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

Limited Data Radon Transforms Stability estimates for limited data Radon transforms Hans Rullgård, Todd Quinto 1 Department of Mathematics Stockholm University 2 Department 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. Such estimates provide one way to measure this 1 ill-posedness. Such estimates can make quantitative the qualitative 2 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 university-logo Wavelets and curvelets (and WF ): Candès, Donoho, et al.

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. Such estimates provide one way to measure this 1 ill-posedness. Such estimates can make quantitative the qualitative 2 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 university-logo Wavelets and curvelets (and WF ): Candès, Donoho, et al.

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. Such estimates provide one way to measure this 1 ill-posedness. Such estimates can make quantitative the qualitative 2 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 university-logo Wavelets and curvelets (and WF ): Candès, Donoho, et al.

Limited Data Radon Transforms Goals and Notation The Radon Hyperplane Transform Hyperplane: ω ∈ S n − 1 , p ∈ R , H ( ω, p ) = { x ∈ R n � � x · ω = p } is the hyperplane perpendicular to ω and containing p ω . Definition (Radon Hyperplane Transform) For f ∈ L 1 ( R n ) and ( ω, p ) ∈ S n − 1 × R , the Radon transform, � Rf ( ω, p ) = f ( x ) dx H , x ∈ H ( ω, p ) is the integral of f over H ( ω, p ) In R 2 , 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 ∈ S n − 1 , p ∈ R , H ( ω, p ) = { x ∈ R n � � x · ω = p } is the hyperplane perpendicular to ω and containing p ω . Definition (Radon Hyperplane Transform) For f ∈ L 1 ( R n ) and ( ω, p ) ∈ S n − 1 × R , the Radon transform, � Rf ( ω, p ) = f ( x ) dx H , x ∈ H ( ω, p ) is the integral of f over H ( ω, p ) In R 2 , 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: � ( F f )( ξ ) = ˆ x ∈ R n e − ix · ξ f ( x ) dx , f ( ξ ) = 1 � ξ ∈ R n e ix · ξ ˆ f ( x ) = f ( ξ ) d ξ ( 2 π ) n Key idea: decrease at ∞ of F f ∼ smoothness of f Definition Let s ∈ R . f ∈ H s ( R n ) if its Sobolev norm � 1 / 2 �� ξ ∈ R n | ˆ f ( ξ ) | 2 ( 1 + | ξ | 2 ) s d ξ � f � s = university-logo is finite.

Limited Data Radon Transforms Microlocal Analysis Fourier Transforms and Sobolev Spaces The Fourier Transform: � ( F f )( ξ ) = ˆ x ∈ R n e − ix · ξ f ( x ) dx , f ( ξ ) = 1 � ξ ∈ R n e ix · ξ ˆ f ( x ) = f ( ξ ) d ξ ( 2 π ) n Key idea: decrease at ∞ of F f ∼ smoothness of f Definition Let s ∈ R . f ∈ H s ( R n ) if its Sobolev norm � 1 / 2 �� ξ ∈ R n | ˆ f ( ξ ) | 2 ( 1 + | ξ | 2 ) s d ξ � f � s = university-logo is finite.

Limited Data Radon Transforms Microlocal Analysis Sobolev seminorms in limited directions. H s norm measures global L 2 smoothness of order s , what about directional smoothness?? Definition Let V be an open cone in R n and let f be a distribution with locally square-integrable Fourier transform. We define � 1 / 2 �� | ˆ f ( ξ ) | 2 ( 1 + | ξ | 2 ) s d ξ � f � V , s = . V university-logo

Limited Data Radon Transforms Microlocal Analysis Sobolev seminorms in limited directions. H s norm measures global L 2 smoothness of order s , what about directional smoothness?? Definition Let V be an open cone in R n and let f be a distribution with locally square-integrable Fourier transform. We define � 1 / 2 �� | ˆ f ( ξ ) | 2 ( 1 + | ξ | 2 ) s d ξ � f � V , s = . V university-logo

Limited Data Radon Transforms Microlocal Analysis Localize + Microlocalize � · � V , s microlocalizes in ξ . Now, localize near x 0 ∈ R n : Multiply f by a smooth cutoff function ϕ ( ϕ ( x 0 ) � = 0) and see if the localized Fourier transform is in H s for certain microlocal directions. Definition (Sobolev Wavefront Set) Let s ∈ R , x 0 ∈ R n and ξ 0 ∈ R n \ 0. The function f is in H s at x 0 in direction ξ 0 if ∃ a cut-off function ϕ near x 0 and an open cone V ∋ ξ 0 such that � ϕ f � s , V < ∞ . On the other hand, ( x 0 , ξ 0 ) ∈ WF s ( f ) if f is not in H s at x 0 in direction ξ 0 . university-logo NOTE: usually ξ 0 is a covector.

Limited Data Radon Transforms Microlocal Analysis Localize + Microlocalize � · � V , s microlocalizes in ξ . Now, localize near x 0 ∈ R n : Multiply f by a smooth cutoff function ϕ ( ϕ ( x 0 ) � = 0) and see if the localized Fourier transform is in H s for certain microlocal directions. Definition (Sobolev Wavefront Set) Let s ∈ R , x 0 ∈ R n and ξ 0 ∈ R n \ 0. The function f is in H s at x 0 in direction ξ 0 if ∃ a cut-off function ϕ near x 0 and an open cone V ∋ ξ 0 such that � ϕ f � s , V < ∞ . On the other hand, ( x 0 , ξ 0 ) ∈ WF s ( f ) if f is not in H s at x 0 in direction ξ 0 . university-logo NOTE: usually ξ 0 is a covector.

Limited Data Radon Transforms Microlocal Analysis Localize + Microlocalize � · � V , s microlocalizes in ξ . Now, localize near x 0 ∈ R n : Multiply f by a smooth cutoff function ϕ ( ϕ ( x 0 ) � = 0) and see if the localized Fourier transform is in H s for certain microlocal directions. Definition (Sobolev Wavefront Set) Let s ∈ R , x 0 ∈ R n and ξ 0 ∈ R n \ 0. The function f is in H s at x 0 in direction ξ 0 if ∃ a cut-off function ϕ near x 0 and an open cone V ∋ ξ 0 such that � ϕ f � s , V < ∞ . On the other hand, ( x 0 , ξ 0 ) ∈ WF s ( f ) if f is not in H s at x 0 in direction ξ 0 . university-logo NOTE: usually ξ 0 is a covector.

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 H 0 = H ( ω 0 , p 0 ) If Rf is in H s + n − 1 near ( ω 0 , p 0 ) (Rf times cutoff is in H s + n − 1 2 ), then f is in 2 H s in direction ± ω 0 at every point on H 0 . In fact, there is a one-to-one correspondence between wavefront of f and wavefront of Rf. So, if Rf is not in H s + n − 1 2 near ( ω 0 , p 0 ) , then for some x 0 ∈ H 0 , ξ 0 parallel ω 0 , ( x 0 , ξ 0 ) ∈ WF s ( f ) . Reasons: See [Q 1993]. Guillemin [1975] showed R is an elliptic FIO of order ( 1 − n ) / 2, and microlocal regularity follows university-logo from the FIO calculus.

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 H 0 = H ( ω 0 , p 0 ) If Rf is in H s + n − 1 near ( ω 0 , p 0 ) (Rf times cutoff is in H s + n − 1 2 ), then f is in 2 H s in direction ± ω 0 at every point on H 0 . In fact, there is a one-to-one correspondence between wavefront of f and wavefront of Rf. So, if Rf is not in H s + n − 1 2 near ( ω 0 , p 0 ) , then for some x 0 ∈ H 0 , ξ 0 parallel ω 0 , ( x 0 , ξ 0 ) ∈ WF s ( f ) . Reasons: See [Q 1993]. Guillemin [1975] showed R is an elliptic FIO of order ( 1 − n ) / 2, and microlocal regularity follows university-logo from the FIO calculus.