LECTURES ON MICROLOCAL CHARACTERIZATIONS IN LIMITED-ANGLE

Transcription

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

2 4 LECTURES 1 Today: Introduction to the mathematics of computerized tomography 2 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

3 4 LECTURES 1 Today: Introduction to the mathematics of computerized tomography 2 Nov. 18: Introduction to the basic concepts of microlocal analysis 2 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

4 4 LECTURES 1 Today: Introduction to the mathematics of computerized tomography 2 Nov. 18: Introduction to the basic concepts of microlocal analysis 3 Nov. 25: Microlocal analysis of limited angle reconstructions in tomography I 2 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

5 4 LECTURES 1 Today: Introduction to the mathematics of computerized tomography 2 Nov. 18: 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)

6 4 LECTURES 1 Today: Introduction to the mathematics of computerized tomography 2 Nov. 18: 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: F. Natterer, The mathematics of computerized tomography. Stuttgart: B. G. Teubner, 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)

7 ORIGINS OF TOMOGRAPHY Image taken from Johan Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte Längs gewisser Manningsfaltigkeiten, Berichte Sächsische Akadamie der Wissenschaften, Leipzig, Math.-Phys. Kl., 69, pp , Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

8 ORIGINS OF TOMOGRAPHY Allan Cormack Godfrey Hounsfield The Nobel Prize in Physiology or Medicine 1979 was awarded jointly to Allan M. Cormack and Godfrey N. Hounsfield for the development of computer assisted tomography 4 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

9 ORIGINS OF TOMOGRAPHY First scanner, $300 Modern scanner, > $1 million 4 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

10 ORIGINS OF TOMOGRAPHY First clinical scan 1971 Modern scan Case courtesy of Dr Maxime St-Amant, Radiopaedia.org 4 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

11 TODAY Analytical approach to computerized tomography Principle of tomography Radon transform - a mathematical model of tomography Reconstruction via backprojection Fourier slice theorem Inversion formulas & Filtered backprojection Ill-posedness & regularization 5 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

12 PRINCIPLE OF COMPUTED TOMOGRAPHY Beer Lambert law X-rays are attenuated when traveling through object according to di = f (γ(t)) I(t) dt for t R I(0) = I 0 Source f I 0 f = attenuation coefficient, γ = x-ray path γ(t) = s θ + t θ = line with direction θ starting at x detector = sθ Detector I(θ, p) 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

13 PRINCIPLE OF COMPUTED TOMOGRAPHY Beer Lambert law X-rays are attenuated when traveling through object according to di = f (γ(t)) I(t) dt for t R I(0) = I 0 Source f I 0 f = attenuation coefficient, γ = x-ray path γ(t) = s θ + t θ = line with direction θ starting at x detector = sθ Detector I(θ, p) Solution of the initial value problem is given by { tdetector } I(θ, s) = I 0 exp f (s θ + t θ ) dt 0 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

14 PRINCIPLE OF COMPUTED TOMOGRAPHY Beer Lambert law X-rays are attenuated when traveling through object according to di = f (γ(t)) I(t) dt for t R I(0) = I 0 Source f I 0 f = attenuation coefficient, γ = x-ray path γ(t) = s θ + t θ = line with direction θ starting at x detector = sθ Detector I(θ, p) Solution of the initial value problem is given by { tdetector } I(θ, s) = I 0 exp f (s θ + t θ ) dt 0 { } = I 0 exp f (x) dx. L(θ,s) 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

15 PRINCIPLE OF COMPUTED TOMOGRAPHY Beer Lambert law X-rays are attenuated when traveling through object according to di = f (γ(t)) I(t) dt for t R I(0) = I 0 Source f I 0 f = attenuation coefficient, γ = x-ray path γ(t) = s θ + t θ = line with direction θ starting at x detector = sθ Detector I(θ, p) Solution of the initial value problem is given by { tdetector } I(θ, s) = I 0 exp f (s θ + t θ ) dt 0 { } = I 0 exp f (x) dx. L(θ,s) Mathematical model of the measurement process ( ) I 0 R f (θ, s) = f (x) dx = ln L(θ,s) I(θ, s) 6 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

16 RECONSTRUCTION PROBLEM??? Data / Sinogram Sought image Mathematical problem Solve the integral equation y = R f 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

17 RECONSTRUCTION PROBLEM??? Data / Sinogram Sought image Mathematical problem Solve the integral equation y = R f I. Algebraic reconstruction Fully discretize formulation of the problem Linear system of equations Rx = y This is an algebraic problem! Examples: ART, SART, SIRT, statistical reconstruction methods such as ML-EM, variational methods, etc. 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

18 RECONSTRUCTION PROBLEM??? Data / Sinogram Sought image Mathematical problem Solve the integral equation y = R f I. Algebraic reconstruction Fully discretize formulation of the problem Linear system of equations Rx = y This is an algebraic problem! Examples: ART, SART, SIRT, statistical reconstruction methods such as ML-EM, variational methods, etc. II. Analytical reconstruction Study of the continuous problem (above) Derivation of inversion formulas Discretization of analytic reconstruction formulas Examples: Filtered backprojection (FBP), Fourier inversion, etc. 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

19 RECONSTRUCTION PROBLEM??? Data / Sinogram Sought image Mathematical problem Solve the integral equation y = R f I. Algebraic reconstruction Fully discretize formulation of the problem Linear system of equations Rx = y This is an algebraic problem! Examples: ART, SART, SIRT, statistical reconstruction methods such as ML-EM, variational methods, etc. II. Analytical reconstruction Study of the continuous problem (above) Derivation of inversion formulas Discretization of analytic reconstruction formulas Examples: Filtered backprojection (FBP), Fourier inversion, etc. 7 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

20 RADON TRANSFORM IN R n Definition Let f : R n R be a suitably chosen function. The Radon transform of f, denoted by R f, is defined as (1) R f (θ, s) = f (x) dσ(x), (θ, s) S n 1 R, H(θ,s) where H(θ, s) = { x R n : x, θ = s } is the hyperplane with normal vector θ and the signed distance from the origin s. 8 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

21 RADON TRANSFORM IN R n Definition Let f : R n R be a suitably chosen function. The Radon transform of f, denoted by R f, is defined as (1) R f (θ, s) = f (x) dσ(x), (θ, s) S n 1 R, H(θ,s) where H(θ, s) = { x R n : x, θ = s } is the hyperplane with normal vector θ and the signed distance from the origin s. Notation: R θ f (s) = R f (θ, s). This function called projections along direction θ. 8 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

22 RADON TRANSFORM IN R n Definition Let f : R n R be a suitably chosen function. The Radon transform of f, denoted by R f, is defined as (1) R f (θ, s) = f (x) dσ(x), (θ, s) S n 1 R, H(θ,s) where H(θ, s) = { x R n : x, θ = s } is the hyperplane with normal vector θ and the signed distance from the origin s. Notation: R θ f (s) = R f (θ, s). This function called projections along direction θ. Radon transform maps functions to its hyperplane itegrals, 8 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

23 RADON TRANSFORM IN R n Definition Let f : R n R be a suitably chosen function. The Radon transform of f, denoted by R f, is defined as (1) R f (θ, s) = f (x) dσ(x), (θ, s) S n 1 R, H(θ,s) where H(θ, s) = { x R n : x, θ = s } is the hyperplane with normal vector θ and the signed distance from the origin s. Notation: R θ f (s) = R f (θ, s). This function called projections along direction θ. Radon transform maps functions to its hyperplane itegrals, Radon transform is an even function R f ( θ, s) = R f (θ, s) Measurment only w.r.t. half-sphere 8 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

24 RADON TRANSFORM IN R n Definition Let f : R n R be a suitably chosen function. The Radon transform of f, denoted by R f, is defined as (1) R f (θ, s) = f (x) dσ(x), (θ, s) S n 1 R, H(θ,s) where H(θ, s) = { x R n : x, θ = s } is the hyperplane with normal vector θ and the signed distance from the origin s. Notation: R θ f (s) = R f (θ, s). This function called projections along direction θ. Radon transform maps functions to its hyperplane itegrals, Radon transform is an even function R f ( θ, s) = R f (θ, s) Measurment only w.r.t. half-sphere Continuity of the Radon transform = Stability of the measurement process R is continuous on many standard function spaces, such as L 1 (R n ), L 2 (Ω), S(R n ) and many distributional spaces. 8 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

25 RADON TRANSFORM OF RADIAL FUNCTIONS f : R 2 R is radial if the value f (x) only depends on x, i.e., if there is a function ϕ : [0, ) R such that f (x) = ϕ( x ). 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

26 RADON TRANSFORM OF RADIAL FUNCTIONS f : R 2 R is radial if the value f (x) only depends on x, i.e., if there is a function ϕ : [0, ) R such that f (x) = ϕ( x ). In this case we have R f (θ, s) = R = f (sθ + tθ ) dt ϕ( s 2 + t 2 ) dt 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

27 RADON TRANSFORM OF RADIAL FUNCTIONS f : R 2 R is radial if the value f (x) only depends on x, i.e., if there is a function ϕ : [0, ) R such that f (x) = ϕ( x ). In this case we have R f (θ, s) = R = = 2 0 f (sθ + tθ ) dt ϕ( s 2 + t 2 ) dt ϕ( s 2 + t 2 ) dt 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

28 RADON TRANSFORM OF RADIAL FUNCTIONS f : R 2 R is radial if the value f (x) only depends on x, i.e., if there is a function ϕ : [0, ) R such that f (x) = ϕ( x ). In this case we have R f (θ, s) = R = = 2 0 f (sθ + tθ ) dt ϕ( s 2 + t 2 ) dt ϕ( s 2 + t 2 ) dt substitution r 2 = s 2 + t 2 gives 2r dr = 2t dt, and hence ϕ(r)r = 2 s r 2 s dr 2 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

29 RADON TRANSFORM OF RADIAL FUNCTIONS f : R 2 R is radial if the value f (x) only depends on x, i.e., if there is a function ϕ : [0, ) R such that f (x) = ϕ( x ). In this case we have R f (θ, s) = R = = 2 0 f (sθ + tθ ) dt ϕ( s 2 + t 2 ) dt ϕ( s 2 + t 2 ) dt substitution r 2 = s 2 + t 2 gives 2r dr = 2t dt, and hence ϕ(r)r = 2 s r 2 s dr 2 The Radon transform of radial functions is independent of θ! 1 Projection enough to reconstruct a radial function 9 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

30 RADON TRANSFORM OF RADIAL FUNCTIONS Example: f (x) = χ B(0,1) (x) = χ [0,1] ( x ): R f (θ, s) = 2 s χ [0,1] (r)r r 2 s 2 dr 10 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

31 RADON TRANSFORM OF RADIAL FUNCTIONS Example: f (x) = χ B(0,1) (x) = χ [0,1] ( x ): R f (θ, s) = 2 s χ [0,1] (r)r r 2 s 2 dr s > 1 : R f (θ, s) = 0 10 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

32 RADON TRANSFORM OF RADIAL FUNCTIONS Example: f (x) = χ B(0,1) (x) = χ [0,1] ( x ): R f (θ, s) = 2 s χ [0,1] (r)r r 2 s 2 dr s > 1 : R f (θ, s) = 0 1 s < 1 : R f (θ, s) = 2 s r r 2 s dr = 2[ r 1/2] 1 s 2 = 2 1 s Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

33 RADON TRANSFORM OF RADIAL FUNCTIONS Example: f (x) = χ B(0,1) (x) = χ [0,1] ( x ): R f (θ, s) = 2 s χ [0,1] (r)r r 2 s 2 dr s > 1 : R f (θ, s) = 0 1 s < 1 : R f (θ, s) = 2 s r r 2 s dr = 2[ r 1/2] 1 s 2 = 2 1 s So 2 1 s 2, s < 1 R f (θ, s) = 0, otherwise R θ f (s) s 10 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

34 BACKPROJECTION OPERATOR Definition Let g be a (sinogram) function on S n 1 R. Given a projection along the direction θ, we define the backprojection operators along direction θ via Rθ g(x) = g(θ, x θ). 11 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

35 BACKPROJECTION OPERATOR Definition Let g be a (sinogram) function on S n 1 R. Given a projection along the direction θ, we define the backprojection operators along direction θ via Rθ g(x) = g(θ, x θ). The backprojection operator R as follows: R g(x) = R S n 1 θg(x) dσ(θ) = g(θ, x θ) dσ(θ). S n 1 11 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

36 BACKPROJECTION OPERATOR Definition Let g be a (sinogram) function on S n 1 R. Given a projection along the direction θ, we define the backprojection operators along direction θ via Rθ g(x) = g(θ, x θ). The backprojection operator R as follows: R g(x) = R S n 1 θg(x) dσ(θ) = g(θ, x θ) dσ(θ). S n 1 The value R g(x) is an average of all measurements g = R f (θ, s) which correspond to hyperplanes passing through the point x R n. 11 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

37 BACKPROJECTION OPERATOR Definition Let g be a (sinogram) function on S n 1 R. Given a projection along the direction θ, we define the backprojection operators along direction θ via Rθ g(x) = g(θ, x θ). The backprojection operator R as follows: R g(x) = R S n 1 θg(x) dσ(θ) = g(θ, x θ) dσ(θ). S n 1 The value R g(x) is an average of all measurements g = R f (θ, s) which correspond to hyperplanes passing through the point x R n. The operator R is the L 2 Hilbert space adjoint of the Radon transform R, i.e., for f L 2 (Ω) and g L 2 (S n 1 R) R f (θ, s)g(θ, s) dθ ds = f (x)r g(x) dx. R R n S n 1 11 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

38 BACKPROJECTION OPERATOR Definition Let g be a (sinogram) function on S n 1 R. Given a projection along the direction θ, we define the backprojection operators along direction θ via Rθ g(x) = g(θ, x θ). The backprojection operator R as follows: R g(x) = R S n 1 θg(x) dσ(θ) = g(θ, x θ) dσ(θ). S n 1 The value R g(x) is an average of all measurements g = R f (θ, s) which correspond to hyperplanes passing through the point x R n. The operator R is the L 2 Hilbert space adjoint of the Radon transform R, i.e., for f L 2 (Ω) and g L 2 (S n 1 R) R f (θ, s)g(θ, s) dθ ds = f (x)r g(x) dx. R R n S n 1 As an adjoint operator, R is continuous whenever R is 11 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

39 RECONSTRUCTION VIA BACKPROJECTION Theorem For f S(R n ) we have R R f (x) = S n 2 ( f 1 ) (x) 12 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

40 RECONSTRUCTION VIA BACKPROJECTION Theorem For f S(R n ) we have R R f (x) = S n 2 ( f 1 ) (x) Original Backprojection reconstruction 12 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

41 RECONSTRUCTION VIA BACKPROJECTION Theorem For f S(R n ) we have R R f (x) = S n 2 ( f 1 ) (x) Backprojection reconstruction produces a function that is smoother than the original function (by 1 order): high frequencies are damped 12 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

42 RECONSTRUCTION VIA BACKPROJECTION Theorem For f S(R n ) we have R R f (x) = S n 2 ( f 1 ) (x) Backprojection reconstruction produces a function that is smoother than the original function (by 1 order): high frequencies are damped Since R is a continuous operator, the backprojection reconstruction is stable 12 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

43 RECONSTRUCTION VIA BACKPROJECTION Theorem For f S(R n ) we have R R f (x) = S n 2 ( f 1 ) (x) Backprojection reconstruction produces a function that is smoother than the original function (by 1 order): high frequencies are damped Since R is a continuous operator, the backprojection reconstruction is stable To reconstruct the original function f, we need to restore (amplify) high frequencies filtering (sharpening) step 12 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

44 RECONSTRUCTION VIA BACKPROJECTION Theorem For f S(R n ) we have R R f (x) = S n 2 ( f 1 ) (x) Backprojection reconstruction produces a function that is smoother than the original function (by 1 order): high frequencies are damped Since R is a continuous operator, the backprojection reconstruction is stable To reconstruct the original function f, we need to restore (amplify) high frequencies filtering (sharpening) step Using the above theorem, the normal equation R R f = R g (up to a constant) reads f 1 = R g, for data g = R f tomographic reconstruction can be interpreted as a deconvolution problem 12 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

45 FOURIER TRANSFORM Fourier transform turns out to be a very useful tool for studying the Radon transform. Definition (Fourier transform and its inverse) Let f L 1 (R n ). The Fourier transform of f is defined via F f (ξ) ˆ f (ξ) = (2π) n/2 R n f (x)e ix ξ dx. Let g L 1 (R n ). The inverse Fourier transform of f is defined via F 1 g(x) ǧ(x) = (2π) n/2 R n f (ξ)e ix ξ dξ. 13 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

46 FOURIER TRANSFORM Fourier transform turns out to be a very useful tool for studying the Radon transform. Definition (Fourier transform and its inverse) Let f L 1 (R n ). The Fourier transform of f is defined via F f (ξ) ˆ f (ξ) = (2π) n/2 R n f (x)e ix ξ dx. Let g L 1 (R n ). The inverse Fourier transform of f is defined via F 1 g(x) ǧ(x) = (2π) n/2 R n f (ξ)e ix ξ dξ. Sometimes it s useful to calculate the 1D Fourier transform of the projection function g(θ, s) = R f (θ, s) with respect to the second variable s. To make that clear, we will write F s g(θ, σ) = (2π) 1/2 g(θ, s)e is σ ds R for the Fourier transform of g(θ, s) with respect to the variable s (here θ is considered to be a fixed parameter). Whenever we write R f (θ, σ) the Fourier transform of R f has to be understood in that sense. Same holds for the inverse Fourier transform. 13 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

47 RELATION TO THE FOURIER TRANSFORM Theorem (Fourier slice theorem) Let f S(R n ). Then, for σ R, F s R f (θ, σ) = (2π) (n 1)/2 f (σθ). 14 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

48 RELATION TO THE FOURIER TRANSFORM Theorem (Fourier slice theorem) Let f S(R n ). Then, for σ R, F s R f (θ, σ) = (2π) (n 1)/2 f (σθ). f 1D Fourier transform fˆ θ R θ f 14 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

49 RELATION TO THE FOURIER TRANSFORM Theorem (Fourier slice theorem) Let f S(R n ). Then, for σ R, F s R f (θ, σ) = (2π) (n 1)/2 f (σθ). f 1D Fourier transform fˆ θ R θ f This theorem can be used derive a reconstruction procedure Fourier reconstructions 14 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

50 INJECTIVITY OF THE RADON TRANSFORM Many proprities of the Radon transform can be derived from the properties of the Fourier transform. Theorem The Radon transform R : L 1 (R n ) L 1 (S n 1 R) is an injective operator. 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

51 INJECTIVITY OF THE RADON TRANSFORM Many proprities of the Radon transform can be derived from the properties of the Fourier transform. Theorem The Radon transform R : L 1 (R n ) L 1 (S n 1 R) is an injective operator. Proof. Suppose R f 0 for f L 1 (R n ). Then, the Fourier slice theorem implies f (σθ) = (2π) (1 n)/2 F s R f (θ, σ) = 0 for all (θ, σ) S n 1 R. Hence, f 0 and the injectivity of the Fourier transform implies that f Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

52 INJECTIVITY OF THE RADON TRANSFORM Many proprities of the Radon transform can be derived from the properties of the Fourier transform. Theorem The Radon transform R : L 1 (R n ) L 1 (S n 1 R) is an injective operator. So far: There is a unique solution to the reconstruction problem R f = y, formally given as f = R 1 y. 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

53 INJECTIVITY OF THE RADON TRANSFORM Many proprities of the Radon transform can be derived from the properties of the Fourier transform. Theorem The Radon transform R : L 1 (R n ) L 1 (S n 1 R) is an injective operator. So far: There is a unique solution to the reconstruction problem R f = y, formally given as f = R 1 y. Are there explicit expressions for R 1? 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

54 INJECTIVITY OF THE RADON TRANSFORM Many proprities of the Radon transform can be derived from the properties of the Fourier transform. Theorem The Radon transform R : L 1 (R n ) L 1 (S n 1 R) is an injective operator. So far: There is a unique solution to the reconstruction problem R f = y, formally given as f = R 1 y. Are there explicit expressions for R 1? YES! We will derive them in a minitute. 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

55 INJECTIVITY OF THE RADON TRANSFORM Many proprities of the Radon transform can be derived from the properties of the Fourier transform. Theorem The Radon transform R : L 1 (R n ) L 1 (S n 1 R) is an injective operator. So far: There is a unique solution to the reconstruction problem R f = y, formally given as f = R 1 y. Are there explicit expressions for R 1? YES! We will derive them in a minitute. Is R 1 continuous? 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

56 INJECTIVITY OF THE RADON TRANSFORM Many proprities of the Radon transform can be derived from the properties of the Fourier transform. Theorem The Radon transform R : L 1 (R n ) L 1 (S n 1 R) is an injective operator. So far: There is a unique solution to the reconstruction problem R f = y, formally given as f = R 1 y. Are there explicit expressions for R 1? YES! We will derive them in a minitute. Is R 1 continuous? Unfortuntely, NO. It can be shown that R 1 is not a continuous operator and, hence, that the reconstruction problem R f = y is ill-posed. However, the ill-posedness is mild (maybe later). 15 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

57 INVERSION FORMULAS For f S(R n ) and α < n we define the Riesz potential (which is a linear operator) via I α f = ( ) α/2 f = F 1 ( ξ α f (ξ) ). For g (S n 1 R) we analogously define the Riesz-potential with respect to the second variable Is α g(θ, s) = ( ) 2 α/2 ( s R f (θ, s) = F 1 s σ α F s g(θ, σ) ). 16 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

58 INVERSION FORMULAS For f S(R n ) and α < n we define the Riesz potential (which is a linear operator) via I α f = ( ) α/2 f = F 1 ( ξ α f (ξ) ). For g (S n 1 R) we analogously define the Riesz-potential with respect to the second variable Is α g(θ, s) = ( ) 2 α/2 ( s R f (θ, s) = F 1 s σ α F s g(θ, σ) ). Theorem Lef f S(R n ). Then, for any α < n, the following inversion formulas hold: f = 1 2 (2π)1 n I α R I α n+1 s R f. 16 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

59 VARIATIONS OF INVERSION FORMULAS n = 3 and α = 2: f = 1 8π 2 R R f 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

60 VARIATIONS OF INVERSION FORMULAS n = 3 and α = 2: f = 1 8π 2 R R f n = 3 and α = 0: f = 1 8π 2 R I 2 s R f = 1 8π 2 R ( 2 s)r f 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

61 VARIATIONS OF INVERSION FORMULAS n = 3 and α = 2: f = 1 8π 2 R R f n = 3 and α = 0: f = 1 8π 2 R I 2 s R f = 1 8π 2 R ( 2 s)r f n = 2 and α = 0: f = 1 4π R I 1 s R f = 1 4π R ( 2 s) 1/2 R f 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

62 VARIATIONS OF INVERSION FORMULAS n = 3 and α = 2: f = 1 8π 2 R R f n = 3 and α = 0: f = 1 8π 2 R I 2 s R f = 1 8π 2 R ( 2 s)r f n = 2 and α = 0: f = 1 4π R I 1 s R f = 1 4π R ( 2 s) 1/2 R f Structure: Filter & Backproject 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

63 VARIATIONS OF INVERSION FORMULAS n = 3 and α = 2: f = 1 8π 2 R R f n = 3 and α = 0: f = 1 8π 2 R I 2 s R f = 1 8π 2 R ( 2 s)r f n = 2 and α = 0: f = 1 4π R I 1 s R f = 1 4π R ( 2 s) 1/2 R f Structure: Filter & Backproject Sinogram Filtered sinogram 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

64 VARIATIONS OF INVERSION FORMULAS n = 3 and α = 2: f = 1 8π 2 R R f n = 3 and α = 0: f = 1 8π 2 R I 2 s R f = 1 8π 2 R ( 2 s)r f n = 2 and α = 0: f = 1 4π R I 1 s R f = 1 4π R ( 2 s) 1/2 R f Structure: Filter & Backproject Backprojection Filtered backprojection 17 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

65 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS For α = 0 we obtain a classical filtered backprojection form f = 1 2 (2π)1 n R I 1 n s R f, where the operator I 1 n s acts as a filter on the data g = R f. 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

66 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS For α = 0 we obtain a classical filtered backprojection form f = 1 2 (2π)1 n R I 1 n s R f, where the operator I 1 n s acts as a filter on the data g = R f. F s [Is 1 n R θ f ](σ) = σ n 1 F s R θ (σ) = (sgn(σ) σ) n 1 F s R θ (σ) 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

67 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS For α = 0 we obtain a classical filtered backprojection form f = 1 2 (2π)1 n R I 1 n s R f, where the operator I 1 n s acts as a filter on the data g = R f. F s [Is 1 n R θ f ](σ) = σ n 1 F s R θ (σ) = (sgn(σ) σ) n 1 F s R θ (σ) = (sgn(σ)) n 1 σ n 1 F s R θ (σ) 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

68 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS For α = 0 we obtain a classical filtered backprojection form f = 1 2 (2π)1 n R I 1 n s R f, where the operator I 1 n s acts as a filter on the data g = R f. F s [Is 1 n R θ f ](σ) = σ n 1 F s R θ (σ) = (sgn(σ) σ) n 1 F s R θ (σ) = (sgn(σ)) n 1 σ n 1 F s R θ (σ) = (sgn(σ)) n 1 i (n 1) i n 1 σ n 1 F s R θ (σ) 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

69 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS For α = 0 we obtain a classical filtered backprojection form f = 1 2 (2π)1 n R I 1 n s R f, where the operator I 1 n s acts as a filter on the data g = R f. F s [Is 1 n R θ f ](σ) = σ n 1 F s R θ (σ) = (sgn(σ) σ) n 1 F s R θ (σ) = (sgn(σ)) n 1 σ n 1 F s R θ (σ) = (sgn(σ)) n 1 i (n 1) i n 1 σ n 1 F s R θ (σ) = ( i sgn(σ)) n 1 (iσ) n 1 F s R θ (σ) 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

70 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS For α = 0 we obtain a classical filtered backprojection form f = 1 2 (2π)1 n R I 1 n s R f, where the operator I 1 n s acts as a filter on the data g = R f. F s [Is 1 n R θ f ](σ) = σ n 1 F s R θ (σ) because F s [ n sg](σ) = (iσ) n ĝ(σ) we have = (sgn(σ) σ) n 1 F s R θ (σ) = (sgn(σ)) n 1 σ n 1 F s R θ (σ) = (sgn(σ)) n 1 i (n 1) i n 1 σ n 1 F s R θ (σ) = ( i sgn(σ)) n 1 (iσ) n 1 F s R θ (σ) = ( i sgn(σ)) n 1 F s [ n 1 s R θ ](σ) 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

71 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS For α = 0 we obtain a classical filtered backprojection form f = 1 2 (2π)1 n R I 1 n s R f, where the operator I 1 n s acts as a filter on the data g = R f. F s [Is 1 n R θ f ](σ) = σ n 1 F s R θ (σ) because F s [ n sg](σ) = (iσ) n ĝ(σ) we have where H is defined in the Fourier domain via = (sgn(σ) σ) n 1 F s R θ (σ) = (sgn(σ)) n 1 σ n 1 F s R θ (σ) = (sgn(σ)) n 1 i (n 1) i n 1 σ n 1 F s R θ (σ) = ( i sgn(σ)) n 1 (iσ) n 1 F s R θ (σ) = ( i sgn(σ)) n 1 F s [ n 1 s R θ ](σ) = F s [H n 1 n 1 s R θ ](σ), Ĥg(σ) = i sgn(σ) ĝ(σ). 18 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

72 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS Now observe that for n 2 we have ( i sgn(σ)) n 1 ( 1) (n 2)/2 ( i sgn(σ)), = ( 1) (n 1)/2, for n even for n odd 19 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

73 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS Now observe that for n 2 we have ( i sgn(σ)) n 1 ( 1) (n 2)/2 ( i sgn(σ)), = ( 1) (n 1)/2, for n even for n odd Therefore f = 1 2 (2π)1 n R ( 1) (n 2)/2 H n 1 s R f (θ, x θ), for n even ( 1) (n 1)/2 n 1 s R f (θ, x θ), for n odd 19 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

74 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS Now observe that for n 2 we have ( i sgn(σ)) n 1 ( 1) (n 2)/2 ( i sgn(σ)), = ( 1) (n 1)/2, for n even for n odd Therefore f = 1 2 (2π)1 n R ( 1) (n 2)/2 H n 1 s R f (θ, x θ), for n even ( 1) (n 1)/2 n 1 s R f (θ, x θ), for n odd If n is odd, the filter is a differential operator and, hence, a local operator. As a result, the inversion formula is local, i.e., the inversion formula can be evaluated at point x if the data R f (θ, y θ) is known for all θ S n 1 and for y in a neighbourhood of x. 19 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

75 INVERSION FORMULAS IN EVEN AND ODD DIMENSIONS Now observe that for n 2 we have ( i sgn(σ)) n 1 ( 1) (n 2)/2 ( i sgn(σ)), = ( 1) (n 1)/2, for n even for n odd Therefore f = 1 2 (2π)1 n R ( 1) (n 2)/2 H n 1 s R f (θ, x θ), for n even ( 1) (n 1)/2 n 1 s R f (θ, x θ), for n odd If n is odd, the filter is a differential operator and, hence, a local operator. As a result, the inversion formula is local, i.e., the inversion formula can be evaluated at point x if the data R f (θ, y θ) is known for all θ S n 1 and for y in a neighbourhood of x. If n is even, the inversion formula is not local, since H is an integral operator Hg(s) = 1 g(t) π R s t dt H is the so-called Hilbert transform. 19 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

76 LAMBDA RECONSTRUCTION To make the reconstruction formula local for n = 2, the strategy is to have filter which is local (differential operator). 20 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

77 LAMBDA RECONSTRUCTION To make the reconstruction formula local for n = 2, the strategy is to have filter which is local (differential operator). Choosing n = 2 and α = 1 gives I 1 f = 1 4π R Is 2 R f = 1 4π R ( 2 s)r f Instead of reconstructing f we reconstruct Λ f := I 1 f. This formula is local and of filtered backprojection type. 20 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

78 LAMBDA RECONSTRUCTION To make the reconstruction formula local for n = 2, the strategy is to have filter which is local (differential operator). Choosing n = 2 and α = 1 gives I 1 f = 1 4π R Is 2 R f = 1 4π R ( 2 s)r f Instead of reconstructing f we reconstruct Λ f := I 1 f. This formula is local and of filtered backprojection type. In the implementation, the standard FBP filter σ has to be replaces by σ 2 high frequecies are amplified even more, low frequencies are damped. 20 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

79 LAMBDA RECONSTRUCTION To make the reconstruction formula local for n = 2, the strategy is to have filter which is local (differential operator). Choosing n = 2 and α = 1 gives I 1 f = 1 4π R Is 2 R f = 1 4π R ( 2 s)r f Instead of reconstructing f we reconstruct Λ f := I 1 f. This formula is local and of filtered backprojection type. In the implementation, the standard FBP filter σ has to be replaces by σ 2 high frequecies are amplified even more, low frequencies are damped. Instead of values f (x), it reconstructs the singularities of f. 20 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

80 LAMBDA RECONSTRUCTION To make the reconstruction formula local for n = 2, the strategy is to have filter which is local (differential operator). Choosing n = 2 and α = 1 gives I 1 f = 1 4π R Is 2 R f = 1 4π R ( 2 s)r f Instead of reconstructing f we reconstruct Λ f := I 1 f. This formula is local and of filtered backprojection type. In the implementation, the standard FBP filter σ has to be replaces by σ 2 high frequecies are amplified even more, low frequencies are damped. Instead of values f (x), it reconstructs the singularities of f. Original FBP Lambda 20 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

81 ILL-POSEDNESS OF COMPUTED TOMOGRAPHY Reconstruction is mildly ill-posed (of order (n-1)/2) c f H α 0 R f H α+(n 1)/2 C f H α 0 21 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

82 ILL-POSEDNESS OF COMPUTED TOMOGRAPHY Reconstruction is mildly ill-posed (of order (n-1)/2) c f H α 0 R f H α+(n 1)/2 C f H α 0 What causes instabilities in the reconstruction??? 21 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

83 ILL-POSEDNESS OF COMPUTED TOMOGRAPHY Reconstruction is mildly ill-posed (of order (n-1)/2) c f H α 0 R f H α+(n 1)/2 C f H α 0 What causes instabilities in the reconstruction??? We know R and R continuous (stable) operations. 21 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

84 ILL-POSEDNESS OF COMPUTED TOMOGRAPHY Reconstruction is mildly ill-posed (of order (n-1)/2) c f H α 0 R f H α+(n 1)/2 C f H α 0 What causes instabilities in the reconstruction??? We know R and R continuous (stable) operations. Given the inversion formula f = 1/2(2π) 1 n R Is 1 n R f the instabilities must come from filtering: 21 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

85 ILL-POSEDNESS OF COMPUTED TOMOGRAPHY Reconstruction is mildly ill-posed (of order (n-1)/2) c f H α 0 R f H α+(n 1)/2 C f H α 0 What causes instabilities in the reconstruction??? We know R and R continuous (stable) operations. Given the inversion formula f = 1/2(2π) 1 n R Is 1 n R f the instabilities must come from filtering: F s [Is 1 n R f ](θ, σ) = σ n 1 F s R f (θ, σ) = (2π) (n 1)/2 σ n 1 f ˆ(σθ) 21 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

86 ILL-POSEDNESS OF COMPUTED TOMOGRAPHY Reconstruction is mildly ill-posed (of order (n-1)/2) c f H α 0 R f H α+(n 1)/2 C f H α 0 What causes instabilities in the reconstruction??? We know R and R continuous (stable) operations. Given the inversion formula f = 1/2(2π) 1 n R Is 1 n R f the instabilities must come from filtering: F s [Is 1 n R f ](θ, σ) = σ n 1 F s R f (θ, σ) = (2π) (n 1)/2 σ n 1 f ˆ(σθ) High frequencies are amplified! 21 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

87 ILL-POSEDNESS OF COMPUTED TOMOGRAPHY Reconstruction is mildly ill-posed (of order (n-1)/2) c f H α 0 R f H α+(n 1)/2 C f H α 0 What causes instabilities in the reconstruction??? We know R and R continuous (stable) operations. Given the inversion formula f = 1/2(2π) 1 n R Is 1 n R f the instabilities must come from filtering: F s [Is 1 n R f ](θ, σ) = σ n 1 F s R f (θ, σ) = (2π) (n 1)/2 σ n 1 f ˆ(σθ) High frequencies are amplified! This causes instabilities since noise is a high frequency phenomenon. 21 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

88 ILL-POSEDNESS OF COMPUTED TOMOGRAPHY Reconstruction is mildly ill-posed (of order (n-1)/2) c f H α 0 R f H α+(n 1)/2 C f H α 0 What causes instabilities in the reconstruction??? We know R and R continuous (stable) operations. Given the inversion formula f = 1/2(2π) 1 n R Is 1 n R f the instabilities must come from filtering: F s [Is 1 n R f ](θ, σ) = σ n 1 F s R f (θ, σ) = (2π) (n 1)/2 σ n 1 f ˆ(σθ) High frequencies are amplified! This causes instabilities since noise is a high frequency phenomenon. Regularizationation by replacing the filtered with a band-limited version: σ n 1 w(σ) σ n 1 21 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

89 REGULARIZED FBP For α > 0, let ω α : ( 1/α, 1/α) [0, ) be smooth such that ω α (σ) σ as α 0 ( σ), and set ψ α (σ) := F 1 (ω α (σ) σ ). Stabilized FBP inversion R α (g)(x) := 1 (g θ ψ α )(x θ) dσ(θ) 4π S 1 Remark f α = R α (g) is a low-pass filtered version of f Plot of ˆψ α = ω α(σ) σ 22 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)

90 See you next week! 23 Mathematics of computerized tomography Jürgen Frikel (DTU Compute)