RADON TRANSFORM, SCATTERING, AND IMAGING

RADON TRANSFORM,SCATTERING, AND IMAGING

J. Hector Morales Barcenas

[email protected]

Departamento de Matematicas

Universidad Autonoma Metropolitana

Unidad Iztapalapa, Ciudad de Mexico

Seminario Control en Tiempos de Crisis 2020

22 de junio de 2020

J. Hector Morales Barcenas

Seminario Control en Tiempos de Crisis 2020 [1/60]

Itinerary

1. Preamble: Applying the Fourier Transform

Solving the 3D Structure of the DNA

2. X-Ray Tomography

Computed Tomography and the Radon Transform

3. Wave-Based Scattering Models

Reconstructive Tomography with Diffracting Wavefields

J. Hector Morales Barcenas UAM Iztapalapa, Mexico


PART I

A Prelude: Structure of DNA



Solved Molecular Structure of Nucleic Acids

• “This structure has novel features which are of considerable biological

interest.”

Figure 1: J. D. Watson and F. H. C. Crick, April 1953 Nature, showing Photo 51.

• “We have also been stimulated by a knowledge of the general nature of

the unplublished experimental results and ideas of Dr. M. H. F. Wilkins,

Dr. R. E. Franklin and their co-workers at King’s College, London.”



Solved Molecular Structure of Nucleic Acids

Figure 2: Rosalind E. Franklin, 1950.

• Interpretation of theX-Ray Photograph

“For a smooth single-strand helixstructure factor on the nth layer lineis given by Fn = Jn(2πR) exp in(ψ+π/2) . . . ”

• Diffraction by Helices

“. . . the intensity distribution inthe diffraction pattern of aseries of points equally spacedalong a helix is given bythe squares of Bessel functions.”

• Rosalind E. Franklin and R. G. Gosling, Molecular Configuration in SodiumThymonucleate, Nature, April 25, 1953.



Inverse Problem solved by Cochran et al., 1952

• Uniform helix of infinite length, radius r and axial spacing P , defined by

(x, y, z) = (r cos(2πz/P ), r sin(2πz/P ), z),

T (ξ, η, ζ) =

ê2πi(xξ+yη+zζ)dV =

ˆ P

0

e2πi[rξ cos(2π zP )+rη sin(2π zP )+zζ]dz.

• In reciprocal coordinates with R2 = ξ2 + η2, tanψ = η/ξ, and ζ = n/P with n

integer

T (R,ψ, n/P ) =

ˆ P

0

e2πi[Rr cos(2πz/P−ψ)+nz/P ]dz.



Inverse Problem solved by Cochran et al., 1952

• With the help of the integral´ 2π

0exp(iX cosϕ) exp(inϕ)dϕ = 2πinJn(X),

with X = 2πRr and ϕ = 2πz/P . The result is

T (R,ψ, n/P ) = Jn(2πRr) exp[in(ψ + π/2)],

where Jn denotes the nth-order Bessel function.

• “The formulæ (for scattering patterns) are given for the Fourier transforms of anumber of helical structures . . . ”

Figure 3: W. Cochran, F. H. C. Crick and V. Vand. The Structure of Synthetic Polypeptides. I.

The Transform of Atoms on a Helix. Acta Cryst. (1952). 5, 581.



Crystallography in a Nutshell

• The relationship between the diffraction pattern (reciprocal space) and thecrystal structure (direct space) is mediated by a Fourier transform representedby the electron density function:

real o direct space︷︸︸︷ρ(x, y, z)︸︷︷︸

electronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic densityelectronic density

=1

V

∑h,k,`

amplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudesamplitudes︷︸︸︷F (h, k, `) exp(−2πi[hx+ ky + `z −

phasesphasesphasesphasesphasesphasesphasesphasesphasesphasesphasesphasesphasesphasesphasesphasesphases︷︸︸︷φ(h, k, `)])︸︷︷︸

reciprocal space: structure factors

,

where V is the volume of the crystals unit cell.

• The main contribution of the diffracted X-rays is that the intensity pattern inthe reciprocal space is proportional to the square of the amplitude

I(h, k, `) ∝ |F (h, k, `)|2.

There are physical factors that influence this intensity. However, a majorproblem is to determine the phases.




• The very first solution to the phase problem was introduced by Arthur Lindo

Patterson (1902-1966). After his training under Norbert Wiener, on Fourier

transforms convolution, Patterson introduced a new function P (u, v, w) in 1934,

which defines a new space (the Patterson space).

• To obtain Patterson’s function we cross-correlate the density function at two

different points over a unit cell

1

0

ρ(x, y, z)ρ(x+ u, y + v, z + w)dxdydz.

For continuous functions, the cross-correlation operator is the adjoint of the

convolution operator. After substituting the density function we get

P (u, v, w) =1

V

∑h,k,`

|F (h, k, `)|2 cos(2π[hu+ kv + `w]).




(a) (b)

Figure 4: (a) Two unit cells of a structure containing four atoms. (b) The Patterson map. The

positions at these maxima (u, v, w) represent the differences between the coordinates of each pair

of atoms in the crystal, i.e. u = x1 − x2, v = y1 − y2, w = z1 − z2 (Sands, 1975).

• The Patterson Space can be considered as the most important singledevelopment in crystal-structure analysis since the discovery of X-rays byRontgen in 1895 or X-ray diffraction by Laue in 1914.



PART II

X-Ray Tomography



X-Ray Tomography

• CT is the prime example of medical imaging.

• It comprises different techniques for (tomographic) imaging 2D

cross sections of 3D objects.

Applied and/or developed in

• X-Ray Tomography, Cormack (1963) & Hounsfield (1971).

• Electronic Microscopy, DeRosier y Klug (1968).

• Radioastronomy, Bracewell (1956).



In 2017 we celebrated a 100 Years of Radon Transform

Figure 5: Berichteder Sachsischen Akadamieder Wissenschaft (69), 262-277, 1917, Johann Radon (1887-1956)



Determination of a Point Function in the Plane fromits Straight Line Integral Values

• Radon introduced the abstract problem to determine functions from their

integral values along certain manifolds.

Theorem 1 (Radon). Consider a continuous function f equal to zero outside of

a disc D ∈ R2. The straight line integral value of f along the line L having the

equation x1 cosϕ+ x2 sinϕ = s is given by

P (s, ϕ) = P (−s, ϕ+ π) =

ˆRf(s cosϕ− r sinϕ, s sinϕ+ r cosϕ)dr

and exists almost everywhere.

Theorem 2 (Radon). The value of f is uniquely determined from P and can be

calculated as

f(p) = −1

π

∞0

dPp(q)

q.



Determination of a Point Function in the Plane fromits Straight Line Integral Values

• This is not all about imaging, but among other things it is the starting point

for developing numerical algorithms.

• In practical inversion, it’s not always possible to measure data for all directions,

that it’s one of the basic assumptions made by Radon, leading us to questions

of sampling, uniqueness and stability.

• Nonetheless, output signaling in CT, Ultrasound, MRI, etc., can be cast in

terms of a Generalized Radon Transform (GRT).



X-Ray Tomography: Beer’s Law

D

detector

θ

L

emitter

θ⊥

s

If I(x) is the number of the X-ray

photons in the beam when it arrives

at x, then the intensity in a small

segment of length ∆x is decreased in

proportion to −∆I ≈ f(x)∆xI(x).

It follows the integral transform

Rf :=

ˆL

f(x)dxL = − ln

(I(emitter)

I(detector)

)

We call R is the Radon transform, Rf the projection and dxL is the arc-length

measured on the straight line L.



X-Ray Tomography (polar coordinates)

ϕ

x1

r

x2

(x1, x2) = x = (s cosϕ− r sinϕ, s sinϕ+ r cosϕ) = sθ + rθ⊥

s

ϕ

θ ∈ S1 ⊂ R2

θ = (cosϕ, sinϕ), θ⊥ = (− sinϕ, cosϕ)

L(ϕ, s) := x ∈ R2 |x · θ = s > 0

s− x1 cosϕ− x2 sinϕ = s− x · θ = 0

• We define the Radon transform of a function f ∈ L1(D ⊂ R2) to be

(Rf)(θ(ϕ), s) =

ˆRf(sθ + rθ⊥)dr =

ˆR2f(x)δ(s− x · θ)dx.

It is a continuous map from L1(D ⊂ R2) to L1([0, 2π]× R) and δ denotes the

Dirac distribution.



X-Ray Tomography (numerical examples)

Figure 6: Left: A ‘blob’. Right: The Radon transform or sinogram. 360 view, 5 resolution.

Figure 7: Radon transform of Shepp-Logan phantom. 360 view, 1 resolution.



X-Ray Tomography

Theorem 3 (Projection Theorem or Fourier Slice Theorem).

F [Rf ](θ, σ) =√

2πF (σθ), where σ is the conjugate for s for fixed θ.

Dem. 1.

1√

2πF [Rθf ](σ) =

ˆR1

(ˆL

f(x1, x2)dx1dx2

)exp(−iσs)ds

=

ˆ +∞

s=−∞

ˆ +∞

r=−∞f(sθ + rθ

⊥) exp(−iσs)drds

=

ˆR1

ˆR1f(x1, x2) exp(−iσx · θ)dx1dx2

=

¨R2f(x) exp(−ik · x)dx, with k = σθ. 2

Where we have used the definitions in RN :

• F [f ](k) = F (k) := (2π)−N/2´RN f(x) exp(−ik · x)dx (direct)

• F−1[F ](x) = f(x) := (2π)−N/2´RN F (k) exp(ik · x)dk (inverse)



X-Ray Tomography Inversion

• Based on the Fourier Slice Theorem, we find the inversion formula for the

Radon transform.

• We write the FT in polar coordinates k = σθ to get

f(x) = (2π)−1

ˆR2F (k)eik·xdk = (2π)−1

ˆS1

ˆ ∞0

F (σθ)eiσθ·xσdσdθ.

• From the Theorem we replace F :

f(x) = (2π)−3/2

ˆS1

ˆR

F [Rf ](θ, σ)eiσθ·x|σ|dσdθ.




• Given the data g = Rf on the cylinder S1 × R, the function f is computed

as back-projection; i. e., taking the average values of g over those lines that

exactly pass through the point x ∈ L, i.e., s = x · θ:

f(x) =1

4π

ˆS1g(θ,x · θ)dθ.

Proposition 1 (Adjoint). Consider a continuos function f , equal zero outside

some disc ⊂ R2, and an integrable function g; then

(Rf, g) = (f,R∗g).

• Where we define the bilinear form (inner product) for f, g ∈ S (RN),

such that (f, g) =´RN

´RN f(x)g(x)d2x, and in polar coordinates (f, g) =´

S1

´R f(θ, s)g(θ, s)dsdθ.




Dem. 2.

(Rf, g) =

ˆS1

ˆR(Rf)(θ, s)g(θ, s)dsdθ

=

ˆS1

ˆR

ˆR2f(x)δ(s− x · θ)g(θ, s)dxdsdθ

=

ˆS1

ˆR2f(x)

ˆRg(θ, s)δ(s− x · θ)dsdxdθ

=

ˆS1

ˆR2f(x)g(θ,x · θ)dxdθ

=

ˆR2f(x)

ˆS1g(θ,x · θ)dθdx

=

ˆR2f(x)(R

∗g)(x)dx = (f,R

∗g). 2

• Therefore, the operator R∗ defines the adjoint and an inverse of the Radontransform:

(R∗g)(x) =1

4π

ˆS1g(θ,x · θ)dθ.

But surprisingly, it doesn’t properly reconstruct f , though!




• The operator R∗ is called back-projector since it throws the projections backinto the image space where f lies.

Remark 1. The reconstruction is non-local!

• To show this fact, let’s investigate the composition R∗Rf :

(R∗Rf)(x) =

ˆS1

(Rf)(θ,x · θ)dθ =

ˆS1

ˆRf(sθ + rθ⊥)drdθ.

• Let’s split the inner integral as follows:


(ˆ x·θ

−∞+

ˆ +∞

x·θ

)f((x · θ)θ + rθ⊥)dr

=

(ˆ 0

−∞+

ˆ +∞

0

)f((x · θ)θ + (t+ x · θ)θ⊥)dt,

where we’ve substituted r = t+ x · θ⊥.




• We rearrange the argument of the integrand to obtain


(ˆ 0

−∞+

ˆ +∞

0

)f(x+ tθ⊥)dt.

• We applied another transformation y = x+ tθ⊥, taking into account the proper

sign for t and the corresponding limits on the two integrals, and finally we

integrate over S1 to get

(R∗Rf)(x) = 2

ˆR2

f(y)

|y − x|dy =

2

|x|? f(x);

i.e., back-projecting the data Rf means that we convolve f with a singular

function |x|−1.

• It is a blurred (‘starry’) version of the original function!




(a) (b) (c)

Figure 8: G. B. Saha, Basics of PET Imaging, Springer, 2005.

(a) Acquired data are backprojected for image reconstruction at three projection

angles (at 120 angle).

(b) When many views are obtained, the reconstructed image represents the activity

distribution with “hot” spots, but the activity is still smeared around the spots.

(c) Blurring effect described by 1/r function, where r is the distance away from

the central point.




Remark 2. Radon transform is a linear operator that maps function from R2 to

the cylinder S1 ×R, but it is not necessarily defined on L2(R2). Actually, R is

a smoothing operator.

• To be more precise with the smoothness properties of R∗, let’s take a look to

the back-projection or the adjoint:

f(x) = (R∗g)(x) =1

4π

ˆS1g(θ,x · θ)dθ

=1

(2π)3/2

ˆS1

ˆR

F [Rf ](θ, σ)eiσθ·x|σ|dσdθ.

What is the meaning of the term |σ|?

• For fixed θ, σ, s are conjugate variables in Fourier space, and σ has the same

units of the scalar wave number k (spatial frequency).




• In mathematical terms, this factor comes out from the FT

F [g](θ, σ) = G(θ, σ) = |σ|F (Rf)(θ, σ).

Therefore, G is the FT of the product of F (Rf) and |σ| and therefore |σ| acts

as a filter function F [H](σ) = |σ|.

• In imaging processing we define the ramp filter or Riesz potential, I−1, by

I−1g = F−1(|σ|Fg) = F−1G for g ∈ C∞c (S1 × R).

• G(θ, σ) is called the filtered projection and it is the product of two Fourier

transforms, therefore g(θ, s) we can express it as a convolution:

G(θ, σ) = F [H](σ)F [Rf ](θ, σ).

g(θ, s) =

ˆ +∞

−∞(Rf)(θ, s′)h(s− s′)ds′.




• If we use a finite bandwidth |σ| ≤ B we can obtain

h(s) =1

2π

ˆ B

−B|σ|e−iσsdσ =

B2

π

[sinBs

Bs− 1

2

(sin(Bs/2)

Bs/2

)2],

where B is the highest spatial frequency.

• In brief, once we calculate I−1g = F−1G we applied the adjoint R∗ and the

object is reconstructed.

• Therefore, we speak of the entire process as the back-projection of the filtered

projection.

• The theorem follows . . .




Theorem 4 (Filtered Backprojection (FBP)).

Let f ∈ C∞c (R2), then f = 14πR

∗(I−1Rf)(x).

Dem. 3. In polar coordinates in k plane, with the scaling k = σθ and dk = σdθ

2πf(x) =

ˆR2eik·x

F (k)dk =

ˆ 2π

ϕ=0

ˆ +∞

σ=0

eiσθ·x

F (σθ)σdσdϕ

= (2π)−1/2

ˆ 2π

ϕ=0

ˆ +∞

σ=0

eiσθ·x

(Rθf)(σ)σdσdϕ

= (2π)−1/2

ˆ 2π

ϕ=0

ˆ +∞

σ=0

eiσθ·x

σG(θ, σ)dσdϕ

=1

2(2π)

−1/2ˆ 2π

ϕ=0

(ˆReiσθ·x|σ|G(θ, σ)dσ

)dϕ

=1

2

ˆ 2π

ϕ=0

(I−1g)(θ(ϕ), θ(ϕ) · x)dϕ

=1

2(R∗I−1

)g(x) =1

2R∗(I−1Rf)(x). 2



Forward and Inverse Radon Transform

Figure 9: A ‘blob’. R∗R turns a blob into a “starry” version in the reconstructed image.

Figure 10: Shepp-Logan phantom. Original, unfiltered backprojection, filtered backprojection.



X-Ray Tomography Pseudo-Inversion

• Actually, this filter resembles a bit a differential operator in the sense that

F [(H H)f ](σ) = |σ|2F (σ).

It turns out that |σ|2 is the (top-order) symbol of the Laplace operator −∆.

The reason is that FT replaces derivatives for multiplications.

• In the inversion formula the ‘annoying’ factor |σ| means that we differentiate

the data twice, then “change the sign and finally take the square root”.



X-Ray Tomography Pseudo-Inversion

• In that way we don’t directly compute f but Λf which is defined through

F (Λf)(σ) = |σ|F (σ),

or Λf = (−∆)−1/2f .

• The same result is found in RN by

Λf(x) = cN∆N

ˆSN−1

(Rf)(θ,x · θ)dθ.

A deep mathematical result states that this pseudo-differential operator Λ

preserves the singular support, i. e. Λf has jumps whenever f has jumps

(Louis, 1992).

§



X-Ray Tomography Inversion via Hilbert Transform

Theorem 5 (Inversion Formula RN). (Louis, 1992). Given Rf on the cylinder

SN−1 × R the function f is computed as back-projection

f(x) = 2−1(2π)1−NˆSN−1

g(θ,x · θ)dθ

of the function g given as

G(θ, σ) = |σ|N−1F [Rf ](θ, σ).

• If we avoid the use of the FT in g we note that, for odd N we have

|σ|N−1 = σN−1, and the multiplication of the FT with its argument correspond

to differentiation.

• In even dimensions we have |σ|N−1 = sign(σ)σN−1.



X-Ray Tomography Inversion via Hilbert Transform

• The multiplication with the sign is the Hilbert transform

H [φ](s) =1

π

φ(t)

s− tdt.

This gives the reformulation of the inversion formula.

Lemma 1.

f(x) =1

2(2π)1−N

ˆSN−1

g(θ,x · θ)dθ,

where

g(θ, s) =

(−1)N/2−1 ∂N−1

∂sN−1g(θ, s) N odd,

(−1)(N−1)/2H ∂N−1

∂sN−1g(θ, s) N even.



X-Ray Tomography: Regularization

• We can generalize the FBP in terms of the filter I−α in R2, such that

F [I−αf ](σ) = |σ|−αF (σ) (Natterer, 2001):

f =1

4πI−αR∗Iα−1g.

• For α > 0, the image is smoothed (high frequencies suppressed) and high

frequencies in the data g are sharpened.

• For α = 1, f = (4π)−1I−1(R∗R)f , so (R∗R)f = 4πI1f , for which we can

foresee the Tikhonov formula

fµ = (R∗R+ µ2)−1R∗g, or fµ = R∗(RR∗ + µ2)−1g.



X-Ray Tomography: PET, SPECT and beyond

• Single-photon emission computed tomography (SPECT) is a nuclear medicine

tomographic imaging technique using gamma rays:

I =

ˆL

f(x) exp

(−ˆL(x)

µ(y)dy

)dx.

• Positron emission tomography (PET) is an imaging technique that uses

radioactive substances to visualize and measure metabolic processes in the

body:

I =

ˆL

f(x) exp

(−ˆL+(x)

µ(y)dy −ˆL−(x)

µ(y)dy

)dx.

• The attenuation coefficient µ depends on the energy E of the X-rays:

Ie

Id=

ˆT (E) exp

(−ˆL

µ(x, E)dx

)dE,

where T (E) is the energy spectrum of the source (Natterer, 2001).



X-Ray Tomography

• The Nobel Prize in Physiology or Medicine 1979:

“for the development of computer assisted tomography”.

EARLY TWO-DIMENSIONALRECONSTRUCTIONANDRECENT TOPICS STEMMING FROM IT

Nobel Lecture, 8 December, 1979

byALLAN M . CORMACKPhysics Department, Tufts University, Medford, Mass., U.S.A.

In 1955 I was a Lecturer in Physics at the University of Cape Town whenthe Hospital Physicist at the Groote Schuur Hospital resigned. SouthAfrican law required that a properly qualified physicist supervise the useof any radioactive isotopes, and since I was the only nuclear physicist in

Cape Town, I was asked to spend 1 1 / 2 days a week at the hospitalattending to the use of isotopes, and I did so for the first half of 1956. I wasplaced in the Radiology Department under Dr. J. Muir Grieve, and in thecourse of my work I observed the planning of radiotherapy treatments. Agirl would superpose isodose charts and come up with isodose contourswhich the physician would then examine and adjust, and the processwould be repeated until a satisfactory dose-distribution was found. Theisodose charts were for homogeneous materials, and it occurred to me thatsince the human body is quite inhomogeneous these results would be quitedistorted by the inhomogeneities - a fact that physicians were, of course,well aware of. It occurred to me that in order to improve treatmentplanning one had to know the distribution of the attenuation coefficient oftissues in the body, and that this distribution had to be found by measure-ments made external to the body. It soon occurred to me that this informa-tion would be useful for diagnostic purposes and would constitute atomogram or ser ies of tomograms , though I did not learn the word“tomogram” for many years.

At that time the exponential attenuation of X- and gamma-rays had beenknown and used for over sixty years with parallel sided homogeneous slabsof material. I assumed that the generalization to inhomogeneous materialshad been made in those sixty years, but a search of the pertinent literaturedid not reveal that it had been done, so I was forced to look at the problemab initio. It was immediately evident that the problem was a mathematicalone which can be seen from Fig. 1. If a fine beam of gamma-rays ofintensity I, is incident on the body and the emerging intensity is I, then themeasurable quantity g = In(I0/ I) = SLfds, where f is the variable absorp-

tion coefficient along the line L. Hence if f is a function in two dimensions,and g is known for all lines intersecting the body, the question is: “Can f bedetermined if g is known ?“. Again this seemed like a problem which would

55 I

COMPUTED MEDICAL IMAGING

Nobel Lecture, 8 December, 1979

BYGODFREY N . HOUNSFIELDThe Medical Systems Department of Central Research Laboratories EMI,London, England

In preparing this paper I realised that I would be speaking to a generalaudience and have therefore included a description of computed tomo-graphy (CT) and some of my early experiments that led up to the develop-ment of the new technique. I have concluded with an overall picture of theCT scene and of projected developments in both CT and other types ofsystems, such as Nuclear Magnetic Resonance (NMR).

Although it is barely 8 years since the first brain scanner was construct-ed, computed tomography is now relatively widely used and has beenextensively demonstrated. At the present time this new system is operatingin some 1000 hospitals throughout the world. The technique has succesful-ly overcome many of the limitations which are inherent in conventional X-ray technology.

When we consider the capabilities of conventional X-ray methods, threemain limitations become obvious. Firstly, it is impossible to display withinthe framework of a two-dimensional X-ray picture all the informationcontained in the three-dimensional scene under view. Objects situated indepth, i. e. in the third dimension, superimpose, causing confusion to theviewer.

Secondly, conventional X-rays cannot distinguish between soft tissues. Ingeneral, a radiogram differentiates only between bone and air, as in thelungs. Variations in soft tissues such as the liver and pancreas are notdiscernible at all and certain other organs may be rendered visible onlythrough the use of radio-opaque dyes.

Thirdly, when conventional X-ray methods are used, it is not possible tomeasure in a quantitative way the separate densities of the individualsubstances through which the X-ray has passed. The radiogram recordsthe mean absorption by all the various tissues which the X-ray has penetrat-ed. This is of little use for quantitative measurement.

Computed tomography, on the other hand, measures the attenuation ofX-ray beams passing through sections of the body from hundreds ofdifferent angles, and then , from the evidence of these measurements, a

computer is able to reconstruct pictures of the body’s interior.Pictures are based on the separate examination of a series of contiguous

cross sections, as though we looked at the body separated into a series ofthin “slices”. By doing so, we virtually obtain total three-dimensional infor-

mation about the body.

568



X-Ray Tomography

• Despite major advances in X-ray sources, detector arrays, gantry mechanical

design and especially computer performance, one component of CT scanners

has remained virtually constant for the past 25 years—the reconstruction

algorithm1.

• CT is a global business with several major manufacturers and many minor

providers, especially of niche systems. Worldwide sales of CT scanners is more

than $2.3 billion per year, despite economic slowdown.

• “Global Market for Medical Imaging Equipment Worth $11.4 Billion by

2012 Says BCC Research Study.”

http://www.businesswire.com/news/home/20071030006295/en

1X. Pan, E. Y. Sidky and M. Vannier. Why do commercial CT scaners still employ traditional, filtered back-projection for imagereconstruction? Inverse Problems 25 (2009) 123009 (36pp)



PART III

Wave-Based Scattering Models



Wave-Based Scattering Models

• In diffraction tomography the goal is to form an image of the interior of a

body from the echo of a wavefield.

• An image is a mapping of the locus of the singularities of the wave speed.

Figure 11: Typical experiment of travel-time tomography (Tarantola, 2005).



Seismic Imaging

It is the process through which Earth’s interior properties are mapped

into seismograms recorded on its surface.

Figure 12: Seismogram showing accompanying ray paths (Stein and Wysession, 2003).

• Seismograms are representations of PDE’s solutions of scattering problems in

variable-parameter elastic medium.



Synthetic-Aperture Radar Imaging

Teledetection and Monitoring: Ocean waves, soil humidity, forest ecology,

planets’ cartography, etc.

~γ(s)

x1

x2

h

x3

~xv

εT

• Moving antenna transmitting and collecting electromagnetic pulses.

• Synthesis of the antenna aperture (measurements).

• Signal processing for imaging (inverse problem).



Acoustic Scattering Models

• We are given the wave number k ∈ C\0 with Re k ≥ 0, Im k ≥ 0, a

bounded region Ω ⊂ RN with bdry ∂Ω ∈ C 2, and solution (incident wave) uI

of the Helmholtz equation

∆u+ k2u = 0

in a region containing Ω in its interior. We assume that RN\Ω is connected.

Problem 1 (Direct Scattering Problem). Determine u ∈ C 2(RN\Ω) ∩ C 2(Ω) ∩C 1(RN) with

(a) ∆u+ k2n(x)u = 0 in RN with n = n(x) ∈ C2(RN) such that the support of

n− 1 is contained in Ω (penetrable scatterer).

(b) uS = u − uI uniformly satisfies, with respect to x = x/|x|, Sommerfeld’s

radiation condition

lim|x|→+∞

|x|(N−1)/2

(∂uS(x)

∂|x|− ikuS(x)

)= 0.




• From the Representation Theorem (fundamental solution), one derives the

Lippmann-Schwinger integral equation for the total wavefield u:

u(x) = uI(x) + k2

¨Ω

(1− n(y))u(y)g(x,y)dy, x ∈ Ω.

This is a Fredholm integral equation of the second kind.

Problem 2 (Inverse Scattering Problem (ISP)). To find an obstacle, characterized

by its index of refraction n(x), from partial knowledge of data uS(x, y; k).

• The scattered, radiating or outgoing wavefield uS is the measured echo in x

after the monochromatic waves (fixed k) are scattered from the direction y.

• We are interested in weak singularities of the wave speed c(x) represented by

a small perturbation to the index of refraction n(x) = 1 + T (x).




• This function T plays the role of a potential (Schrodinger Eqn.), a reflectivity

(Radar imaging), an obstacle (ultrasound tomography), an attenuation

coefficient (CT tomography):

k2n(x) =ω2

c20n(x) = k2(1 + T (x)),

where c0 is the background wave speed.

• We will transform the Lippmann-Schwinger equation as follows:

1. Decompose the total field u = uI + uS.

2. Substitute n(x) = 1 + T (x).

3. Use the far-field approximation, |x − y| = |x| − x · y + O(|x|−1), where x

is an unitary vector that denotes the direction of the incident wavefield.

4. Linearize using the Born approximation: |TuS| |TuI|.




• Rearranging the expression in terms of new variables for sources and receivers.

Aditionally, all wave variables depend on the frequency ω:

uS(xg,xs, ω, T ) =ω2

c20

ˆΩ

T (y)uI(y,xs, ω)g(y,xg, ω)dy.

xs

source

xg1xg2

receivers

F (ω)

n

Ω

Figure 13: Geometry of one source xs and multiple receiver positions xgi.

• Now we reshape this formula introducing Kirchhoff Approximation to represent

the upward scattered data from a single reflector T .




• By definition, g is the fundamental solution and satisfies

(∆ + ω2/c20)g(y,xg, ω) = −δ(y − xg),

therefore uI(y,xs, ω) = F (ω)g(y,xs, ω) and g is given by its WKBJ

approximation (geometrical optics):

g(x0,xs, ω) = A(x0,xs)eiωτS(x0,xs),

with τS being the travel-time in heterogeneous medium, and A being the

corresponding WKBJ amplitude (rays).

• We finally get

uS(xg,xs, ω, T ) ≈ ω2

c20F (ω)

ˆΩ

T (y)a(y, ξ)eiωΦ(y,ξ)dydξ,

where we introduced a phase function Φ(y, ξ) = τ(y,xs(ξ)) + τ(xg(ξ),y) and

an amplitude a(y, ξ) = A(y,xs(ξ))A(xg(ξ),y).




• To solve the inverse scattering problem means that we are able to estimate the

reflector T .

• We denote by T our model for the inverse imaging map. It is suggested by

the expression

TQ(x) =

¨Q(x, ξ)e−iωΦ(x,ξ)uS(xg,xs, ω, T )dωdξ,

where the filter Q will become a regularization term.

• The complex-conjugate phase recalls the adjoint of uS.

• Substituting uS into this expression we obtain

TQ(x) =

˚ω2

c20F (ω)Q(x, ξ)eiω[Φ(y,ξ)−Φ(x,ξ)]T (y)a(y, ξ)dωdξdy




• Key observation: To get the reflector’s maximum fidelity, it is desirable thatthe image map must have the sift property of a Dirac delta, such that

TQ(x) ∼ˆδ(y − x)T (y)dΩ ≡

êik·(y−x)T (y)dΩ;

in other words, we must be able to get an asymptotic representation

δ(y − x) ∼¨

ω2

c20F (ω)Q(x, ξ)eiω[Φ(y,ξ)−Φ(x,ξ)]a(y, ξ)dωdξ.

• For this purpose, we linearize the phase at the critical dominant points wherey = x (stationary phase approximation). We obtain

iω[Φ(y, ξ)− Φ(x, ξ)] ≈ ik · (y − x),

where k := ω∇yΦ(y, ξ)|y=x is the Stolt change of variables (ω, ξ)→ k.




• Therefore, in terms of variable k, we have

δ(y − x) ∼ˆQ(x, ξ)

ω2(k)

c20F (ω(k))a(x, ξ)J(ω, ξ,k)eik·(y−x)dk,

where ω(k) = k · ∇xΦ(x, ξ)/|∇xΦ(x, ξ)|2 and J(ω, ξ,k) = |∂(ω, ξ)/∂(k)| is

the Beylkin determinant.

• With a suitable choice of Q, we can rearrange the inversion formula in terms of

uS to obtain

TQ(x) =

ë−ik·(y−x)P (x, ξ)uS(xg,xs, ω, T )dxdk,

where P = 8π3Q(x, ξ) and Q = c20/ω2(k)F (ω(k))a(x), ξ)J(ω, ξ,k).

THIS IS NOT A FOURIER TRANSFORM OF uS, THIS IS A ΨDO



Microlocal Analysis and Inversion

• The singularities, T and f , are mapped into the forward data by

pseudodifferential operators, ΨDOs, such as uS = A[T ] and g = R[f ].

• ΨDOs can be explained in the case of constant coefficient differential operators

on Euclidean space.

• Constant coefficient differential operators can be diagonalized by the Fourier

transforms into a multiplication of polynomials.

• ΨDOs are generalizations of the Fourier transforms since we multiply by more

general functions.

• We estimate T or f from an inverse mapping (adjoint) of A or R, with the help

of microlocal analysis in phase space (of distributions).




What are the singularities of T (x) or f(x)?

• Physically, they correspond to changes in the material composition and its

geometry: cracks, spikes, peaks, edges, wedges, etc.

• They have both location and orientation in phase space (x,k) or (s, σθ).

• How to quantify its smoothness? Let α ≥ 0, we define the scale of Hilbert

spaces Hα(RN) as the space of measurable functions f(x) such that

‖f‖2Hs(RN) =

ˆRN

(1 + |k|2)α|Ff |2(k)dk <∞.

• The parameter α characterizes the degree of smoothness of a function f(x).

The larger α, the smoother the function f ∈ Hα(RN), and the faster the decay

of its Fourier transform Ff(k).




• Therefore, L2(R2, (1 + |k|)2)α) integrability or rapid decrease of Ff , its

equivalent to smoothness of f , or L2 derivatives (Sobolev).

• Localize in Fourier space: Multiply f by a cutt-off function ψ ∈ C∞c (R2):

F (ψf)(k) =1

2π

ˆx∈R2

e−ik·xψ(x)f(x)dx.

Singular support: f is not smooth at x0 iff for every smooth cut-off ψ near

x0 (ψ(x0) 6= 0), the F (ψf) is not rapidly decreasing.

• Microlocalize: Find in the phase space the directions where F (ψf) is not

rapidly decreasing.

Def. Let x0 ∈ R2 and k0 ∈ R2\0. The function f is smooth at x0 in direction

k0 iff ∃ a cut-off function ψ near x0 such that F (ψf)(k) is rapidly decreasing

in some open cone Γ from the origin containing k0.

• On the other hand, (x0,k0) ∈ WF(f) iff f is not smooth at x0 in direction k0.




• Previous mappings for Radon transform, Rf , and scattering waves, uS(T ), are

both integral operators or, more precisely, Fourier Integral Operators (FIOs):

uS = A[T ] =

ê−iΦ(k,x,z)a(k, x, z)T (z)dz.

• These output signals uS are weighted projections of T , as well as the output

signal g (projections).

• For a large class of FIOs, we can model the Inverse Mapping:

A†Q[uS](z) =

êiΦ(k′,x′,z)Q(k′, x′, z)uS(k′, x′, T )dk′dx′,

that it’s the adjoint of A in L2 with a regularization filter Q.




• So, A†Q[uS] is an inverse approximation T of the reflectivity T , in the sense that

T = A†Q[uS] = A†Q[A(T )] = (A†Q A)[T ].

Ideal fidelity means that A†Q A ≡ I when its kernel is a Dirac delta, otherwise

it is desirable that it behaves like one

T ∼ˆδ(x− y)T (y)dΩy.

• Why? The Dirac delta provides us with an assertive statement of the

orthogonality property of the Fourier’s kernel

δ(x− y) =

ˆRne−2πik·(x−y)dk.

The analogy applies to the kernel of A A†Q.



Numerical Set-up of the Forward Problem

−100−80

−60−40

−200

2040

6080

100

−100−80

−60−40

−200

2040

6080

1000

5

10

T2T1

T2T1

T2T1

T2T1

T2T1

T2T1

T2

T1

T2

T1

T2

T1

T2

T1

T2

T1

T2

T1

T2

T1

h=

10

m

m m

r=100 m

γ(s)

(a) Scenario

4 4.1 4.2 4.3 4.4 4.5 4.6x 10−8

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (s)

Ampl

itude

(V/m

)

Transmitted Wavefield p(t)

(b) Square Sinusoid

Fast−Time (s)

Slow−T

ime

(rad)

Dispersive Material

0 1 2 3 4 5 6 7 8x 10−8

0

1

2

3

4

5

6

(c) Sinograms

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5x 10−8

−3

−2

−1

0

1

2

3

x 10−9

Fast−Time (s)

V/m

Brillouin Precursors

(d) Precursors



Numerical Simulations




• The FIO A†Q is called

Backprojection (X-Ray and Emission Tomography)

Migration Operator (Geophysics)

Matched Filter (Radar/Sonar, Ultrasound, Imaging Processing)



References

1. Editorial, “The first 100 years of the Radon transform”, Inv. Prob. 34 (2018) 090201 (4pp)

2. R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, World

Scientific, 2003.

3. A. K. Louis, “Medical imaging: state of the art and future development”, Inv. Prob. 8 (1992)

709-738.

4. F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction,

SIAM, 2001.

5. F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons,

1985.

6. E. T. Quinto, “An Introduction to X-ray tomography and Radon Transforms”, Proceedings of

Symposia in Applied Mathematics, (2005) AMS.

7. Y. Nievergelt, “Elementary inversion of Radon’s transform”, SIAM Review 28 (1) March 1986.

8. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering,

Prentice-Hall, 1991.

9. A. J. Devaney, Mathematical Foundations of Imaging Tomography and

Wavefield Inversion, Cambridge, 2012.

10. A. Tarantola, Inverse Problem Theory, SIAM, 2005.

11. L. Tenorio, An Introduction to Data Analysis and Uncertainty Quantification

for Inverse Problems, SIAM, 2017.

12. J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with

Practical Applications, SIAM, 2012.



13. W. Allison, Fundamental Physics for Probing and Imaging, Oxford, 2006.

14. Blahut, Theory of Remote Image Formation, Cambridge, 2004.

15. G. T. Herman (Ed.), Image Reconstruction from Projections Implementation

and Applications, Springer, 1979.

16. Varslot, T., M., J. H., and Cheney, M. Synthetic-aperture radar imaging through dispersive

media, Inverse Problems 26 (2010) 025008 (27 pages).

17. E. Candes, L. Demanet, and L. Ying, (2007), Fast Computation of Fourier Integral

Operators, SIAM, J. Sci. Comput., 29 (6) 2464-2493.

18. Bleistein, N., Cohen, J. K., and Stockwell, J. W. Jr., 2001, Mathematics

of Multidimensional Seismic Imaging, Migration, and Inversion, Springer-Verlag

New York.

19. Lamoureux, M. P. and Margrave, G. F., 2008, An Introduction to Numerical Methods of

Pseudodifferential Operators, in Pseudo-Differential Operators Quantization and

Signals, Lecture Notes in Mathematics 1949, Springer-Verlag, Berlin.

20. Scales, J. A., 1995, Theory of Seismic Imaging, Lecture Notes in Earth Sciences 55,

Springer, Berlin.

21. Isakov, V., 1997, Inverse Problems for Partial Differential Equations, Springer.

22. Stein, S., and Wysession, M., 2003, An Introduction to Seismology, Earthquakes,

and Earth Structure, Wiley-Blackwell.

23. Chapman, C. H., 2004, Fundamentals of seismic wave propagation, Cambridge.



Obrigado!

¡Gracias!

Thank you!


RADON TRANSFORM, SCATTERING, AND IMAGING

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