inverse problems of tomographic typeaipc.tamu.edu/pre/aip11_kuchment.pdf · dual radon radon...
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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
Literature
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
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
Literature
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
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
Literature
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
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
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
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
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
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
Examples of tomographic reconstructions
In medicine
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
Examples of tomographic reconstructions
Industry
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
Examples of tomographic reconstructions
Geophysics
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
CAT scan
CAT=Computer Assisted Tomography
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
Medical tomographic scanner inventors
A. Cormack and G. Hounsfield built the first computedtomography scanners in 1960s, won 1979 Nobel prize in medicine.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
CAT scan (= X ray tomography) procedure
The X-ray tomography (=CAT scan) procedure:
I0, I1 - initial and terminal intensities.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
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:
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
Radon transform in coordinates
Rf (ω, s) :=
∫x ·ω=s
f (x)dx , (ω, s) ∈ C := S1 × R
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
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
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Inverse problemsX-ray tomography = CAT scanRadon transformInvariances of the Radon transform
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transformProjection slice formulaDual Radon transform (backprojection)
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
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
Literature
Fourier transformProjection slice formulaDual Radon transform (backprojection)
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transformProjection slice formulaDual Radon transform (backprojection)
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
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transformProjection slice formulaDual Radon transform (backprojection)
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
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
Literature
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
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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)
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transform inversionFiltered backprojection inversionFourier series inversion (Cormack’s formulas)Range conditions
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).
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transform inversionFiltered backprojection inversionFourier series inversion (Cormack’s formulas)Range conditions
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 .
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transform inversionFiltered backprojection inversionFourier series inversion (Cormack’s formulas)Range conditions
An example of a Matlab FBP inversion
Phantom (left) and its reconstruction from 128 projections and128 detectors. USE SIMPLE, BUT REVEALING PHANTOMS!
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transform inversionFiltered backprojection inversionFourier series inversion (Cormack’s formulas)Range conditions
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?
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transform inversionFiltered backprojection inversionFourier series inversion (Cormack’s formulas)Range conditions
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
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transform inversionFiltered backprojection inversionFourier series inversion (Cormack’s formulas)Range conditions
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
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
Literature
Fourier transform inversionFiltered backprojection inversionFourier series inversion (Cormack’s formulas)Range conditions
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Fourier transform inversionFiltered backprojection inversionFourier series inversion (Cormack’s formulas)Range conditions
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 .
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Ill posed problemsStability of reconstruction
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Ill posed problemsStability of reconstruction
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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?
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Limited angle problemExterior problemInterior problemSingularity detection
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Limited angle problemExterior problemInterior problemSingularity detection
A limited angle reconstruction
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Limited angle problemExterior problemInterior problemSingularity detection
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Limited angle problemExterior problemInterior problemSingularity detection
Example of exterior reconstruction
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Limited angle problemExterior problemInterior problemSingularity detection
Interior problem
ROI is the interior of a domain:
Reconstruction is non-unique. But ....Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Limited angle problemExterior problemInterior problemSingularity detection
Example of interior reconstruction
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Limited angle problemExterior problemInterior problemSingularity detection
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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 .
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Local reconstruction example - full data
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Local reconstruction example - limited angle
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Local reconstruction example - exterior data
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Local reconstruction example - interior data
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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 ????
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Emission tomography and attenuated X-ray transformMRI
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
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Emission tomography and attenuated X-ray transformMRI
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)
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Emission tomography and attenuated X-ray transformMRI
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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
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
Literature
EIT - Pros and Contras
Great contrast
Low resolution (exponential instability)
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
OT
−∇ · σ(x)∇u + µu = 0 = 0 in Ω
Great contrast Low resolution (exponential instability)
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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!
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Thermo-/ photo-acoustic imaging and spherical mean operatorsHelping EIT: Acousto Electric Tomography, MREIT, CDIIHelping OT: Ultrasound Modulated Optical TomographyMagnetic resonance elastography (MRE)
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
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Thermo-/ photo-acoustic imaging and spherical mean operatorsHelping EIT: Acousto Electric Tomography, MREIT, CDIIHelping OT: Ultrasound Modulated Optical TomographyMagnetic resonance elastography (MRE)
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 .
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Thermo-/ photo-acoustic imaging and spherical mean operatorsHelping EIT: Acousto Electric Tomography, MREIT, CDIIHelping OT: Ultrasound Modulated Optical TomographyMagnetic resonance elastography (MRE)
Some TAT reconstructions
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Thermo-/ photo-acoustic imaging and spherical mean operatorsHelping EIT: Acousto Electric Tomography, MREIT, CDIIHelping OT: Ultrasound Modulated Optical TomographyMagnetic resonance elastography (MRE)
Acousto Electric Tomography (AET)
Ammari, Bal, Bonnetier, Capdebosque, Fink, Kang, Kuchment,Kunyansky, Scherzer, Steinhauer, Wang, Xu, ...
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Thermo-/ photo-acoustic imaging and spherical mean operatorsHelping EIT: Acousto Electric Tomography, MREIT, CDIIHelping OT: Ultrasound Modulated Optical TomographyMagnetic resonance elastography (MRE)
AET - reconstructions
Ammari, Bal, Bonnetier, Capdebosq, Fink, Kang, Kuchment,Kunyansky, Scherzer, Steinhauer, Wang, Xu, ...
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Thermo-/ photo-acoustic imaging and spherical mean operatorsHelping EIT: Acousto Electric Tomography, MREIT, CDIIHelping OT: Ultrasound Modulated Optical TomographyMagnetic resonance elastography (MRE)
UOT
Allmaras, Bal, Bangerth, Dobson, Kuchment, Nam, Oraevsky,Schotland, Steinhauer, Uhlmann, Wang, ...A combination of OT with ultrasound irradiation, similarly to AET.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Thermo-/ photo-acoustic imaging and spherical mean operatorsHelping EIT: Acousto Electric Tomography, MREIT, CDIIHelping OT: Ultrasound Modulated Optical TomographyMagnetic resonance elastography (MRE)
Magnetic resonance elastography (MRE)
Ehman, Manduca, J. McLaughlin, ...Using MRI data to recover mechanical properties of biologicaltissues (e.g., stiffness).
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
Thermo-/ photo-acoustic imaging and spherical mean operatorsHelping EIT: Acousto Electric Tomography, MREIT, CDIIHelping OT: Ultrasound Modulated Optical TomographyMagnetic resonance elastography (MRE)
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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.
Peter Kuchment Inverse Problems of Tomographic Type
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
Literature
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
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
Literature
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
Peter Kuchment Inverse Problems of Tomographic Type