New reconstructions from cone Radon transform Victor Palamodov Tel - - PowerPoint PPT Presentation

New reconstructions from cone Radon transform

Victor Palamodov

Tel Aviv University

March 30, 2017

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

1 / 27

Trajectories of single-scattered photons with …xed income and

• utcome energies in Compton camera form a cone of rotation:

D

scattering site

p detector plate

Scheme of the Compton camera

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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A spherical cone in an Euclidean space E 3 with apex at the origin can be written in the form C (λ) =

• x 2 E 3 : λx1 = s
• , s =

q x2

2 + x2 3 .

The line s = 0 is the axis and λ = tan ψ where ψ is the opening of the cone. In particular C (∞) = fx : x1 = 0g.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

3 / 27

A spherical cone in an Euclidean space E 3 with apex at the origin can be written in the form C (λ) =

• x 2 E 3 : λx1 = s
• , s =

q x2

2 + x2 3 .

The line s = 0 is the axis and λ = tan ψ where ψ is the opening of the cone. In particular C (∞) = fx : x1 = 0g. The integral gC (y) = cos ψ

Z

x2C (λ) f (y + x) w (x) dx2dx3, y 2 E 3

is called weighted cone Radon or Compton transform.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

3 / 27

A spherical cone in an Euclidean space E 3 with apex at the origin can be written in the form C (λ) =

• x 2 E 3 : λx1 = s
• , s =

q x2

2 + x2 3 .

The line s = 0 is the axis and λ = tan ψ where ψ is the opening of the cone. In particular C (∞) = fx : x1 = 0g. The integral gC (y) = cos ψ

Z

x2C (λ) f (y + x) w (x) dx2dx3, y 2 E 3

is called weighted cone Radon or Compton transform. If w (x) = jxjk we call this integral regular in the case k = 0, 1 and singular if k = 2.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

3 / 27

A spherical cone in an Euclidean space E 3 with apex at the origin can be written in the form C (λ) =

• x 2 E 3 : λx1 = s
• , s =

q x2

2 + x2 3 .

The line s = 0 is the axis and λ = tan ψ where ψ is the opening of the cone. In particular C (∞) = fx : x1 = 0g. The integral gC (y) = cos ψ

Z

x2C (λ) f (y + x) w (x) dx2dx3, y 2 E 3

is called weighted cone Radon or Compton transform. If w (x) = jxjk we call this integral regular in the case k = 0, 1 and singular if k = 2. Any regular integral is well de…ned for any continuous f de…ned on E 3 vanishing for x1 > m for some m.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

3 / 27

A spherical cone in an Euclidean space E 3 with apex at the origin can be written in the form C (λ) =

• x 2 E 3 : λx1 = s
• , s =

q x2

2 + x2 3 .

The line s = 0 is the axis and λ = tan ψ where ψ is the opening of the cone. In particular C (∞) = fx : x1 = 0g. The integral gC (y) = cos ψ

Z

x2C (λ) f (y + x) w (x) dx2dx3, y 2 E 3

is called weighted cone Radon or Compton transform. If w (x) = jxjk we call this integral regular in the case k = 0, 1 and singular if k = 2. Any regular integral is well de…ned for any continuous f de…ned on E 3 vanishing for x1 > m for some m. The singular integral is not well de…ned if f (y) 6= 0.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

3 / 27

A spherical cone in an Euclidean space E 3 with apex at the origin can be written in the form C (λ) =

• x 2 E 3 : λx1 = s
• , s =

q x2

2 + x2 3 .

The line s = 0 is the axis and λ = tan ψ where ψ is the opening of the cone. In particular C (∞) = fx : x1 = 0g. The integral gC (y) = cos ψ

Z

x2C (λ) f (y + x) w (x) dx2dx3, y 2 E 3

is called weighted cone Radon or Compton transform. If w (x) = jxjk we call this integral regular in the case k = 0, 1 and singular if k = 2. Any regular integral is well de…ned for any continuous f de…ned on E 3 vanishing for x1 > m for some m. The singular integral is not well de…ned if f (y) 6= 0. Analytic inversion of the regular and singular monochrome (one

• pening) cone Radon transforms is in the focus of this talk.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Single-scattering tomography

The realistic model (SPSF) for single-scattering optical tomography based on the photometric law of scattered radiation modeled by the singular cone transform.

S

D

incident photons scattered photons recoiled electrons

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Polychrome reconstructions

(many openings)

Cree and Bones 1994 proposed reconstruction formulae from data of regular cone transform with apices restricted to a plane orthogonal to the axis.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

5 / 27

Polychrome reconstructions

(many openings)

Cree and Bones 1994 proposed reconstruction formulae from data of regular cone transform with apices restricted to a plane orthogonal to the axis. Analytic reconstructions from the cone transform with restricted apex were obtained by Nguen and Truong 2002, Smith 2005, Nguen, Truong, Grangeat 2005, Maxim et al 2009, Maxim 2014.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

5 / 27

Polychrome reconstructions

(many openings)

Cree and Bones 1994 proposed reconstruction formulae from data of regular cone transform with apices restricted to a plane orthogonal to the axis. Analytic reconstructions from the cone transform with restricted apex were obtained by Nguen and Truong 2002, Smith 2005, Nguen, Truong, Grangeat 2005, Maxim et al 2009, Maxim 2014. Haltmeier 2014, Terzioglu 2015, Moon 2016, Jung and Moon 2016 gave inversion formulae for arbitrary dimension n.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

5 / 27

Polychrome reconstructions

(many openings)

Cree and Bones 1994 proposed reconstruction formulae from data of regular cone transform with apices restricted to a plane orthogonal to the axis. Analytic reconstructions from the cone transform with restricted apex were obtained by Nguen and Truong 2002, Smith 2005, Nguen, Truong, Grangeat 2005, Maxim et al 2009, Maxim 2014. Haltmeier 2014, Terzioglu 2015, Moon 2016, Jung and Moon 2016 gave inversion formulae for arbitrary dimension n. Jung and Moon 2016 proposed the scheme for collecting non redunded data from a line of detectors and rotating axis.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

5 / 27

Monochromatic reconstructions

(one opening)

Basko et al 1998 proposed a numerical method based on developing f in spherical harmonics from cone integrals with swinging axis.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Monochromatic reconstructions

(one opening)

Basko et al 1998 proposed a numerical method based on developing f in spherical harmonics from cone integrals with swinging axis. X-ray transform for a family of broken rays was applied by Eskin 2004 for study of inverse problems for the Schrödinger equation.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

6 / 27

Monochromatic reconstructions

(one opening)

Basko et al 1998 proposed a numerical method based on developing f in spherical harmonics from cone integrals with swinging axis. X-ray transform for a family of broken rays was applied by Eskin 2004 for study of inverse problems for the Schrödinger equation. Florescu, Markel and Schotland 2010, 2011 studied reconstruction of a function on a plane from the broken ray integral transform.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

6 / 27

Monochromatic reconstructions

(one opening)

Basko et al 1998 proposed a numerical method based on developing f in spherical harmonics from cone integrals with swinging axis. X-ray transform for a family of broken rays was applied by Eskin 2004 for study of inverse problems for the Schrödinger equation. Florescu, Markel and Schotland 2010, 2011 studied reconstruction of a function on a plane from the broken ray integral transform. Nguen and Truong 2011 and Ambartsoumian 2012 studied reconstruction of a function on a disc from data of V-line Radon transform.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

6 / 27

Monochromatic reconstructions

(one opening)

Basko et al 1998 proposed a numerical method based on developing f in spherical harmonics from cone integrals with swinging axis. X-ray transform for a family of broken rays was applied by Eskin 2004 for study of inverse problems for the Schrödinger equation. Florescu, Markel and Schotland 2010, 2011 studied reconstruction of a function on a plane from the broken ray integral transform. Nguen and Truong 2011 and Ambartsoumian 2012 studied reconstruction of a function on a disc from data of V-line Radon transform. Katsevich and Krylov 2013 studied reconstruction of the attenuation coe¢cient from of broken ray transform with curved lines of detectors.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

6 / 27

Monochromatic reconstructions

(one opening)

Basko et al 1998 proposed a numerical method based on developing f in spherical harmonics from cone integrals with swinging axis. X-ray transform for a family of broken rays was applied by Eskin 2004 for study of inverse problems for the Schrödinger equation. Florescu, Markel and Schotland 2010, 2011 studied reconstruction of a function on a plane from the broken ray integral transform. Nguen and Truong 2011 and Ambartsoumian 2012 studied reconstruction of a function on a disc from data of V-line Radon transform. Katsevich and Krylov 2013 studied reconstruction of the attenuation coe¢cient from of broken ray transform with curved lines of detectors. Gouia-Zarrad and Ambartsoumian 2014 found the reconstruction formula for the regular cone transform in the half-space with free apex.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

6 / 27

Cone transform with free apex

Cone Radon integral equation can written in the convolution form g = jxjk δC f , (1) where δC (ϕ) =

Z

C ϕdS = cos1 ψ

Z Z

ϕ (λs, x2, x3) dx2dx3, s = q x2

2 + x2 3 .

is a tempered distribution in E 3.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Cone transform with free apex

Cone Radon integral equation can written in the convolution form g = jxjk δC f , (1) where δC (ϕ) =

Z

C ϕdS = cos1 ψ

Z Z

ϕ (λs, x2, x3) dx2dx3, s = q x2

2 + x2 3 .

is a tempered distribution in E 3. The solution f of (1) de…ned on fx1 0g is unique if it vanishes for x1 > m for some m > 0.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

7 / 27

Cone transform with free apex

Cone Radon integral equation can written in the convolution form g = jxjk δC f , (1) where δC (ϕ) =

Z

C ϕdS = cos1 ψ

Z Z

ϕ (λs, x2, x3) dx2dx3, s = q x2

2 + x2 3 .

is a tempered distribution in E 3. The solution f of (1) de…ned on fx1 0g is unique if it vanishes for x1 > m for some m > 0. We focus on the case n = 3 and use the notations ∆0 = δC , ∆1 = jxj1 δC .

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Support of the convolution

For a function f on E n vanishing for x1 > m for some m, the convolution g = ∆k f is well de…ned and supp∆k f suppf C.

C

• C

supp f supp f - C

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Inversion of regular transforms

Case k = 0. The solution of ∆0 f0 = g0, can be found in the form f0 (x) = 1 2π cos3 ψ2∆1 Θ1 g0 (2) = 1 2π cos3 ψ2

Z

t2C

Z ∞

x1

g0 (y t1, x2 t2, x3 t3) dy dS jtj and

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Inversion of regular transforms

Case k = 0. The solution of ∆0 f0 = g0, can be found in the form f0 (x) = 1 2π cos3 ψ2∆1 Θ1 g0 (2) = 1 2π cos3 ψ2

Z

t2C

Z ∞

x1

g0 (y t1, x2 t2, x3 t3) dy dS jtj and = ∂2 ∂x2

1

λ2 ∂2 ∂x2

2

+ ∂2 ∂x2

3

• .

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Case k = 1. The solution of ∆1 f1 = g1 (3) reads f1 (x) = 1 2π cos3 ψ2∆0 Θ1 g1 (4) = 1 2π cos3 ψ2

Z

t2C

Z ∞

x1

g1 (y t1, x2 t2, x3 t3) dydS.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Conclusion: Inversion of any of two regular cone transform is given by the another cone transform followed (or preceded) by the 4 order di¤erential

• perator and additional integration from x1 to ∞ in the vertical variable.

No Fourier transform etc. is necessary.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Support of the solution

Corollary For any function f with support in Em for some m, we have suppf supp∆k f V , k = 0, 1 where V is the convex hull of C.

C

• V

supp g supp f supp g - V

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Proofs

Distributions ∆0 and ∆1 are homogeneous of order 2 and 1. Fourier transforms are equal to (V.P. 2016, P.140) ˆ ∆0 (p) =

• 1

2π cos2 ψ jp1j

• p2

1 λ2

p2

2 + p2 3

3/2 , ˆ ∆1 (p) = 2i cos ψ sgnp1

• p2

1 λ2

p2

2 + p2 3

1/2 for p2

1 > λ2

p2

2 + p2 3

• .

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Proofs

Distributions ∆0 and ∆1 are homogeneous of order 2 and 1. Fourier transforms are equal to (V.P. 2016, P.140) ˆ ∆0 (p) =

• 1

2π cos2 ψ jp1j

• p2

1 λ2

p2

2 + p2 3

3/2 , ˆ ∆1 (p) = 2i cos ψ sgnp1

• p2

1 λ2

p2

2 + p2 3

1/2 for p2

1 > λ2

p2

2 + p2 3

• .

Both have analytical continuation at H+ =

• p 2 C3 : Im p1 0
• .

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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The above calculations results 2πi cos3 ψ (p1 + i0)1 p2

1 λ2

p2

2 + p2 3

2 ˆ ∆0 (p) ˆ ∆1 (p) = 1 since function (p1 + i0)1 admits holomorphic continuation at H+. This equation holds for all p 2 R3.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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The above calculations results 2πi cos3 ψ (p1 + i0)1 p2

1 λ2

p2

2 + p2 3

2 ˆ ∆0 (p) ˆ ∆1 (p) = 1 since function (p1 + i0)1 admits holomorphic continuation at H+. This equation holds for all p 2 R3. Calculating the inverse Fourier transform we obtain F 1 p2

1 λ2

p2

2 + p2 3

= 1 4π2 δ0, and F 1 (p1 + i0)1 = 2πi Θ1, where Θ1 = θ (x1) δ0 (x2, x3) , θ (t) = 1 for t < 0 and θ (t) = 0 for t > 0.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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The above calculations results 2πi cos3 ψ (p1 + i0)1 p2

1 λ2

p2

2 + p2 3

2 ˆ ∆0 (p) ˆ ∆1 (p) = 1 since function (p1 + i0)1 admits holomorphic continuation at H+. This equation holds for all p 2 R3. Calculating the inverse Fourier transform we obtain F 1 p2

1 λ2

p2

2 + p2 3

= 1 4π2 δ0, and F 1 (p1 + i0)1 = 2πi Θ1, where Θ1 = θ (x1) δ0 (x2, x3) , θ (t) = 1 for t < 0 and θ (t) = 0 for t > 0. Finally cos3 ψ2δ0 Θ1 ∆1 ∆0 = δ0, (5) where the convolutions of distributions Θ1, ∆1 and 2δ0 are well de…ned and commute.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Applying (5) to f0 gives f0 = cos3 ψ2 ∆1 Θ1 ∆0 f0 = cos3 ψ2 ∆1 Θ1 g0 which is equivalent to (2).

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Applying (5) to f0 gives f0 = cos3 ψ2 ∆1 Θ1 ∆0 f0 = cos3 ψ2 ∆1 Θ1 g0 which is equivalent to (2). Commuting factors in (5) yields f1 = cos3 ψ2 ∆0 Θ1 ∆1 f1 = cos3 ψ2 ∆0 Θ1 g1 and (4) follows.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Applying (5) to f0 gives f0 = cos3 ψ2 ∆1 Θ1 ∆0 f0 = cos3 ψ2 ∆1 Θ1 g0 which is equivalent to (2). Commuting factors in (5) yields f1 = cos3 ψ2 ∆0 Θ1 ∆1 f1 = cos3 ψ2 ∆0 Θ1 g1 and (4) follows. Remark 1. Constant attenuation can be included in this method.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Remark 2. Solution of (1) could be done in form ˆ g (p) / ˆ ∆k (p) in the frequency domain. Iimplementation of this method supposes cutting out the "plumes" of g which causes the artifacts in the reconstruction as in the following picture

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Remark 2. Solution of (1) could be done in form ˆ g (p) / ˆ ∆k (p) in the frequency domain. Iimplementation of this method supposes cutting out the "plumes" of g which causes the artifacts in the reconstruction as in the following picture which is due to the courtesy of Gouia-Zarrad, Ambartsoumian 2014.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Inversion of the singular cone transform

Fix λ > 0 and consider the singular integral transform G (q, θ) =

Z

Cλ(θ) f (q + x) dS

jxj2 , θ 2 S2, q 2 E 3, (6) where Cλ (θ) means for the spherical cone with apex x = 0, axis θ 2 S2 and opening λ.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

17 / 27

Inversion of the singular cone transform

Fix λ > 0 and consider the singular integral transform G (q, θ) =

Z

Cλ(θ) f (q + x) dS

jxj2 , θ 2 S2, q 2 E 3, (6) where Cλ (θ) means for the spherical cone with apex x = 0, axis θ 2 S2 and opening λ. The integral is well de…ned if f is smooth and f (q) = 0

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

17 / 27

Inversion of the singular cone transform

Fix λ > 0 and consider the singular integral transform G (q, θ) =

Z

Cλ(θ) f (q + x) dS

jxj2 , θ 2 S2, q 2 E 3, (6) where Cλ (θ) means for the spherical cone with apex x = 0, axis θ 2 S2 and opening λ. The integral is well de…ned if f is smooth and f (q) = 0 Theorem For any λ > 0 and any set Q E 3, an arbitrary function f 2 C 2 with compact support can be recovered from data of integrals (6) for q 2 Q provided

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

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Inversion of the singular cone transform

Fix λ > 0 and consider the singular integral transform G (q, θ) =

Z

Cλ(θ) f (q + x) dS

jxj2 , θ 2 S2, q 2 E 3, (6) where Cλ (θ) means for the spherical cone with apex x = 0, axis θ 2 S2 and opening λ. The integral is well de…ned if f is smooth and f (q) = 0 Theorem For any λ > 0 and any set Q E 3, an arbitrary function f 2 C 2 with compact support can be recovered from data of integrals (6) for q 2 Q provided (i) any plane H which meets suppf has a common point with Q,

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

March 30, 2017

17 / 27

Inversion of the singular cone transform

Fix λ > 0 and consider the singular integral transform G (q, θ) =

Z

Cλ(θ) f (q + x) dS

jxj2 , θ 2 S2, q 2 E 3, (6) where Cλ (θ) means for the spherical cone with apex x = 0, axis θ 2 S2 and opening λ. The integral is well de…ned if f is smooth and f (q) = 0 Theorem For any λ > 0 and any set Q E 3, an arbitrary function f 2 C 2 with compact support can be recovered from data of integrals (6) for q 2 Q provided (i) any plane H which meets suppf has a common point with Q, (ii) for any point q 2 Q, there exists a unit vector θ (q) such that suppf q + Cλ (θ (q)).

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Compton cones with swinging axis

q S

scattering detectors absorption detectors

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Proof

Step 1. The singular ray transform Xf (q, ξ) =

Z ∞

f (q + rξ) dr r , ξ 2 S2, q 2 Q (7) is wel de…ned since f vanishes on Q since of (ii).

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Proof

Step 1. The singular ray transform Xf (q, ξ) =

Z ∞

f (q + rξ) dr r , ξ 2 S2, q 2 Q (7) is wel de…ned since f vanishes on Q since of (ii). By Fubini’s theorem G (q, θ) =

Z

Sλ(θ)

Z ∞

f (q + ξ (σ) r) dr r dσ =

Z

Sλ(θ) Xf (q, ξ (σ)) dσ,

where ξ (σ) runs over the circle Sλ (θ) = Cλ (θ) \ S2.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Proof

Step 1. The singular ray transform Xf (q, ξ) =

Z ∞

f (q + rξ) dr r , ξ 2 S2, q 2 Q (7) is wel de…ned since f vanishes on Q since of (ii). By Fubini’s theorem G (q, θ) =

Z

Sλ(θ)

Z ∞

f (q + ξ (σ) r) dr r dσ =

Z

Sλ(θ) Xf (q, ξ (σ)) dσ,

where ξ (σ) runs over the circle Sλ (θ) = Cλ (θ) \ S2. Circles Sλ (θ) have the same radius r = λ

• 1 + λ21/2 .

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Proof

Step 1. The singular ray transform Xf (q, ξ) =

Z ∞

f (q + rξ) dr r , ξ 2 S2, q 2 Q (7) is wel de…ned since f vanishes on Q since of (ii). By Fubini’s theorem G (q, θ) =

Z

Sλ(θ)

Z ∞

f (q + ξ (σ) r) dr r dσ =

Z

Sλ(θ) Xf (q, ξ (σ)) dσ,

where ξ (σ) runs over the circle Sλ (θ) = Cλ (θ) \ S2. Circles Sλ (θ) have the same radius r = λ

• 1 + λ21/2 .

The planes containing these circles are tangent to the central ball B

• f radius ρ =
• 1 + λ21/2 .

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Step 2: Nongeodesic Funk transform

Nongeodesic circles on sphere

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Theorem For any ρ, 0 ρ < 1, α 2 E, jαj 1, an arbitrary function g 2 C 2 S2 can be reconstructed from data of integrals Γ (θ) =

Z

ξ2S2,hξα,θi=ρ g (ξ) dσ, θ 2 S2

(8)

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Theorem For any ρ, 0 ρ < 1, α 2 E, jαj 1, an arbitrary function g 2 C 2 S2 can be reconstructed from data of integrals Γ (θ) =

Z

ξ2S2,hξα,θi=ρ g (ξ) dσ, θ 2 S2

(8) by g (ξ) = jξ αj2 2π2

• jξ αj2 ρ2

1/2

Z

S2

Γ (θ) (hξ α, θi ρ)2 dS (9) provided there exists a vector θ0 2 S2 such that suppg

• ξ 2 S2 : hξ α, θ0i ρ
• .

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Theorem For any ρ, 0 ρ < 1, α 2 E, jαj 1, an arbitrary function g 2 C 2 S2 can be reconstructed from data of integrals Γ (θ) =

Z

ξ2S2,hξα,θi=ρ g (ξ) dσ, θ 2 S2

(8) by g (ξ) = jξ αj2 2π2

• jξ αj2 ρ2

1/2

Z

S2

Γ (θ) (hξ α, θi ρ)2 dS (9) provided there exists a vector θ0 2 S2 such that suppg

• ξ 2 S2 : hξ α, θ0i ρ
• .

The singular integral is regularized as follows

Z

S2

Γ (θ) (hξ α, θi ρ)2 dS = ∆ (θ)

Z

S2 Γ (θ) log (hξ α, θi ρ) dS.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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References

Hermann Minkowski 1905 stated uniqueness of an even functions on S2 with given big circle integrals. For n = 2, ρ = 0, α = 0, The analytic reconstruction of an even function is due to Paul Funk’s 1913 (student of David Hilbert).

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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References

Hermann Minkowski 1905 stated uniqueness of an even functions on S2 with given big circle integrals. For n = 2, ρ = 0, α = 0, The analytic reconstruction of an even function is due to Paul Funk’s 1913 (student of David Hilbert). Funk’s result and his method encouraged Johann Radon 1917 for his famous reconstruction in the ‡at plane.

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References

Hermann Minkowski 1905 stated uniqueness of an even functions on S2 with given big circle integrals. For n = 2, ρ = 0, α = 0, The analytic reconstruction of an even function is due to Paul Funk’s 1913 (student of David Hilbert). Funk’s result and his method encouraged Johann Radon 1917 for his famous reconstruction in the ‡at plane. Generalizations for higher dimensions: S.Helgason 1959, 1990, 2006 and V.Semjanistyi 1961.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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References

Hermann Minkowski 1905 stated uniqueness of an even functions on S2 with given big circle integrals. For n = 2, ρ = 0, α = 0, The analytic reconstruction of an even function is due to Paul Funk’s 1913 (student of David Hilbert). Funk’s result and his method encouraged Johann Radon 1917 for his famous reconstruction in the ‡at plane. Generalizations for higher dimensions: S.Helgason 1959, 1990, 2006 and V.Semjanistyi 1961. The case ρ = 0, jαj = 1 of the above theorem follows from Radon’s reconstruction by means of the stereographic projection.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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References

Hermann Minkowski 1905 stated uniqueness of an even functions on S2 with given big circle integrals. For n = 2, ρ = 0, α = 0, The analytic reconstruction of an even function is due to Paul Funk’s 1913 (student of David Hilbert). Funk’s result and his method encouraged Johann Radon 1917 for his famous reconstruction in the ‡at plane. Generalizations for higher dimensions: S.Helgason 1959, 1990, 2006 and V.Semjanistyi 1961. The case ρ = 0, jαj = 1 of the above theorem follows from Radon’s reconstruction by means of the stereographic projection. Y.Salman 2016 obtained the particular case for n = 2, ρ = 0, jαj < 1. This result was also published by M.Quellmalz 2017.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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References

Hermann Minkowski 1905 stated uniqueness of an even functions on S2 with given big circle integrals. For n = 2, ρ = 0, α = 0, The analytic reconstruction of an even function is due to Paul Funk’s 1913 (student of David Hilbert). Funk’s result and his method encouraged Johann Radon 1917 for his famous reconstruction in the ‡at plane. Generalizations for higher dimensions: S.Helgason 1959, 1990, 2006 and V.Semjanistyi 1961. The case ρ = 0, jαj = 1 of the above theorem follows from Radon’s reconstruction by means of the stereographic projection. Y.Salman 2016 obtained the particular case for n = 2, ρ = 0, jαj < 1. This result was also published by M.Quellmalz 2017. The reconstruction from spherical integrals (8) on Sn was stated in V.P. 2016 for arbitrary n, 0 ρ < 1, jαj 1.

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Step 3. By (i) formula (9) can be applied to Γ (q, θ) for α = 0, ρ =

• 1 + λ21/2 which provides the reconstruction of

g (q, ξ) = Xf (q, ξ) for any q 2 Q and all ξ.

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Step 3. By (i) formula (9) can be applied to Γ (q, θ) for α = 0, ρ =

• 1 + λ21/2 which provides the reconstruction of

g (q, ξ) = Xf (q, ξ) for any q 2 Q and all ξ. For any x 2 E 3 and any unit orthogonal vectors ω, ξ, we have

• ω, rξ

2 f (q + rξ) = r2 hω, rqi2 f (q + rξ) which yields (by Grangeat’s method) for any p,

Z

hω,ξi=0

• ω, rξ

2 Xf (q, ξ) dϕ =

Z

ω, rξ 2 Z ∞ f (q + rξ) dr r dϕ =

Z Z ∞

hω, rqi2 f (q + rξ) rdrdϕ = ∂2 ∂p2

Z

hω,qi=p f (q) dS,

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Step 3. By (i) formula (9) can be applied to Γ (q, θ) for α = 0, ρ =

• 1 + λ21/2 which provides the reconstruction of

g (q, ξ) = Xf (q, ξ) for any q 2 Q and all ξ. For any x 2 E 3 and any unit orthogonal vectors ω, ξ, we have

• ω, rξ

2 f (q + rξ) = r2 hω, rqi2 f (q + rξ) which yields (by Grangeat’s method) for any p,

Z

hω,ξi=0

• ω, rξ

2 Xf (q, ξ) dϕ =

Z

ω, rξ 2 Z ∞ f (q + rξ) dr r dϕ =

Z Z ∞

hω, rqi2 f (q + rξ) rdrdϕ = ∂2 ∂p2

Z

hω,qi=p f (q) dS,

where the left hand side can be calculated from Xf .

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Step 4 By (ii) we can use the Lorentz-Radon formula for any x 2 suppf , f (x) = 1 8π2

Z

ω2S2

∂2 ∂p2

Z

hω,qxi=0 f (q) dqdΩ

= 1 8π2

Z

ω2S2

Z

hω,ξi=0

• ω, rξ

2 Xf (q (ω) , ξ) dϕdΩ if we choose for any ω 2 S2, a point q = q (ω) 2 Q such that hq (ω) x, ωi = 0.

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Step 4 By (ii) we can use the Lorentz-Radon formula for any x 2 suppf , f (x) = 1 8π2

Z

ω2S2

∂2 ∂p2

Z

hω,qxi=0 f (q) dqdΩ

= 1 8π2

Z

ω2S2

Z

hω,ξi=0

• ω, rξ

2 Xf (q (ω) , ξ) dϕdΩ if we choose for any ω 2 S2, a point q = q (ω) 2 Q such that hq (ω) x, ωi = 0. This completes the reconstruction of f .

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Other reconstructions from the singular cone beam transform

Let Γ = fy = y (s)g be a closed C 2 smooth curve.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Other reconstructions from the singular cone beam transform

Let Γ = fy = y (s)g be a closed C 2 smooth curve. Let σ : Γ S2 ! R S2; σ (y, ξ) = (hy, ξi , ξ) . All critical points of the map σ are supposed of Morse type.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Other reconstructions from the singular cone beam transform

Let Γ = fy = y (s)g be a closed C 2 smooth curve. Let σ : Γ S2 ! R S2; σ (y, ξ) = (hy, ξi , ξ) . All critical points of the map σ are supposed of Morse type. Let ε : Γ S2 ! R be a smooth function such that

∑

y;hy,ξi=p

• y 0, ξ
• ε (y, ξ) = 1, (p, ξ) 2 Im σ.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Other reconstructions from the singular cone beam transform

Let Γ = fy = y (s)g be a closed C 2 smooth curve. Let σ : Γ S2 ! R S2; σ (y, ξ) = (hy, ξi , ξ) . All critical points of the map σ are supposed of Morse type. Let ε : Γ S2 ! R be a smooth function such that

∑

y;hy,ξi=p

• y 0, ξ
• ε (y, ξ) = 1, (p, ξ) 2 Im σ.

Theorem For any function f 2 C 2

• E 3

and any x 2 suppf nΓ such that any plane P through x meets Γ, the equation holds f (x) =

• 1

32π4

Z

y2Γ

Z

hyx,ξi=0 ∂2 s

ε (y, ξ) jy xjds

• Z

hξ,vi=0 hξ, rv i2 ∂sg (y, v) dθdϕ.

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Some references

Cree M J and Bones P J 1994 IEEE Trans. of Medical Imag. 13 398-407 Basko R, Zeng G L and Gullberg G T 1998 Phys. Med. Biol. 43 887–894 Nguyen M K and Truong T T 2002 Inverse Probl. 18 265–277 Eskin G 2004 Inverse Probl. 20 1497-1516 Smith B 2005 J. Opt. Soc. Am. A 22 445–459 Nguyen M K, Truong T T and Grangeat P 2005 J. Phys. A: Math.

• Gen. 38 8003–8015

Maxim V, Frandes M and Prost R 2009 Inverse Problems 25 095001 Florescu L, Markel V A and Schotland J C 2011 Inverse Prob. 27 025002 Truong T T and Nguyen M K 2011 J. Phys. A: Math. Theor. 44 075206

Victor Palamodov (Tel Aviv University) New reconstructions from cone Radon transform

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Ambartsoumian G 2012 Comput. Math. Appl. 64 260–5 Katsevich A and Krylov R 2013 Inverse Probl. 29 075008 Maxim V 2014 IEEE Trans. Image Proc. 23 332-341 Haltmeier M 2014 Inverse Probl. 30 03500 Gouia-Zarrad R 2014 Comput. Math. Appl., 68 1016–1023. Gouia-Zarrad R and Ambartsoumian G 2014 Inverse Probl. 30 045007 Terzioglu F 2015 31 115010 Jung Ch-Y and Moon S 2016 SIAM J. Imaging Sci. 9 520–536 Moon S 2016 SIAM J Math. Anal. 48 1833–1847 Palamodov V 2016 CRC Press Salman Y 2016 Anal. Math. Phys. 6, no. 1, 43–58. Quellmalz M 2017 Inverse Probl. 33 035016

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