Inverse Problems of Tomographic Typeaipc.tamu.edu/pre/AIP11_kuchment.pdf · Dual Radon Radon...

CAT scan and Radon/X-ray transformRelations with the Fourier transform. Dual Radon

Radon transform over d dimensional planes. X-ray transform.Inversion formulas. Range

Stability of inversionIncomplete data and singularities

Local (high frequency) tomographyOther types of tomography involving integral geometric transforms

A classification of tomographic proceduresElectrical Impedance Tomography

Optical TomographyNovel Hybrid methods

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Inverse Problems of Tomographic Type

Peter Kuchment

Applied Inverse Problems, Pre-conference workshopTexas A&M, May 21–22, 2011

Peter Kuchment Inverse Problems of Tomographic Type







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Table of contents

1 Inverse problems and what questions one may want to ask2 Tomography3 CAT scan and Radon/X-ray transform4 Relations with Fourier transform. Inversion. Stability. Range5 Incomplete data. Singularity detection. Local (high

frequency) tomography6 Other tomographic methods that involve integral geometry:

Emission tomography, MRI7 Electrical Impedance Tomography. Diffuse optical

tomography. Ultrasound imaging.8 Novel hybrid methods: MREIT, CDI, MRE, TAT/PAT, AET,

UOTPeter Kuchment Inverse Problems of Tomographic Type







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Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform

Direct and inverse problems

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








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Questions to ask

Injectivity of the direct transform input f ⇒ output g .(Uniqueness of reconstruction)

Inversion algorithms

Stability of inversion

Range of the direct operator

Incomplete data effects

Contrast

Resolution








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Tomography

Derived from the Greek tomos (part) and graphein (to write).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!Peter Kuchment Inverse Problems of Tomographic Type







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Examples of tomographic reconstructions

In medicine








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Industry








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Geophysics








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CAT scan

CAT=Computer Assisted Tomography








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Medical tomographic scanner inventors

A. Cormack and G. Hounsfield built the first computedtomography scanners in 1960s, won 1979 Nobel prize in medicine.








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CAT scan (= X ray tomography) procedure

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

I0, I1 - initial and terminal intensities.








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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.








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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.








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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:








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Radon transform in coordinates

Rf (ω, s) :=

∫x ·ω=s

f (x)dx , (ω, s) ∈ C := S1 × R








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Symmetries

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








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Symmetries continued

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 (check it) ⇒ Melin transform is useful.








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Fourier transformProjection slice formulaDual Radon transform (backprojection)

Fourier transforms

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








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Ridge functions

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)Peter Kuchment Inverse Problems of Tomographic Type







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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.








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Dual Radon

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








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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.Peter Kuchment Inverse Problems of Tomographic Type







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d-plane transform

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








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Fourier transform inversionFiltered backprojection inversionFourier series inversion (Cormack’s formulas)Range conditions

Fourier inversion

Projection-slice formula gives an inversion procedure:

f (x) ⇒ Rf (ω, s) ⇒ Rf (ω, σ) =√

2πf (σω)inverse FT︷︸︸︷⇒ f (x)








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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).








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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 .








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An example of a Matlab FBP inversion

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








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Fourier series

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?








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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








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Range

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 ωPeter Kuchment Inverse Problems of Tomographic Type







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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.








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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 .








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Ill posed problemsStability of reconstruction

Ill posed problems

A problem of finding g from f is correctly posed (in Hadamard’ssense), if it has unique solution and small variations in f lead tosmall variations 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.








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Notion of stability

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.








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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.








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Limited angle problemExterior problemInterior problemSingularity detection

Incomplete data

What if the values of Rf (ω, s) are known for a set of points (ω, s)only?

Is the reconstruction possible (uniqueness)?

Reconstruction procedures.

Stability.

Any resulting image deterioration?








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Limited angle

Data known for an open set of ωs and all s.

Any compactly supported function is uniquely reconstructed.

Follows from Paley-Wiener theorem.

Corollary:

Data known for an angle of ωs and all s ⇒ compactly supportedfunctions are uniquely reconstructed.

Reconstruction is unstable.








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A limited angle reconstruction








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Exterior problem

ROI (Region Of Interest) imaging - can we use only beamsintersecting the ROI?Exterior problem: data is known for beams not intersecting a(bounded convex) domain (i.e., ROI is the exterior):

• Reconstruction is unique, if the function decays at ∞ faster thanany power |x |−N (not true for a fixed power, no matter how large).Reconstruction is unstable.








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Example of exterior reconstruction








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Interior problem

ROI is the interior of a domain:

Reconstruction is non-unique. But ....Peter Kuchment Inverse Problems of Tomographic Type







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Example of interior reconstruction








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Visible/audible singularities

Q.: Why did some of the interfaces disappear?A.: Since they are “invisible” with the limited data we had.

Rule of thumb:

Wave front point (x , ξ) is “invisible” if there is no line (surface) ofintegration in the data set passing through x and co-normal to ξ.

Q.: What about uniqueness theorems? Everything should bevisible?A.: “Visible” theoretically, but unstable (and thus impossible) toreconstruct.








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Local tomography

Making the filtration in FBP stronger, emphasizes (but does notmove, create, or eliminate) the singularities of f . E.g., in 2Dconsider the following modified reconstruction operator:

Ag := R# d2g

ds2

(notice the replacement of H dds by d2

ds2 ).

Theorem

AR is an elliptic operator of order 1. Thus WF (ARf ) = WF (f ),but singularities of f are emphasized in ARf .








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Local reconstruction example - full data








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Local reconstruction example - limited angle








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Local reconstruction example - exterior data








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Local reconstruction example - interior data








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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.








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Emission tomography and attenuated X-ray transformMRI

A classification of tomographic procedures

Transmission (X-ray, Ultrasound, Optical, ElectricalImpedance)

Emission (SPECT, PAT, astronomy, homeland security)

Reflection (Ultrasound)

What’s that ????








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SPECT = Single Photon Emission Computed Tomography

Imaging self-radiating objects (medicine, nuclear reactors, security).

f (x) - unknown source distribution, µ(x) - attenuation.Attenuated Radon transform

f 7→ Rµf (ω, s) =

∫L

f (x)e−

∫Lx

µ(y)dy

dx








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Emission tomography - continued

Notice evenness disappear: g(ω, s) 6= g(−ω,−s)

After 30 years long history, uniqueness, inversion, range, andstability issues have finally been resolved very recently.

Q.: Can one recover simultaneously f and µ?A.: ??? (Recent major advances by Bal and Bukhgeim)








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MRI

Physics and mathematics of MRI are too complex to discuss now.The bottom line: 2D or 3D Fourier transform of a function(tomogram) is measured. Then the tomogram is reconstructedeither by direct FT inversion, or by reducing (projection-slice f-la)to the Radon transform and then inverting it.








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EIT

−∇ · σ(x)∇u = 0 in Ω

u|∂Ω = f , σ ∂u∂ν |∂Ω = g

Λ : f → g – Dirichlet-to-Neumann mapPeter Kuchment Inverse Problems of Tomographic Type







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EIT - Pros and Contras

Great contrast

Low resolution (exponential instability)








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OT

−∇ · σ(x)∇u + µu = 0 = 0 in Ω

Great contrast Low resolution (exponential instability)








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No justice in the world!

There seem to be no cheap, safe, high contrast, and high

resolution methods!Q: What can one do?A: Plagiarise! (Tom Lehrer) Sorry! Combine, hybridize.Now the hybrid methods come into play!








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Thermo-/ photo-acoustic imaging and spherical mean operatorsHelping EIT: Acousto Electric Tomography, MREIT, CDIIHelping OT: Ultrasound Modulated Optical TomographyMagnetic resonance elastography (MRE)

TAT/PAT - thermo/photo-acoustic tomography

Agranovsky, Ambartsoumian, Ammari, Arridge, Bal, Burgholzer,Cox, Finch, Haltmeier, Hristova, Kang, Kuchment, Kunyansky,Nguyen, Palamodov, Patch, Quinto, Rakesh, Scherzer, Stefanov,Uhlmann, Wang, Xu, ...

Recovering f (x) from its restricted spherical means MS f (p, r).Standard issues resolved recently: uniqueness, inversion, stability,range.








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The truth about TAT

If c(x) - sound speed, the model is:∂2u∂t2 = c2(x)∆u, in R3 × R+

u(0, x) = f (x), ∂u∂t (0, x) = 0

Inversion of the operator f 7→ g := u |S×R+

Reduces to spherical means for constant sound speed only.Related to spectral theory, transmission eigenvalues, spectralgeometry, number theory








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More about TAT

Reconstruction methods available:

FBP formulas - available in all dimensions, stable; work onlyfor constant speed, known for S - sphere only, do not workwhen f extends beyond S .

Eigenfunction expansions - all dimensions, stable,(theoretically) for variable speed, S -arbitrary, work when fcan extend beyond S ; probably unfeasible numerically forvariable speed.

Time reversal - all dimensions, stable, easy to implement,variable speed, any S , any location of f .








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Some TAT reconstructions








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Acousto Electric Tomography (AET)

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








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AET - reconstructions

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








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UOT

Allmaras, Bal, Bangerth, Dobson, Kuchment, Nam, Oraevsky,Schotland, Steinhauer, Uhlmann, Wang, ...A combination of OT with ultrasound irradiation, similarly to AET.








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Magnetic resonance elastography (MRE)

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








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MREIT, CDI

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








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Bibliography

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

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

L. Borcea, Electrical impedance tomography, Inverse Problems18 (2002), R99R136.

L. Ehrenpreis, Universality of the Radon Transform, 2003.








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Bibliography-2

C. Epstein, Introduction to the Math of Medical Imaging,Prentice Hall 2003T. Freeman, The Mathematics of Medical Imaging, Springer2010.I.M. Gelfand, S.G. Gindikin, M.I. Graev, Selected topics inintegral geometry, AMS 2003.S. Helgason, The Radon transform, Boston : Birkhauser,1980.S. Helgason, Integral Geometry and Radon Transforms, 301p,Springer, First Edition 2010.








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Bibliography-3

P. Kuchment, L. Kunyansky, Mathematics of thermoacoustictomography, European J. Appl. Math. 19(2008), 191-224.P. Kuchment, L. Kunyansky, 2D and 3D reconstructions inacousto-electric tomography, Inverse Problems 27 (2011),055013Mathematics and physics of emerging biomedical imaging,National Academy Press, 1996F. Natterer, The mathematics of computerized tomography,SIAM 2001.F. Natterer and F. Wbbeling, Mathematical methods in imagereconstruction, SIAM 2001.Peter Kuchment Inverse Problems of Tomographic Type







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Bibliography - 4

G. Olafsson, E. T. Quinto (Eds.), The Radon transform,inverse problems, and tomography, AMS 2006

V. Palamodov, Reconstructive integral geometry, Birkhauser2004. Univ. Press 2003.

G. Uhlmann (Editor), Inside out: inverse problems andapplications, MSRI Publ., Cambridge 2003.

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

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


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