(y)dy

dx

e−

∫Lx

µ(y)dy

– the probability of a photon emitted at x in directionLx to reach the detector.




Difficulties

1. The transform is complicated.2. Two unknown functions.




Exponential Radon transform

Attenuation coefficient µ – known constant.Body of interest – of known convex shape.ExerciseShow that Tµ can be written as follows:

(Tµf )(t, ω) = v(x)

∫x ·ω=t

f (x)ex ·ω⊥dl ,

where v(x) – known, ω⊥ – 90o rotation of ω.




Exponential X-ray transform

(Rµf )(t, ω) :=

∫x ·ω=t

f (x)ex ·ω⊥dl ,

ExerciseEstablish an analog of the projection-slice theorem for theexponential X-ray transform and show that values of Rµfdetermine f (ξ) on

ξ = σω + iµω⊥.

Can one reconstruct the function f from these values? Uniqueness(for a compactly supported f ) follows from the Paley-Wienertheorem.




Tretiak-Metz formulas

Reconstruction formulas analogous to FBP were obtained byTretiak and Metz [?]:

f =1

4πR#−µJµ (Rµf ) , (1)

where

(R#−µg)(x) =

∫§1

e−µx ·ω⊥g(ω, x · ω)dω

and Jµ is the operator that multiplies the Fourier transform by thefilter

j(σ) =

{|σ| when |σ| > |µ|0 otherwise.




Range

The range description of the exponential X-ray transform (obtainedin [?, ?]) also happens to be very fascinating. In particular, itproduces an unusual separate analyticity theorem [?, ?] and aninteresting infinite series of identities for sin x [?, ?, ?]




The truly attenuated transform

Such inversions were found in the end of 1990th and beginning of2000s By A. Bukhgeim [?] with co-authors and R. Novikov [?].The Novikov type formula:

f =1

4πRe div R#

−µ(ωe−hHehg), (2)

H – the Hilbert transform, h = 0.5(I + iH)Rµ (R – the Radon

transform), and R#−µ is the weighted backprojection

R#−µg(x) =

∫§1

e−Dµ(x ,ω⊥)g(ω,x ·ω)dω.

Then complete range conditions were also found by Novikov [?].




Simultaneous reconstruction of attenuation and source

Can one recover both functions, using only the data Rµf ?If µ is constant, it can be found simultaneously with f from thedata Rµf , unless f is a radial function.Recent results for variable attenuation in a generic situation.




PET (Positron Emission Tomography)

Radionuclide emits positrons, which annihilate with electronsnearby and emit simultaneous pairs of γ-photons in oppositedirections. Detectors count simultaneous hits.

Probability for the pair emitted at location x to reach bothdetectors is

e−

∫L1x

µ(y)dy

e−

∫L2x

µ(y)dy

= e−

∫L

µ(y)dy

.

mu




PET - continued

The data measured gives the expression∫L

f (x)e−

∫L

µ(y)dy

dx = e−

∫L

µ(y)dy∫L

f (x)dx .

If the attenuation µ(x) is known, the PET data provides, up to

multiplication by the known factor e−

∫L

µ(y)dy

, the Radontransform

∫L

f (x)dx of f (x). The mathematics of PET is

equivalent to the one of X -ray CT.




L. 9: Helping EIT: Acousto Electric Tomography (AET)

Ammari, Bal, Bonnetier, Capdeboscq, Fink, Kang, Kuchment,Kunyansky, Scherzer, Steinhauer, Triki, Wang, Xu, ...

Focusing the ultrasound.




Synthetic focusing

Kuchment & Kunyansky , Wang at alUse a different (unfocused) basis of ultrasound waves and focus(change basis) “synthetically.”The useful (and stable) options:Planar transducers creating planar waves ⇒ Fourier transforminversion.Point transducers creating spherical waves ⇒ TAT inversion (sic!).Narrow US beams ⇒ Radon transform inversion.




AET - reconstructions

Ammari, Bal, Bonnetier, Capdebosq, Fink, Kang, Kuchment,Kunyansky, Scherzer, Steinhauer, Wang, Xu, ...




L. 9. Helping OT: Ultrasound Modulated OpticalTomography

Allmaras, Bal, Bangerth, Dobson, Kuchment, Nam, Oraevsky,Schotland, Steinhauer, Uhlmann, Wang, ...A combination of OT with ultrasound irradiation, similarly to AET.Use of coherent and incoherent light.Internal information measured G (x , d)A2(x)I (x), A - ultrasoundpower, I light intensity, G - Green’s function, d - detector position.

Absorption coefficient reconstruction (coherent light model).Peter Kuchment Dedicated to the memory of Leon Ehrenpreis and Iosif Shneiberg, friends, great mathematicians and human beingsMathematical Methods of Computed Tomography



L. 9. Inverse problems with interior information

A folklore meta-statement: “appropriate” internal informationstabilizes the severely unstable problems like diffused OT or EIT.Particular cases justified in the previously mentioned studies.general question: What kind of a functionF (D(x), σ(x), u(x),∇u(x)), if known, stabilizes the inverseboundary problems for

−∇ · D(x)∇u + σu = o?

A rather general (brand new) answer by Steinhauer & P. K.




L. 10. MRI helping other modalities. Magnetic resonanceelastography (MRE)

Ehman, Manduca, McLaughlinUsing MRI data to recover mechanical properties of biologicaltissues (e.g., stiffness).




MREIT, CDI (Current density imaging)

Seo et al (MREIT), Joy, Nachman, Tamasan, Timonov ...MRI data are used in conjunction with EIT to arrive to a stablemathematical problem and good reconstructions.




Magnetic resonance elastography (MRE)

Ehman, Manduca, McLaughlinUsing MRI data to recover mechanical properties of biologicaltissues (e.g., stiffness).




MREIT, CDI (Current density imaging)

Seo et al (MREIT), Joy, Nachman, Tamasan, Timonov ...MRI data are used in conjunction with EIT to arrive to a stablemathematical problem and good reconstructions.




Thanks

Thank you for your attention!

Thanks, Julianne, Tuncay, Gaik,Matthew, and other organizers for

exceptional organization of thiswonderful conference!!!




Bibliography

Some TAT/PAT books and surveys

M. Agranovsky, P. K., L. Kunyansky, On reconstructionformulas and algorithms for the TAT and PAT tomography,Ch. 8 in ”Photoacoustic imaging and spectroscopy,” CRCPress 2009, pp. 89-101.

G. Bal, D. Finch, P. K., P. Stefanov, G. Uhlmann (Ed.),Tomography and Inverse Transport Theory, AMS, 2011.

D. Finch and Rakesh, The spherical mean value operator withcenters on a sphere, Inverse Problems, 23 (2007), S37S50.

P. K., L. Kunyansky, Mathematics of thermoacoustictomography, European J. Appl. Math. 19(2008), 191-224.




Bibliography

P. Kuchment, L. Kunyansky, Mathematics of thermoacousticand photoacoustic tomography, Ch. 19 in ”Handbook ofMath. Methods in Imaging”, Springer 2010, pp.817 - 866.

S. K. Patch and O. Scherzer, Photo- and thermo-acousticimaging, Inverse Problems, 23 (2007), S01S10.

G. Uhlmann (Editor), Inside out, Vol. 2, MSRI Publ., toappear.

L. H. Wang (Editor) ”Photoacoustic imaging andspectroscopy,” CRC Press 2009, pp. 89-101.

L. Wang and H. Wu, Biomedical Optics, Wiley 2007




Bibliography

Recent (2010 - ) TAT/PAT publications (especially concerningquantitative PAT, time reversal, and speed recovery)

by Arridge et al, Bal et al, Nguyen, Stefanov, Uhlmann

See their Web pages




Bibliography

AET books, surveys and some papers

H. Ammari, An Introduction to Mathematics of EmergingBiomedical Imaging, Springer 2008.

H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M.Fink, Electrical impedance tomography by elastic deformation,SIAM J. Appl. Math., 68 (2008), 15571573.

G. Bal, D. Finch, P. Kuchment, P. Stefanov, G. Uhlmann(Ed.), Tomography and Inverse Transport Theory, AMS, inpreparation

B. Gebauer and O. Scherzer, Impedance-acoustic tomography,SIAM J. Applied Math., 69 (2009), 565576.




Bibliography

P. Kuchment, L. Kunyansky, Synthetic focusing in ultrasoundmodulated tomography, Inverse Problems and Imaging, V 4(2010), Number 4, 665 – 673. (arXiv preprint in January2009)

P. Kuchment, L. Kunyansky, 2D and 3D reconstructions inacousto-electric tomography, Inverse Problems 27 (2011),055013

H. Zhang and L. Wang, Acousto-electric tomography, Proc.SPIE, 5320 (2004), 145149.

UOT books, surveys and some papers




Bibliography

M. Allmaras and W. Bangerth, Reconstructions in UltrasoundModulated Optical Tomography, Preprint, arXiv:0910.2748v3

G. Bal and J.C. Schotland, Inverse Scattering andAcousto-Optic Imaging, Phys. Rev. Letters, 104, 043902,2010

H. Nam, Ultrasound Modulated Optical Tomography, Ph.Dthesis, Texas A&M University, 2002.

H. Nam and D. Dobson, Ultrasound modulated opticaltomography, preprint 2004.

V. V. Tuchin (Editor), Handbook of Optical BiomedicalDiagonstics, SPIE, WA 2002.

T. Vo-Dinh (Editor), Biomedical Photonics Handbook, editedby CRC, 2003.

L. Wang and H. Wu, Biomedical Optics, Wiley 2007


mathematical methods of computed tomography

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