Stability estimates for limited data Radon transforms Hans Rullgrd, - - PowerPoint PPT Presentation

university-logo Limited Data Radon Transforms

Stability estimates for limited data Radon transforms

Hans Rullgård, Todd Quinto

1Department of Mathematics

Stockholm University

2Department of Mathematics

Tufts University

Applied Inverse Problems 2009

university-logo Limited Data Radon Transforms Goals and Notation

Goals

Goal: Develop new quantitative estimates to relate the size of singularities of a function to the size of singularities of its Radon transforms. Reasons: Limited data tomography is, in general, highly ill-posed.

1

Such estimates provide one way to measure this ill-posedness.

2

Such estimates can make quantitative the qualitative microlocal correspondence between singularities. Other ways to understand ill-posedness and singularities SVDs and singular functions: Louis, Louis Rieder, Davison, Grünbaum, Maaß... Precise measurements for specific singularities: Ramm Zaslavsky Wavelets and curvelets (and WF): Candès, Donoho, et al.

university-logo Limited Data Radon Transforms Goals and Notation

Goals

Goal: Develop new quantitative estimates to relate the size of singularities of a function to the size of singularities of its Radon transforms. Reasons: Limited data tomography is, in general, highly ill-posed.

1

Such estimates provide one way to measure this ill-posedness.

2

Such estimates can make quantitative the qualitative microlocal correspondence between singularities. Other ways to understand ill-posedness and singularities SVDs and singular functions: Louis, Louis Rieder, Davison, Grünbaum, Maaß... Precise measurements for specific singularities: Ramm Zaslavsky Wavelets and curvelets (and WF): Candès, Donoho, et al.

university-logo Limited Data Radon Transforms Goals and Notation

Goals

Goal: Develop new quantitative estimates to relate the size of singularities of a function to the size of singularities of its Radon transforms. Reasons: Limited data tomography is, in general, highly ill-posed.

1

Such estimates provide one way to measure this ill-posedness.

2

Such estimates can make quantitative the qualitative microlocal correspondence between singularities. Other ways to understand ill-posedness and singularities SVDs and singular functions: Louis, Louis Rieder, Davison, Grünbaum, Maaß... Precise measurements for specific singularities: Ramm Zaslavsky Wavelets and curvelets (and WF): Candès, Donoho, et al.

university-logo Limited Data Radon Transforms Goals and Notation

The Radon Hyperplane Transform

Hyperplane: ω ∈ Sn−1, p ∈ R, H(ω, p) = {x ∈ Rn x · ω = p} is the hyperplane perpendicular to ω and containing pω. Definition (Radon Hyperplane Transform) For f ∈ L1(Rn) and (ω, p) ∈ Sn−1 × R, the Radon transform, Rf(ω, p) =

• x∈H(ω,p)

f(x)dxH, is the integral of f over H(ω, p) In R2, lines are the “hyperplanes” and R is the X-ray transform.

university-logo Limited Data Radon Transforms Goals and Notation

The Radon Hyperplane Transform

Hyperplane: ω ∈ Sn−1, p ∈ R, H(ω, p) = {x ∈ Rn x · ω = p} is the hyperplane perpendicular to ω and containing pω. Definition (Radon Hyperplane Transform) For f ∈ L1(Rn) and (ω, p) ∈ Sn−1 × R, the Radon transform, Rf(ω, p) =

• x∈H(ω,p)

f(x)dxH, is the integral of f over H(ω, p) In R2, lines are the “hyperplanes” and R is the X-ray transform.

university-logo Limited Data Radon Transforms Microlocal Analysis

Fourier Transforms and Sobolev Spaces

The Fourier Transform: (Ff)(ξ) = ˆ f(ξ) =

• x∈Rn e−ix·ξf(x)dx,

f(x) = 1 (2π)n

• ξ∈Rn eix·ξˆ

f(ξ)dξ Key idea: decrease at ∞ of Ff ∼ smoothness of f Definition Let s ∈ R. f ∈ Hs(Rn) if its Sobolev norm fs =

• ξ∈Rn |ˆ

f(ξ)|2(1 + |ξ|2)sdξ 1/2 is finite.

university-logo Limited Data Radon Transforms Microlocal Analysis

Fourier Transforms and Sobolev Spaces

The Fourier Transform: (Ff)(ξ) = ˆ f(ξ) =

• x∈Rn e−ix·ξf(x)dx,

f(x) = 1 (2π)n

• ξ∈Rn eix·ξˆ

f(ξ)dξ Key idea: decrease at ∞ of Ff ∼ smoothness of f Definition Let s ∈ R. f ∈ Hs(Rn) if its Sobolev norm fs =

• ξ∈Rn |ˆ

f(ξ)|2(1 + |ξ|2)sdξ 1/2 is finite.

university-logo Limited Data Radon Transforms Microlocal Analysis

Sobolev seminorms in limited directions.

Hs norm measures global L2 smoothness of order s, what about directional smoothness?? Definition Let V be an open cone in Rn and let f be a distribution with locally square-integrable Fourier transform. We define fV,s =

• V

|ˆ f(ξ)|2(1 + |ξ|2)sdξ 1/2 .

university-logo Limited Data Radon Transforms Microlocal Analysis

Sobolev seminorms in limited directions.

Hs norm measures global L2 smoothness of order s, what about directional smoothness?? Definition Let V be an open cone in Rn and let f be a distribution with locally square-integrable Fourier transform. We define fV,s =

• V

|ˆ f(ξ)|2(1 + |ξ|2)sdξ 1/2 .

university-logo Limited Data Radon Transforms Microlocal Analysis

Localize + Microlocalize

· V,s microlocalizes in ξ. Now, localize near x0 ∈ Rn: Multiply f by a smooth cutoff function ϕ (ϕ(x0) = 0) and see if the localized Fourier transform is in Hs for certain microlocal directions. Definition (Sobolev Wavefront Set) Let s ∈ R, x0 ∈ Rn and ξ0 ∈ Rn \ 0. The function f is in Hs at x0 in direction ξ0 if ∃ a cut-off function ϕ near x0 and an open cone V ∋ ξ0 such that ϕfs,V < ∞. On the other hand, (x0, ξ0) ∈ WFs(f) if f is not in Hs at x0 in direction ξ0. NOTE: usually ξ0 is a covector.

university-logo Limited Data Radon Transforms Microlocal Analysis

Localize + Microlocalize

· V,s microlocalizes in ξ. Now, localize near x0 ∈ Rn: Multiply f by a smooth cutoff function ϕ (ϕ(x0) = 0) and see if the localized Fourier transform is in Hs for certain microlocal directions. Definition (Sobolev Wavefront Set) Let s ∈ R, x0 ∈ Rn and ξ0 ∈ Rn \ 0. The function f is in Hs at x0 in direction ξ0 if ∃ a cut-off function ϕ near x0 and an open cone V ∋ ξ0 such that ϕfs,V < ∞. On the other hand, (x0, ξ0) ∈ WFs(f) if f is not in Hs at x0 in direction ξ0. NOTE: usually ξ0 is a covector.

university-logo Limited Data Radon Transforms Microlocal Analysis

Localize + Microlocalize

· V,s microlocalizes in ξ. Now, localize near x0 ∈ Rn: Multiply f by a smooth cutoff function ϕ (ϕ(x0) = 0) and see if the localized Fourier transform is in Hs for certain microlocal directions. Definition (Sobolev Wavefront Set) Let s ∈ R, x0 ∈ Rn and ξ0 ∈ Rn \ 0. The function f is in Hs at x0 in direction ξ0 if ∃ a cut-off function ϕ near x0 and an open cone V ∋ ξ0 such that ϕfs,V < ∞. On the other hand, (x0, ξ0) ∈ WFs(f) if f is not in Hs at x0 in direction ξ0. NOTE: usually ξ0 is a covector.

university-logo Limited Data Radon Transforms Microlocal Analysis

Key Correspondence of singularities under R:

BIG IDEA: Radon transforms detect singularities perpendicular to the surface (hyperplane) being integrated over. Theorem (Microlocal Regularity of R) Assume f has compact support. Let H0 = H(ω0, p0) If Rf is in Hs+ n−1

2

near (ω0, p0) (Rf times cutoff is in Hs+ n−1

2 ), then f is in

Hs in direction ±ω0 at every point on H0. In fact, there is a one-to-one correspondence between wavefront of f and wavefront of Rf. So, if Rf is not in Hs+ n−1

2

near (ω0, p0), then for some x0 ∈ H0, ξ0 parallel ω0, (x0, ξ0) ∈ WFs(f). Reasons: See [Q 1993]. Guillemin [1975] showed R is an elliptic FIO of order (1 − n)/2, and microlocal regularity follows from the FIO calculus.

university-logo Limited Data Radon Transforms Microlocal Analysis

Key Correspondence of singularities under R:

BIG IDEA: Radon transforms detect singularities perpendicular to the surface (hyperplane) being integrated over. Theorem (Microlocal Regularity of R) Assume f has compact support. Let H0 = H(ω0, p0) If Rf is in Hs+ n−1

2

near (ω0, p0) (Rf times cutoff is in Hs+ n−1

2 ), then f is in

Hs in direction ±ω0 at every point on H0. In fact, there is a one-to-one correspondence between wavefront of f and wavefront of Rf. So, if Rf is not in Hs+ n−1

2

near (ω0, p0), then for some x0 ∈ H0, ξ0 parallel ω0, (x0, ξ0) ∈ WFs(f). Reasons: See [Q 1993]. Guillemin [1975] showed R is an elliptic FIO of order (1 − n)/2, and microlocal regularity follows from the FIO calculus.

university-logo Limited Data Radon Transforms Microlocal Analysis

Key Correspondence of singularities under R:

BIG IDEA: Radon transforms detect singularities perpendicular to the surface (hyperplane) being integrated over. Theorem (Microlocal Regularity of R) Assume f has compact support. Let H0 = H(ω0, p0) If Rf is in Hs+ n−1

2

near (ω0, p0) (Rf times cutoff is in Hs+ n−1

2 ), then f is in

Hs in direction ±ω0 at every point on H0. In fact, there is a one-to-one correspondence between wavefront of f and wavefront of Rf. So, if Rf is not in Hs+ n−1

2

near (ω0, p0), then for some x0 ∈ H0, ξ0 parallel ω0, (x0, ξ0) ∈ WFs(f). Reasons: See [Q 1993]. Guillemin [1975] showed R is an elliptic FIO of order (1 − n)/2, and microlocal regularity follows from the FIO calculus.

university-logo Limited Data Radon Transforms Microlocal Analysis

Key Correspondence of singularities under R:

BIG IDEA: Radon transforms detect singularities perpendicular to the surface (hyperplane) being integrated over. Theorem (Microlocal Regularity of R) Assume f has compact support. Let H0 = H(ω0, p0) If Rf is in Hs+ n−1

2

near (ω0, p0) (Rf times cutoff is in Hs+ n−1

2 ), then f is in

Hs in direction ±ω0 at every point on H0. In fact, there is a one-to-one correspondence between wavefront of f and wavefront of Rf. So, if Rf is not in Hs+ n−1

2

near (ω0, p0), then for some x0 ∈ H0, ξ0 parallel ω0, (x0, ξ0) ∈ WFs(f). Reasons: See [Q 1993]. Guillemin [1975] showed R is an elliptic FIO of order (1 − n)/2, and microlocal regularity follows from the FIO calculus.

university-logo Limited Data Radon Transforms Microlocal Analysis

Problems! This result is qualitative. How can we measure it? Numerical data are finite and so arbitrarily smooth but can have large numerical derivatives. Sobolev wavefront singularities are defined by integrability (asymptotics) at ∞ of Ff. Are asymptotics measurable from smooth data? Larry Shepp, Frank Natterer, Adel Faridani, Alfred Louis .... ( . .

⌣): For the standard discontinuities for many tomographic

problem, this slow decay occurs sufficiently close to zero to be numerically visible. Our Goals: Develop seminorms to measure local and microlocal Sobolev scales. (Long term) apply to Electron microscopy, other modalities, develop band limited versions.

university-logo Limited Data Radon Transforms Microlocal Analysis

Problems! This result is qualitative. How can we measure it? Numerical data are finite and so arbitrarily smooth but can have large numerical derivatives. Sobolev wavefront singularities are defined by integrability (asymptotics) at ∞ of Ff. Are asymptotics measurable from smooth data? Larry Shepp, Frank Natterer, Adel Faridani, Alfred Louis .... ( . .

⌣): For the standard discontinuities for many tomographic

problem, this slow decay occurs sufficiently close to zero to be numerically visible. Our Goals: Develop seminorms to measure local and microlocal Sobolev scales. (Long term) apply to Electron microscopy, other modalities, develop band limited versions.

university-logo Limited Data Radon Transforms Microlocal Analysis

Problems! This result is qualitative. How can we measure it? Numerical data are finite and so arbitrarily smooth but can have large numerical derivatives. Sobolev wavefront singularities are defined by integrability (asymptotics) at ∞ of Ff. Are asymptotics measurable from smooth data? Larry Shepp, Frank Natterer, Adel Faridani, Alfred Louis .... ( . .

⌣): For the standard discontinuities for many tomographic

problem, this slow decay occurs sufficiently close to zero to be numerically visible. Our Goals: Develop seminorms to measure local and microlocal Sobolev scales. (Long term) apply to Electron microscopy, other modalities, develop band limited versions.

university-logo Limited Data Radon Transforms Microlocal Analysis

Problems! This result is qualitative. How can we measure it? Numerical data are finite and so arbitrarily smooth but can have large numerical derivatives. Sobolev wavefront singularities are defined by integrability (asymptotics) at ∞ of Ff. Are asymptotics measurable from smooth data? Larry Shepp, Frank Natterer, Adel Faridani, Alfred Louis .... ( . .

⌣): For the standard discontinuities for many tomographic

problem, this slow decay occurs sufficiently close to zero to be numerically visible. Our Goals: Develop seminorms to measure local and microlocal Sobolev scales. (Long term) apply to Electron microscopy, other modalities, develop band limited versions.

university-logo Limited Data Radon Transforms Microlocal Analysis

Problems! This result is qualitative. How can we measure it? Numerical data are finite and so arbitrarily smooth but can have large numerical derivatives. Sobolev wavefront singularities are defined by integrability (asymptotics) at ∞ of Ff. Are asymptotics measurable from smooth data? Larry Shepp, Frank Natterer, Adel Faridani, Alfred Louis .... ( . .

⌣): For the standard discontinuities for many tomographic

problem, this slow decay occurs sufficiently close to zero to be numerically visible. Our Goals: Develop seminorms to measure local and microlocal Sobolev scales. (Long term) apply to Electron microscopy, other modalities, develop band limited versions.

university-logo Limited Data Radon Transforms Microlocal Analysis

Problems! This result is qualitative. How can we measure it? Numerical data are finite and so arbitrarily smooth but can have large numerical derivatives. Sobolev wavefront singularities are defined by integrability (asymptotics) at ∞ of Ff. Are asymptotics measurable from smooth data? Larry Shepp, Frank Natterer, Adel Faridani, Alfred Louis .... ( . .

⌣): For the standard discontinuities for many tomographic

problem, this slow decay occurs sufficiently close to zero to be numerically visible. Our Goals: Develop seminorms to measure local and microlocal Sobolev scales. (Long term) apply to Electron microscopy, other modalities, develop band limited versions.

university-logo Limited Data Radon Transforms Microlocal Analysis

Problems! This result is qualitative. How can we measure it? Numerical data are finite and so arbitrarily smooth but can have large numerical derivatives. Sobolev wavefront singularities are defined by integrability (asymptotics) at ∞ of Ff. Are asymptotics measurable from smooth data? Larry Shepp, Frank Natterer, Adel Faridani, Alfred Louis .... ( . .

⌣): For the standard discontinuities for many tomographic

problem, this slow decay occurs sufficiently close to zero to be numerically visible. Our Goals: Develop seminorms to measure local and microlocal Sobolev scales. (Long term) apply to Electron microscopy, other modalities, develop band limited versions.

university-logo Limited Data Radon Transforms Results

Local and Microlocal Measurements

Definition (Local seminorm [Triebel]) Let f be a distribution. Let Ω ⊂ Rn be open and let s ∈ R. Define fΩ,s = inf{˜ fs : ˜ f ∈ Hs(Rn),˜ f|Ω = f}. Definition (Microlocal Seminorm) Let s and s′ be real numbers with s > s′. Let Ω ⊂ Rn be open and let V ⊂ Rn be an open cone. We define the seminorm fΩ,V,s,s′ = inf{(˜ f2

V,s + ˜

f2

s′)1/2 : ˜

f ∈ Hs′(Rn),˜ f|Ω = f}. (1) Without the s′ norm, (1) would be trivial.

university-logo Limited Data Radon Transforms Results

Local and Microlocal Measurements

Definition (Local seminorm [Triebel]) Let f be a distribution. Let Ω ⊂ Rn be open and let s ∈ R. Define fΩ,s = inf{˜ fs : ˜ f ∈ Hs(Rn),˜ f|Ω = f}. Definition (Microlocal Seminorm) Let s and s′ be real numbers with s > s′. Let Ω ⊂ Rn be open and let V ⊂ Rn be an open cone. We define the seminorm fΩ,V,s,s′ = inf{(˜ f2

V,s + ˜

f2

s′)1/2 : ˜

f ∈ Hs′(Rn),˜ f|Ω = f}. (1) Without the s′ norm, (1) would be trivial.

university-logo Limited Data Radon Transforms Results

Hyperplane Space

Definition Let Ω ⊂ Rn be open, let Ω′ denote the set of all (ω, p) corresponding to hyperplanes intersecting Ω. If ǫ > 0, let Ω′

ǫ

denote the set of (ω, p) corresponding to hyperplanes passing within a distance ǫ from Ω. For an integrable function u on Sn−1 × R, define the Fourier transform of u with respect to the second variable as ˆ u(ω, σ) =

• R

u(ω, p)e−ipσdp.

university-logo Limited Data Radon Transforms Results

Seminorms in Hyperplane Space

Definition Let s ∈ R, u a distribution. u ∈ H0,s(Sn−1 × R) if the Sobolev norm us =

• Sn−1×R

|ˆ u(ω, σ)|2(1 + σ2)sdωdσ 1/2 and for Λ an open set in Sn−1 × R we define the seminorm uΛ,s = inf{˜ us : ˜ u ∈ H0,s(Sn−1 × R), ˜ u|Λ = u}.

university-logo Limited Data Radon Transforms Results

Results I

Theorem (Local Estimate [Rullgård and Q.]) Let s ∈ R, s′ < s, and let Ω be a bounded open subset of Rn. Let f be a distribution of compact support. Then for n odd: fΩ,s ≤ 1 √ 2 RfΩ′,s+(n−1)/2, for n even: fΩ,s ≤ CnRfΩ′

ǫ,s+(n−1)/2 + C′

n,s,s′,ǫfs′.

So, if n is odd and the s seminorm of f on Ω is big, then the s + (n − 1)/2 seminorm of Rf on Ω′ must be big. For n even, the fs′ norm is needed on the right (interior problem).

university-logo Limited Data Radon Transforms Results

Results I

Theorem (Local Estimate [Rullgård and Q.]) Let s ∈ R, s′ < s, and let Ω be a bounded open subset of Rn. Let f be a distribution of compact support. Then for n odd: fΩ,s ≤ 1 √ 2 RfΩ′,s+(n−1)/2, for n even: fΩ,s ≤ CnRfΩ′

ǫ,s+(n−1)/2 + C′

n,s,s′,ǫfs′.

So, if n is odd and the s seminorm of f on Ω is big, then the s + (n − 1)/2 seminorm of Rf on Ω′ must be big. For n even, the fs′ norm is needed on the right (interior problem).

university-logo Limited Data Radon Transforms Results

Results II

Theorem (Microlocal Estimate [Rullgård and Q.]) Let n be odd, let Ω be a bounded open subset of Rn, and let V be a symmetric open cone in Rn. Let s > s′ be real numbers. Let f be a distribution of compact support. Then fΩ,V,s,s′ ≤ RfΩ′∩(V×R),s+(n−1)/2 + fs′. fΩ,V,s,s′ = inf{(˜ f2

V,s + ˜

f2

s′)1/2 : ˜

f ∈ Hs′(Rn),˜ f|Ω = f}. uΛ,s = inf{˜ us : ˜ f ∈ Hs(Sn−1 × R), ˜ u|Λ = u}. A similar theorem holds (with constants) for n even.

university-logo Limited Data Radon Transforms Results

Discussion

fΩ,V,s,s′ is a local (restr: Ω) and microlocal (restr: V) seminorm of f. Our result (n odd): fΩ,V,s,s′ ≤ RfΩ′∩(V×R),s+(n−1)/2 + fs′ shows that if the Ω, V, s, s′ seminorm of f is large, then the corresponding seminorm of Rf on Ω′ ∩ (V × R) must also be large (if fs′ is small). For functions of fixed compact support, global bounds are well known [Hertle, Louis, Hahn Q...]: C1fs ≤ Rfs+ n−1

2

≤ C2fs. Proof idea: Use the inversion formula for the Radon transform, estimates for specific convolution ΨDOs, and estimates on the various norms. For odd n the inverse is local. Similar results can be proven using general ΨDO/FIO theorems but they would not give the constants.

university-logo Limited Data Radon Transforms Results

Discussion

fΩ,V,s,s′ is a local (restr: Ω) and microlocal (restr: V) seminorm of f. Our result (n odd): fΩ,V,s,s′ ≤ RfΩ′∩(V×R),s+(n−1)/2 + fs′ shows that if the Ω, V, s, s′ seminorm of f is large, then the corresponding seminorm of Rf on Ω′ ∩ (V × R) must also be large (if fs′ is small). For functions of fixed compact support, global bounds are well known [Hertle, Louis, Hahn Q...]: C1fs ≤ Rfs+ n−1

2

≤ C2fs. Proof idea: Use the inversion formula for the Radon transform, estimates for specific convolution ΨDOs, and estimates on the various norms. For odd n the inverse is local. Similar results can be proven using general ΨDO/FIO theorems but they would not give the constants.

university-logo Limited Data Radon Transforms Results

Discussion

fΩ,V,s,s′ is a local (restr: Ω) and microlocal (restr: V) seminorm of f. Our result (n odd): fΩ,V,s,s′ ≤ RfΩ′∩(V×R),s+(n−1)/2 + fs′ shows that if the Ω, V, s, s′ seminorm of f is large, then the corresponding seminorm of Rf on Ω′ ∩ (V × R) must also be large (if fs′ is small). For functions of fixed compact support, global bounds are well known [Hertle, Louis, Hahn Q...]: C1fs ≤ Rfs+ n−1

2

≤ C2fs. Proof idea: Use the inversion formula for the Radon transform, estimates for specific convolution ΨDOs, and estimates on the various norms. For odd n the inverse is local. Similar results can be proven using general ΨDO/FIO theorems but they would not give the constants.

university-logo Limited Data Radon Transforms Results

Discussion

fΩ,V,s,s′ is a local (restr: Ω) and microlocal (restr: V) seminorm of f. Our result (n odd): fΩ,V,s,s′ ≤ RfΩ′∩(V×R),s+(n−1)/2 + fs′ shows that if the Ω, V, s, s′ seminorm of f is large, then the corresponding seminorm of Rf on Ω′ ∩ (V × R) must also be large (if fs′ is small). For functions of fixed compact support, global bounds are well known [Hertle, Louis, Hahn Q...]: C1fs ≤ Rfs+ n−1

2

≤ C2fs. Proof idea: Use the inversion formula for the Radon transform, estimates for specific convolution ΨDOs, and estimates on the various norms. For odd n the inverse is local. Similar results can be proven using general ΨDO/FIO theorems but they would not give the constants.

university-logo Limited Data Radon Transforms Results

Discussion

fΩ,V,s,s′ is a local (restr: Ω) and microlocal (restr: V) seminorm of f. Our result (n odd): fΩ,V,s,s′ ≤ RfΩ′∩(V×R),s+(n−1)/2 + fs′ shows that if the Ω, V, s, s′ seminorm of f is large, then the corresponding seminorm of Rf on Ω′ ∩ (V × R) must also be large (if fs′ is small). For functions of fixed compact support, global bounds are well known [Hertle, Louis, Hahn Q...]: C1fs ≤ Rfs+ n−1

2

≤ C2fs. Proof idea: Use the inversion formula for the Radon transform, estimates for specific convolution ΨDOs, and estimates on the various norms. For odd n the inverse is local. Similar results can be proven using general ΨDO/FIO theorems but they would not give the constants.

university-logo Limited Data Radon Transforms Appendix For Further Reading

For Further Reading I

F . Natterer, The Mathematics of Computerized Tomography Teubner, 1986, SIAM, 2001. F . Natterer and F . Wübbling, Mathematical Methods in Image Reconstruction, SIAM, 2001.

• A. Faridani, E.L. Ritman, and K.T. Smith, SIAM J. Appl.
• Math. 52(1992), 459–484,

+Finch II: 57(1997) 1095–1127.

• A. Greenleaf and G. Uhlmann, Duke Math. J. (1989),

205-240. E.T. Quinto, SIAM J. Math. Anal. 24(1993), 1215-1225.

• H. Rullgård and E.T. Quinto, Inverse Problems and

Imaging, to appear.