LECTURES ON MICROLOCAL CHARACTERIZATIONS IN LIMITED-ANGLE

Transcription

1 LECTURES ON MICROLOCAL CHARACTERIZATIONS IN LIMITED-ANGLE TOMOGRAPHY Jürgen Frikel

2 4 LECTURES 1 Nov. 11: Introduction to the mathematics of computerized tomography 2 Today: Introduction to the basic concepts of microlocal analysis 3 Nov. 25: Microlocal analysis of limited angle reconstructions in tomography I 4 Dec. 02: Microlocal analysis of limited angle reconstructions in tomography I 2 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

3 4 LECTURES 1 Nov. 11: Introduction to the mathematics of computerized tomography 2 Today: Introduction to the basic concepts of microlocal analysis 3 Nov. 25: Microlocal analysis of limited angle reconstructions in tomography I 4 Dec. 02: Microlocal analysis of limited angle reconstructions in tomography I References: L. Hörmander, The analysis of linear partial differential operators I: Distribution theory and Fourier analysis, vol Berlin: Springer-Verlag, JF and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems 29(12):125007, December Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

4 A REMINDER ON FBP TYPE INVERSION FORMULAS We consider the Radon transform R in 2D and filtered backprojection inversion formulas. Data: g = R f. Classical reconstruction: f (x) = 1 4π R Is 1 g = 1 4π R ( 2 s) 1/2 g = 1 I 1 4π S 1 s g(θ, x θ) dθ = 1 [ψ s g](θ, x θ) dθ, 4π S 1 where the filter ψ is defined in the Fourier domain via ψ(σ) = σ. 3 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

5 A REMINDER ON FBP TYPE INVERSION FORMULAS We consider the Radon transform R in 2D and filtered backprojection inversion formulas. Data: g = R f. Classical reconstruction: f (x) = 1 4π R Is 1 g = 1 4π R ( 2 s) 1/2 g = 1 I 1 4π S 1 s g(θ, x θ) dθ = 1 [ψ s g](θ, x θ) dθ, 4π S 1 where the filter ψ is defined in the Fourier domain via ψ(σ) = σ. Lambda reconstruction: Λ f (x) = 1 4π R ( 2 s)g = 1 [ψ Λ s g](θ, x θ) dθ 4π S 1 where the filter ψ is defined in the Fourier domain via ψ Λ (σ) = σ 2. 3 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

6 GENERAL FORM FBP TYPE INVERSION FORMULAS In Lambda reconstruction a differential operator ( 2 s) is used as filter, which can be written in terms of the Fourier transform as follows (recall F ( s g) = iσ ĝ): ( 2 s)g(s) = F ( 1 σ 2 ĝ(σ) ) = (2π) 1/2 σ 2 ĝ(σ)e isσ ds. R 4 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

7 GENERAL FORM FBP TYPE INVERSION FORMULAS In Lambda reconstruction a differential operator ( 2 s) is used as filter, which can be written in terms of the Fourier transform as follows (recall F ( s g) = iσ ĝ): ( 2 s)g(s) = F ( 1 σ 2 ĝ(σ) ) = (2π) 1/2 σ 2 ĝ(σ)e isσ ds. R In classical reconstructions a fractional power of a differential operator ( 2 s) 1/2 is used as filter. We have a similar representation in the Fourier domain ( 2 s) 1/2 g(s) = F 1 ( σ ĝ(σ)) = (2π) 1/2 σ ĝ(σ)e isσ ds. R However, the difference is huge as this operator is not local any more (involves Hilbert transforms). This is a so-called pseudodifferential operator. 4 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

8 GENERAL FORM FBP TYPE INVERSION FORMULAS In Lambda reconstruction a differential operator ( 2 s) is used as filter, which can be written in terms of the Fourier transform as follows (recall F ( s g) = iσ ĝ): ( 2 s)g(s) = F ( 1 σ 2 ĝ(σ) ) = (2π) 1/2 σ 2 ĝ(σ)e isσ ds. R In classical reconstructions a fractional power of a differential operator ( 2 s) 1/2 is used as filter. We have a similar representation in the Fourier domain ( 2 s) 1/2 g(s) = F 1 ( σ ĝ(σ)) = (2π) 1/2 σ ĝ(σ)e isσ ds. R However, the difference is huge as this operator is not local any more (involves Hilbert transforms). This is a so-called pseudodifferential operator. Let P be any of the above filters, then reconstruction means applying the operator to data g = R f. R := R P 4 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

9 GENERAL FORM FBP TYPE INVERSION FORMULAS In Lambda reconstruction a differential operator ( 2 s) is used as filter, which can be written in terms of the Fourier transform as follows (recall F ( s g) = iσ ĝ): ( 2 s)g(s) = F ( 1 σ 2 ĝ(σ) ) = (2π) 1/2 σ 2 ĝ(σ)e isσ ds. R In classical reconstructions a fractional power of a differential operator ( 2 s) 1/2 is used as filter. We have a similar representation in the Fourier domain ( 2 s) 1/2 g(s) = F 1 ( σ ĝ(σ)) = (2π) 1/2 σ ĝ(σ)e isσ ds. R However, the difference is huge as this operator is not local any more (involves Hilbert transforms). This is a so-called pseudodifferential operator. Let P be any of the above filters, then reconstruction means applying the operator to data g = R f. R := R P General FBP type reconstruction formulas use general pseudodifferential operators P as filters. 4 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

10 GENERAL FORM FBP TYPE INVERSION FORMULAS Since FBP type reconstructions are so easy to implement and since the yield very fast reconstructions, the are applied to all kinds of tomographic data, in particular to incomplete data 5 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

11 GENERAL FORM FBP TYPE INVERSION FORMULAS Since FBP type reconstructions are so easy to implement and since the yield very fast reconstructions, the are applied to all kinds of tomographic data, in particular to incomplete data Limited angle data Sparse angle data Interior data Exterior data... 5 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

12 GENERAL FORM FBP TYPE INVERSION FORMULAS Since FBP type reconstructions are so easy to implement and since the yield very fast reconstructions, the are applied to all kinds of tomographic data, in particular to incomplete data Limited angle data Sparse angle data Interior data Exterior data... In limited angle tomography, the projectios g θ = R θ f are known only for certain directions θ S 1 Φ S 1, for other directions θ the projections g θ = R θ f are unknown. FBP Reconstruction of limited angle data R ( R Φ f )(x) = 1 4π S 1 Φ [ψ s g](θ, x θ) dθ =??? 5 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

13 GENERAL FORM FBP TYPE INVERSION FORMULAS Since FBP type reconstructions are so easy to implement and since the yield very fast reconstructions, the are applied to all kinds of tomographic data, in particular to incomplete data Limited angle data Sparse angle data Interior data Exterior data... In limited angle tomography, the projectios g θ = R θ f are known only for certain directions θ S 1 Φ S 1, for other directions θ the projections g θ = R θ f are unknown. FBP Reconstruction of limited angle data R ( R Φ f )(x) = 1 4π S 1 Φ [ψ s g](θ, x θ) dθ =??? What do we reconstruct? 5 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

14 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 1 Original FBP Lambda [0, 20 ] 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

15 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 1 Original FBP Lambda [0, 40 ] 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

16 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 1 Original FBP Lambda [0, 60 ] 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

17 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 1 Original FBP Lambda [0, 80 ] 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

18 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 1 Original FBP Lambda [0, 100 ] 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

19 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 1 Original FBP Lambda [0, 120 ] 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

20 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 1 Original FBP Lambda [0, 140 ] 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

21 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 1 Original FBP Lambda [0, 160 ] 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

22 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 1 Original FBP Lambda [0, 180 ] 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

23 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 2 Original FBP Lambda [0, 90 ] 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

24 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 2 Original FBP Lambda [20, 110 ] 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

25 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 2 Original FBP Lambda [40, 130 ] 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

26 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 2 Original FBP Lambda [60, 150 ] 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

27 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 2 Original FBP Lambda [80, 170 ] 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

28 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 2 Original FBP Lambda [100, 190 ] 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

29 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 2 Original FBP Lambda [120, 210 ] 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

30 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 2 Original FBP Lambda [140, 230 ] 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

31 LIMITED ANGLE RECONSTRUCTIONS: EXPERIMENT 2 Original FBP Lambda [160, 250 ] 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

32 CHALLENGES IN LIMITED VIEW TOMOGRAPHY Observations at a first glance: Only certain features of the original object can be reconstructed Artifacts are generated 8 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

33 CHALLENGES IN LIMITED VIEW TOMOGRAPHY Observations at a first glance: Only certain features of the original object can be reconstructed Artifacts are generated Goal: Characterize image features that can be reliably reconstructed? Develop strategy to reduce artifacts? 8 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

34 CHALLENGES IN LIMITED VIEW TOMOGRAPHY Observations at a first glance: Only certain features of the original object can be reconstructed Artifacts are generated Goal: Characterize image features that can be reliably reconstructed? Develop strategy to reduce artifacts? Observations at a second glance: Common characteristic features for both types of reconstructions are edges Information about edges is given in terms of projection directions 8 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

35 WHAT ARE EDGES? Practically: Density jumps, boundaries between regions, etc. f 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

36 WHAT ARE EDGES? Practically: Density jumps, boundaries between regions, etc. f Singularities of f 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

37 WHAT ARE EDGES? Practically: Density jumps, boundaries between regions, etc. Mathematically: Where the function is not smooth singularities. f Singularities of f 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

38 WHAT ARE EDGES? Practically: Density jumps, boundaries between regions, etc. Mathematically: Where the function is not smooth singularities. Edges are local and oriented: What s an appropriate mathematical framework? f Singularities of f 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

39 WHAT ARE EDGES? Practically: Density jumps, boundaries between regions, etc. Mathematically: Where the function is not smooth singularities. Edges are local and oriented: What s an appropriate mathematical framework? Image processing: Location of singularities = locations of high gradient magnitude; Direction of singularities = gradient directions f Singularities of f 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

40 WHAT ARE EDGES? Practically: Density jumps, boundaries between regions, etc. Mathematically: Where the function is not smooth singularities. Edges are local and oriented: What s an appropriate mathematical framework? More powerful framework: Microlocal Analysis f Singularities of f 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

41 TODAY Global smoothness vs. decay of Fourier transform Singular locations Singular directions Wavefront sets: simultaneous description of singular locations and directions Examples 10 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

42 GLOBAL SMOOTHNESS VS. DECAY OF FOURIER TRANSFORM In what follows, we only consider real functions f : R 2 R. The smoothness of any function f can be charachterized by means of the decay of its Fourier transform f ˆ: f C k f ˆ(ξ) = O( ξ k ) for ξ. In particular, f C ˆ f (ξ) = O( ξ k ) for all k N. 11 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

43 GLOBAL SMOOTHNESS VS. DECAY OF FOURIER TRANSFORM In what follows, we only consider real functions f : R 2 R. The smoothness of any function f can be charachterized by means of the decay of its Fourier transform f ˆ: f C k f ˆ(ξ) = O( ξ k ) for ξ. In particular, f C ˆ f (ξ) = O( ξ k ) for all k N. Notation: A function which satisfies f (x) = O( x k ) for all k N is called rapidly decreasing. In all other cases slowly decreasing. Definition A function f which is not C is called singular. Equivalently, f is singular if the Fourier transform does not decay rapidly. 11 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

44 GLOBAL SMOOTHNESS VS. DECAY OF FOURIER TRANSFORM In what follows, we only consider real functions f : R 2 R. The smoothness of any function f can be charachterized by means of the decay of its Fourier transform f ˆ: f C k f ˆ(ξ) = O( ξ k ) for ξ. In particular, f C ˆ f (ξ) = O( ξ k ) for all k N. Notation: A function which satisfies f (x) = O( x k ) for all k N is called rapidly decreasing. In all other cases slowly decreasing. Definition A function f which is not C is called singular. Equivalently, f is singular if the Fourier transform does not decay rapidly. Above relations can be proven by using the property F (D α f ) = i α ξ α F f Above relations also hold for distributions in D and S If the Fourier transform of f has compact support (fastest decay one can imagine), then it can be shown that the function f is analytic (Paley-Wiener) One can also use the above relations to define smoothness of order α R (Sobolev-Spaces) 11 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

45 GLOBAL SMOOTHNESS VS. DECAY OF FOURIER TRANSFORM Limitation: No information about locations or directions is provided. Global smoothness is related to a non-directional decay of the Fourier transform. 12 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

46 GLOBAL SMOOTHNESS VS. DECAY OF FOURIER TRANSFORM Limitation: No information about locations or directions is provided. Global smoothness is related to a non-directional decay of the Fourier transform. Example: Let f be a function which is smooth except at the point 0 R 2, i.e. f C (R 2 \ 0). Let y R 2 be arbitrary and let f y (x) = f (x y). Then and hence fˆ y (ξ) = e iξy f ˆ(ξ) ˆ f y (ξ) = ˆ f (ξ). 12 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

47 GLOBAL SMOOTHNESS VS. DECAY OF FOURIER TRANSFORM Limitation: No information about locations or directions is provided. Global smoothness is related to a non-directional decay of the Fourier transform. Example: Let f be a function which is smooth except at the point 0 R 2, i.e. f C (R 2 \ 0). Let y R 2 be arbitrary and let f y (x) = f (x y). Then and hence fˆ y (ξ) = e iξy f ˆ(ξ) ˆ f y (ξ) = ˆ f (ξ). That is, we can generate functions f and f y that have equal decay properties but arbitrarily different locations of jums, discontinuities, etc.. Hence, the decay of the Fourier transform does not provide information about the location of discontinuities. 12 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

48 SINGULAR LOCATIONS How can we gain knowledge about locations of jump, discontinuieties, etc.? What is a proper concept of a singularity that can be used in a distributional setting? 13 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

49 SINGULAR LOCATIONS How can we gain knowledge about locations of jump, discontinuieties, etc.? What is a proper concept of a singularity that can be used in a distributional setting? Idea: Localize and study decay of the Fourier transforms! Definition Let f be a function or distribution. We say that x 0 R 2 is a regular point of f, if there exists a (cut-off) function ϕ C c (R 2 ) with ϕ(x 0 ) 0 such that the Fourier transform F (ϕ f ) decays rapidly (or in other words s.t. ϕ f C (R 2 )). 13 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

50 SINGULAR LOCATIONS How can we gain knowledge about locations of jump, discontinuieties, etc.? What is a proper concept of a singularity that can be used in a distributional setting? Idea: Localize and study decay of the Fourier transforms! Definition Let f be a function or distribution. We say that x 0 R 2 is a regular point of f, if there exists a (cut-off) function ϕ C c (R 2 ) with ϕ(x 0 ) 0 such that the Fourier transform F (ϕ f ) decays rapidly (or in other words s.t. ϕ f C (R 2 )). The complement of the (open) set of all regular points is called the singular support of f and is denoted by sing supp( f ). It can be explicitly stated as sing supp( f ) = { x R 2 : ϕ C c (R 2 ), ϕ(x) 0 : ϕ f C (R 2 ) }. sing supp( f ) supp( f ) sing supp( f ) = iff f C (R 2 ) 13 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

51 SINGULAR DIRECTIONS So far: Locations of singularities are given by the singular support and x sing supp f implies that F (ϕ f ) decays slowly for any cut-off function ϕ(x 0 ) Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

52 SINGULAR DIRECTIONS So far: Locations of singularities are given by the singular support and x sing supp f implies that F (ϕ f ) decays slowly for any cut-off function ϕ(x 0 ) 0. Can there exist directions along which F (ϕ f ) has rapid decay? 14 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

53 SINGULAR DIRECTIONS So far: Locations of singularities are given by the singular support and x sing supp f implies that F (ϕ f ) decays slowly for any cut-off function ϕ(x 0 ) 0. Can there exist directions along which F (ϕ f ) has rapid decay? (Example with a line singularity suggests that this might be the case) 14 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

54 SINGULAR DIRECTIONS So far: Locations of singularities are given by the singular support and x sing supp f implies that F (ϕ f ) decays slowly for any cut-off function ϕ(x 0 ) 0. Can there exist directions along which F (ϕ f ) has rapid decay? (Example with a line singularity suggests that this might be the case) Definition Let f be a be compactly supported distribution. A direction ξ 0 R 2 \ 0 is called regular direction if there exists a conic neighborhood N of ξ 0 such that fˆ decays fast in N. 14 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

55 SINGULAR DIRECTIONS So far: Locations of singularities are given by the singular support and x sing supp f implies that F (ϕ f ) decays slowly for any cut-off function ϕ(x 0 ) 0. Can there exist directions along which F (ϕ f ) has rapid decay? (Example with a line singularity suggests that this might be the case) Definition Let f be a be compactly supported distribution. A direction ξ 0 R 2 \ 0 is called regular direction if there exists a conic neighborhood N of ξ 0 such that fˆ decays fast in N. The complement of the (open) set of all regular directions is called the frequency set of f and is denoted by Σ( f ). It can be explicitly stated as Σ( f ) = { ξ R 2 \ 0 : conic neighb. N of ξ : ˆ f decays slowly in N }. Σ( f ) = iff f C (R 2 ) Singular directions only exist if singularities exist and vice versa. For any compactly supported distribution f and any ϕ Cc (R 2 ) we have Σ(ϕ f ) Σ( f ), i.e., multiplication by compactly supported smooth function does not add singular directions. 14 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

56 LOCATION AND DIRECTION OF A SINGULARITY How can we combine the concepts of location and direction of a singularity? 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

57 LOCATION AND DIRECTION OF A SINGULARITY How can we combine the concepts of location and direction of a singularity? The property Σ(ϕ f ) Σ( f ) is the key. 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

58 LOCATION AND DIRECTION OF A SINGULARITY How can we combine the concepts of location and direction of a singularity? The property Σ(ϕ f ) Σ( f ) is the key. Idea: ZOOM IN 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

59 LOCATION AND DIRECTION OF A SINGULARITY How can we combine the concepts of location and direction of a singularity? The property Σ(ϕ f ) Σ( f ) is the key. Idea: ZOOM IN Choose ϕ k C c (R 2 ) such that supp(ϕ k ) {x} as k, then the idea is that the limit lim k Σ(ϕ f ) will contain only singular directions that correspond to the singularity in x. 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

60 LOCATION AND DIRECTION OF A SINGULARITY How can we combine the concepts of location and direction of a singularity? The property Σ(ϕ f ) Σ( f ) is the key. Idea: ZOOM IN Choose ϕ k C c (R 2 ) such that supp(ϕ k ) {x} as k, then the idea is that the limit lim k Σ(ϕ f ) will contain only singular directions that correspond to the singularity in x. Definition For a distribution f and x 0 R 2 ( f ) we define the localized frequency set as Σ x0 ( f ) = Σ(ϕ f ). ϕ C c,ϕ(x 0 ) 0 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

61 LOCATION AND DIRECTION OF A SINGULARITY How can we combine the concepts of location and direction of a singularity? The property Σ(ϕ f ) Σ( f ) is the key. Idea: ZOOM IN Choose ϕ k C c (R 2 ) such that supp(ϕ k ) {x} as k, then the idea is that the limit lim k Σ(ϕ f ) will contain only singular directions that correspond to the singularity in x. Definition For a distribution f and x 0 R 2 ( f ) we define the localized frequency set as Σ x0 ( f ) = Σ(ϕ f ). ϕ C c,ϕ(x 0 ) 0 Σ x ( f ) Σ( f ) Σ x ( f ) iff x sing supp( f ) Σ x ( f ) is the set of singular directions of f at location x 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

62 THE WAVEFRONT SET Definition The wavefront set of a distribution is the set of all tuples (x, ξ) where x is a singular location of f and ξ is a singular direction of f at x. That is, WF( f ) = { (x, ξ) R 2 (R 2 \ 0) : ξ Σ x ( f ) }. 16 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

63 THE WAVEFRONT SET Definition The wavefront set of a distribution is the set of all tuples (x, ξ) where x is a singular location of f and ξ is a singular direction of f at x. That is, WF( f ) = { (x, ξ) R 2 (R 2 \ 0) : ξ Σ x ( f ) }. WF simultaneously describes locations and directions of singularities: where π 1 (x, ξ) = x and π 2 (x, ξ) = ξ. π 1 (WF( f )) = sing supp( f ), π 2 (WF( f )) = Σ( f ), 16 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

64 THE WAVEFRONT SET Definition The wavefront set of a distribution is the set of all tuples (x, ξ) where x is a singular location of f and ξ is a singular direction of f at x. That is, WF( f ) = { (x, ξ) R 2 (R 2 \ 0) : ξ Σ x ( f ) }. WF simultaneously describes locations and directions of singularities: where π 1 (x, ξ) = x and π 2 (x, ξ) = ξ. π 1 (WF( f )) = sing supp( f ), π 2 (WF( f )) = Σ( f ), Alternative representation: WF( f ) = {x} Σ x ( f ). x sing supp 16 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

65 THE WAVEFRONT SET Definition The wavefront set of a distribution is the set of all tuples (x, ξ) where x is a singular location of f and ξ is a singular direction of f at x. That is, WF( f ) = { (x, ξ) R 2 (R 2 \ 0) : ξ Σ x ( f ) }. WF simultaneously describes locations and directions of singularities: where π 1 (x, ξ) = x and π 2 (x, ξ) = ξ. π 1 (WF( f )) = sing supp( f ), π 2 (WF( f )) = Σ( f ), Alternative representation: WF( f ) = {x} Σ x ( f ). x sing supp A characterization of WF in terms of the complement: (x 0, ξ 0 ) WF( f ) ϕ C c (R 2 ), ϕ(x 0 ) 0 : ξ 0 Σ(ϕ f ) ϕ C c (R 2 ), ϕ(x 0 ) 0 : conic neighb. N(ξ 0 ) : k N : F (ϕ f ) = O( ξ k ) 16 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

66 EXAMPLES 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

67 EXAMPLES Example: Ω R 2 such that the boundary Ω is a smooth manifold: (x, ξ) WF(χ Ω ) x Ω, and ξ N x, where N x is the normal space to Ω at x Ω. 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

68 EXAMPLES Example: Ω R 2 such that the boundary Ω is a smooth manifold: (x, ξ) WF(χ Ω ) x Ω, and ξ N x, where N x is the normal space to Ω at x Ω. ξ x f distribution with supp( f ) D (x, ξdx) WF( f ) 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

69 EXAMPLES Example: Ω R 2 such that the boundary Ω is a smooth manifold: (x, ξ) WF(χ Ω ) x Ω, and ξ N x, where N x is the normal space to Ω at x Ω. x ξ (x, ξ) WF( f ) f Singularities of f 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

70 EXAMPLES Example: Ω R 2 such that the boundary Ω is a smooth manifold: (x, ξ) WF(χ Ω ) x Ω, and ξ N x, where N x is the normal space to Ω at x Ω. What happens at crossings and corners? x ξ (x, ξ) WF( f ) f Singularities of f 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

71 EXAMPLES Example: Ω R 2 such that the boundary Ω is a smooth manifold: (x, ξ) WF(χ Ω ) x Ω, and ξ N x, where N x is the normal space to Ω at x Ω. What happens at crossings and corners? If x is a corner or a crossing singularity (or of similar nature), Σ x ( f ) = R 2 \ 0. x ξ (x, ξ) WF( f ) f Singularities of f 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

72 PSEUDODIFFERENTIAL OPERATORS ΨDO S Theorem (Pseudolocal property of ΨDO s) For any pseudodifferential operator P, we have WF(P f ) WF( f ). If P is elliptic, we even have WF(P f ) = WF( f ). 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

73 PSEUDODIFFERENTIAL OPERATORS ΨDO S Theorem (Pseudolocal property of ΨDO s) For any pseudodifferential operator P, we have WF(P f ) WF( f ). If P is elliptic, we even have WF(P f ) = WF( f ). Pseudodifferential operators do not create new singularities! Example: The Riesz-Potentials are elliptic pseudodifferential operators. I α f = ( ) α/2 f = F 1 ( ξ α ˆ f (ξ)) 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

74 PSEUDODIFFERENTIAL OPERATORS ΨDO S Theorem (Pseudolocal property of ΨDO s) For any pseudodifferential operator P, we have WF(P f ) WF( f ). If P is elliptic, we even have WF(P f ) = WF( f ). Pseudodifferential operators do not create new singularities! The result is also valid for differential operators P as they are special cases of ΨDo s. The result can be also used in order to compute WF, e.g. for Heaviside function H we have H = δ so WF(H) = WF(δ) = {0}. Example: The Riesz-Potentials are elliptic pseudodifferential operators. I α f = ( ) α/2 f = F 1 ( ξ α ˆ f (ξ)) 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

75 CONSEQUENCES FOR TOMOGRAPHY In Λ-tomography we reconstruct Λ f = I 1 f. By ellipticity and the pseudolocal property we get WF(Λ f ) = WF( f ). 19 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

76 CONSEQUENCES FOR TOMOGRAPHY In Λ-tomography we reconstruct Λ f = I 1 f. By ellipticity and the pseudolocal property we get Λ-tomography reconstructs all singularities of f. WF(Λ f ) = WF( f ). 19 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

77 CONSEQUENCES FOR TOMOGRAPHY In Λ-tomography we reconstruct Λ f = I 1 f. By ellipticity and the pseudolocal property we get WF(Λ f ) = WF( f ). Λ-tomography reconstructs all singularities of f. From the inversion formula f = const I 1 n R R f we get similarly WF(R R f ) = WF( f ). 19 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

78 CONSEQUENCES FOR TOMOGRAPHY In Λ-tomography we reconstruct Λ f = I 1 f. By ellipticity and the pseudolocal property we get Λ-tomography reconstructs all singularities of f. WF(Λ f ) = WF( f ). From the inversion formula f = const I 1 n R R f we get similarly WF(R R f ) = WF( f ). Although singularities are differently pronounced in f, R R f and Λ f, we have WF( f ) = WF(R R f ) = WF(Λ f ). WF does not quantify the strength of singularities. To quantify the strength of singularities, a refined notion of the wavefront set can be defined (Sobolev wavefront set). 19 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

79 CONSEQUENCES FOR TOMOGRAPHY In Λ-tomography we reconstruct Λ f = I 1 f. By ellipticity and the pseudolocal property we get Λ-tomography reconstructs all singularities of f. WF(Λ f ) = WF( f ). From the inversion formula f = const I 1 n R R f we get similarly WF(R R f ) = WF( f ). Although singularities are differently pronounced in f, R R f and Λ f, we have WF( f ) = WF(R R f ) = WF(Λ f ). WF does not quantify the strength of singularities. To quantify the strength of singularities, a refined notion of the wavefront set can be defined (Sobolev wavefront set). For general FBP type reconstruction operators R P with general pseudodifferential operators P as filters, a minimum requirement on the filter can be formulated as In general, this is not even if P is elliptic. WF(R PR f )! = WF( f ). 19 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

80 Thanks! Next week: Microlocal analysis in limited angle tomography 20 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)