Tomographic Image Reconstruction in Noisy and Limited Data Settings.

Transcription

1 Tomographic Image Reconstruction in Noisy and Limited Data Settings. Syed Tabish Abbas International Institute of Information Technology, Hyderabad July 1, 2016 Tabish (IIIT-H) July 1, / 39

2 Problem Statement and Contributions We investigate the tomographic reconstruction under 2 scenarios - Noisy Data case. - Limited Data case. and consider the following questions. In linear Radon transform, does reconstruction lattice play a role in quality of reconstructed image? How to reconstruct an image under limited view circular Radon Transform: the Circular arc Radon transform? How to remove the artifacts which arise in the Circular arc Radon transform due to the limited view? Tabish (IIIT-H) July 1, / 39

3 Problem Statement and Contributions We investigate the tomographic reconstruction under 2 scenarios - Noisy Data case. - Limited Data case. and consider the following questions. In linear Radon transform, does reconstruction lattice play a role in quality of reconstructed image? How to reconstruct an image under limited view circular Radon Transform: the Circular arc Radon transform? How to remove the artifacts which arise in the Circular arc Radon transform due to the limited view? Tabish (IIIT-H) July 1, / 39

4 Problem Statement and Contributions We investigate the tomographic reconstruction under 2 scenarios - Noisy Data case. - Limited Data case. and consider the following questions. In linear Radon transform, does reconstruction lattice play a role in quality of reconstructed image? How to reconstruct an image under limited view circular Radon Transform: the Circular arc Radon transform? How to remove the artifacts which arise in the Circular arc Radon transform due to the limited view? Tabish (IIIT-H) July 1, / 39

5 Problem Statement and Contributions We investigate the tomographic reconstruction under 2 scenarios - Noisy Data case. - Limited Data case. and consider the following questions. In linear Radon transform, does reconstruction lattice play a role in quality of reconstructed image? How to reconstruct an image under limited view circular Radon Transform: the Circular arc Radon transform? How to remove the artifacts which arise in the Circular arc Radon transform due to the limited view? Tabish (IIIT-H) July 1, / 39

6 Problem Statement and Contributions We investigate the tomographic reconstruction under 2 scenarios - Noisy Data case. - Limited Data case. and consider the following questions. In linear Radon transform, does reconstruction lattice play a role in quality of reconstructed image? How to reconstruct an image under limited view circular Radon Transform: the Circular arc Radon transform? How to remove the artifacts which arise in the Circular arc Radon transform due to the limited view? Tabish (IIIT-H) July 1, / 39

7 Problem Statement and Contributions We investigate the tomographic reconstruction under 2 scenarios - Noisy Data case. - Limited Data case. and consider the following questions. In linear Radon transform, does reconstruction lattice play a role in quality of reconstructed image? How to reconstruct an image under limited view circular Radon Transform: the Circular arc Radon transform? How to remove the artifacts which arise in the Circular arc Radon transform due to the limited view? Tabish (IIIT-H) July 1, / 39

8 Problem Statement and Contributions We investigate the tomographic reconstruction under 2 scenarios - Noisy Data case. - Limited Data case. and consider the following questions. In linear Radon transform, does reconstruction lattice play a role in quality of reconstructed image? How to reconstruct an image under limited view circular Radon Transform: the Circular arc Radon transform? How to remove the artifacts which arise in the Circular arc Radon transform due to the limited view? Tabish (IIIT-H) July 1, / 39

9 Reconstruction onto Hexagonal Lattices. In linear Radon transform, does reconstruction lattice play a role in quality of reconstructed image? Tabish (IIIT-H) July 1, / 39

10 Positron Emission Tomography(PET) Positron Emission Tomography(PET) is an invasive, nuclear imaging technique involves injecting the patient with a radioactive material(tracer) Tabish (IIIT-H) July 1, / 39

11 Positron Emission Tomography(PET) Positron Emission Tomography(PET) is an invasive, nuclear imaging technique involves injecting the patient with a radioactive material(tracer) PET imaging allows collecting metabolic information about different tissues. Tabish (IIIT-H) July 1, / 39

12 Positron Emission Tomography(PET) Positron Emission Tomography(PET) is an invasive, nuclear imaging technique involves injecting the patient with a radioactive material(tracer) PET imaging allows collecting metabolic information about different tissues. Due to physics of imaging process, PET scans are very noisy. Tabish (IIIT-H) July 1, / 39

13 Positron Emission Tomography(PET) Positron Emission Tomography(PET) is an invasive, nuclear imaging technique involves injecting the patient with a radioactive material(tracer) PET imaging allows collecting metabolic information about different tissues. Due to physics of imaging process, PET scans are very noisy. Figure: Forward Projection. Tabish (IIIT-H) July 1, / 39

14 PET Image Reconstruction PET Images are reconstructed from noisy sinogram data by essentially inverting the forward emission process. Tabish (IIIT-H) July 1, / 39

15 PET Image Reconstruction PET Images are reconstructed from noisy sinogram data by essentially inverting the forward emission process. An approximate inversion is achieved by high pass filtered back projection. Tabish (IIIT-H) July 1, / 39

16 PET Image Reconstruction PET Images are reconstructed from noisy sinogram data by essentially inverting the forward emission process. An approximate inversion is achieved by high pass filtered back projection. Tabish (IIIT-H) July 1, / 39

17 Handling Noisy Data Reconstructed Image 1 1 Herman, 80, 2 Fessler, 00 3 Valiollahzadeh, 13 Tabish (IIIT-H) July 1, / 39

18 Handling Noisy Data More sophisticated methods, like algebraic inversion 1, Statistical inversion 2, etc. have also been proposed Reconstructed Image 1 1 Herman, 80, 2 Fessler, 00 3 Valiollahzadeh, 13 Tabish (IIIT-H) July 1, / 39

19 Handling Noisy Data More sophisticated methods, like algebraic inversion 1, Statistical inversion 2, etc. have also been proposed Other methods, follow a two step process of reconstruction followed by denoising 3. Reconstructed Image 1 1 Herman, 80, 2 Fessler, 00 3 Valiollahzadeh, 13 Tabish (IIIT-H) July 1, / 39

20 Handling Noisy Data More sophisticated methods, like algebraic inversion 1, Statistical inversion 2, etc. have also been proposed Other methods, follow a two step process of reconstruction followed by denoising 3. Reconstruction onto a different lattice has received very little attention. Reconstructed Image 1 1 Herman, 80, 2 Fessler, 00 3 Valiollahzadeh, 13 Tabish (IIIT-H) July 1, / 39

21 Our Pipeline We propose a 2 step reconstruction process onto Hexagonal lattice: Tabish (IIIT-H) July 1, / 39

22 Our Pipeline We propose a 2 step reconstruction process onto Hexagonal lattice: Step 1: Noisy Reconstruction using Filtered Back Projection. Tabish (IIIT-H) July 1, / 39

23 Our Pipeline We propose a 2 step reconstruction process onto Hexagonal lattice: Step 1: Noisy Reconstruction using Filtered Back Projection. Step 2: Denoising using a sparse dictionary learned for the noisy image. Tabish (IIIT-H) July 1, / 39

24 Tiling of Euclidean Plane Figure: Square Tiling of Euclidean Plane Tabish (IIIT-H) July 1, / 39

25 Tiling of Euclidean Plane Figure: Square Tiling of Euclidean Plane Figure: Hexagonal Tiling of Euclidean Plane Tabish (IIIT-H) July 1, / 39

26 Tiling of Euclidean Plane Figure: Square Tiling of Euclidean Plane Figure: Hexagonal Tiling of Euclidean Plane Packing density. Larger, symmetric neighbourhood Tabish (IIIT-H) July 1, / 39

27 Tiling of Euclidean Plane Figure: Square Tiling of Euclidean Plane Packing density. Larger, symmetric neighbourhood Figure: Hexagonal Tiling of Euclidean Plane Irrational Coordinates Tabish (IIIT-H) July 1, / 39

28 Addressing Hexagonal Lattices 2 Figure: Addressing Hexagonal Lattice 2 L Middleton and J Sivaswamy, Tabish (IIIT-H) July 1, / 39

29 Addressing Hexagonal Lattices Use base 7 indices 2 Figure: Addressing Hexagonal Lattice 2 L Middleton and J Sivaswamy, Tabish (IIIT-H) July 1, / 39

30 Addressing Hexagonal Lattices Use base 7 indices Start numbering from center and move out spirally. 2 Figure: Addressing Hexagonal Lattice 2 L Middleton and J Sivaswamy, Tabish (IIIT-H) July 1, / 39

31 Addressing Hexagonal Lattices Use base 7 indices Start numbering from center and move out spirally. 2 Figure: Addressing Hexagonal Lattice 2 L Middleton and J Sivaswamy, Tabish (IIIT-H) July 1, / 39

32 Hexagonal Patch and vectorization Figure: Hexagonal Patch of order 2 Tabish (IIIT-H) July 1, / 39

33 Hexagonal Patch and vectorization Figure: Hexagonal Patch of order 2 Tabish (IIIT-H) July 1, / 39

34 Our Pipeline We propose a 2 step reconstruction process onto Hexagonal lattice: Step 1: Noisy Reconstruction using Filtered Back Projection. Step 2: Denoising using a sparse dictionary learned for the noisy image. Tabish (IIIT-H) July 1, / 39

35 Filtered Back-Projection (FBP) Tabish (IIIT-H) July 1, / 39

36 Filtered Back-Projection (FBP) Image reconstruction (especially in nuclear modalities) is very noisy. Tabish (IIIT-H) July 1, / 39

37 Filtered Back-Projection (FBP) Image reconstruction (especially in nuclear modalities) is very noisy. Back-projection (and also other reconstruction methods) allows a choice of reconstruction lattice. Tabish (IIIT-H) July 1, / 39

38 Our Pipeline We propose a 2 step reconstruction process onto Hexagonal lattice: Step 1: Noisy Reconstruction using Filtered Back Projection. Step 2: Denoising using a sparse dictionary learned for the noisy image. Tabish (IIIT-H) July 1, / 39

39 Dictionary based denoising Learn a dictionary of patches of size 49 (a level 2 patch). 3 3 M Elad and M Aharon, 2006 Tabish (IIIT-H) July 1, / 39

40 Dictionary based denoising Learn a dictionary of patches of size 49 (a level 2 patch). Use the learned dictionary for Denoising. 3 3 M Elad and M Aharon, 2006 Tabish (IIIT-H) July 1, / 39

41 Dictionary based denoising Learn a dictionary of patches of size 49 (a level 2 patch). Use the learned dictionary for Denoising. Dictionary is learned by solving the following optimization problem. 1 min D C,α R k n 2 X Dα 2 F +λ α 1,1 C = {D R m k s.t j = 1,...k, d T j 2 1} 3 3 M Elad and M Aharon, 2006 Tabish (IIIT-H) July 1, / 39

42 Dictionary based denoising Figure: Sample Dictionary atoms Learn a dictionary of patches of size 49 (a level 2 patch). Use the learned dictionary for Denoising. Dictionary is learned by solving the following optimization problem. 1 min D C,α R k n 2 X Dα 2 F +λ α 1,1 C = {D R m k s.t j = 1,...k, d T j 2 1} 3 3 M Elad and M Aharon, 2006 Tabish (IIIT-H) July 1, / 39

43 Qualitative Results Noisy Image Tabish (IIIT-H) July 1, / 39

44 Qualitative Results Square lattice Noisy Image Tabish (IIIT-H) July 1, / 39

45 Qualitative Results Square lattice Noisy Image Hexagonal lattice Tabish (IIIT-H) July 1, / 39

46 Qualitative Results Square lattice Noisy Image Hexagonal lattice Tabish (IIIT-H) July 1, / 39

47 Qualitative Results Square lattice Hexagonal lattice Noisy Image Tabish (IIIT-H) July 1, / 39

48 Qualitative Results Noisy Image Tabish (IIIT-H) July 1, / 39

49 Qualitative Results Square lattice Noisy Image Tabish (IIIT-H) July 1, / 39

50 Qualitative Results Square lattice Noisy Image Hexagonal lattice Tabish (IIIT-H) July 1, / 39

51 Qualitative Results Square lattice Noisy Image Hexagonal lattice Tabish (IIIT-H) July 1, / 39

52 Qualitative Results Square lattice Hexagonal lattice Noisy Image Tabish (IIIT-H) July 1, / 39

53 Quantitative Results Figure: PSNR Comparison Tabish (IIIT-H) July 1, / 39

54 Quantitative Results Figure: PSNR Comparison Figure: Line Profile Tabish (IIIT-H) July 1, / 39

55 Summary & Future Work We Proposed that the change of lattice can improve the reconstruction quality of PET images. The change in lattice improves both the quality and fidelity of the final denoised image. Include the noise model in the denoising step. Provide an analytical explanation for the improvement in reconstruction. Tabish (IIIT-H) July 1, / 39

56 Summary & Future Work We Proposed that the change of lattice can improve the reconstruction quality of PET images. The change in lattice improves both the quality and fidelity of the final denoised image. Include the noise model in the denoising step. Provide an analytical explanation for the improvement in reconstruction. Tabish (IIIT-H) July 1, / 39

57 Summary & Future Work We Proposed that the change of lattice can improve the reconstruction quality of PET images. The change in lattice improves both the quality and fidelity of the final denoised image. Include the noise model in the denoising step. Provide an analytical explanation for the improvement in reconstruction. Tabish (IIIT-H) July 1, / 39

58 Summary & Future Work We Proposed that the change of lattice can improve the reconstruction quality of PET images. The change in lattice improves both the quality and fidelity of the final denoised image. Include the noise model in the denoising step. Provide an analytical explanation for the improvement in reconstruction. Tabish (IIIT-H) July 1, / 39

59 Reconstruction In Limited View Scenario How to reconstruct an image under limited view circular Radon Transform: the Circular arc Radon transform? Tabish (IIIT-H) July 1, / 39

60 Imaging setup Type: Photoacoustic type sensors where source of excitement is EM waves and measurement is acoustic waves. y axis φ α Pφ C(ρ, φ) O x axis Object Figure: Measurement Setup. Tabish (IIIT-H) July 1, / 39

61 Imaging setup Type: Photoacoustic type sensors where source of excitement is EM waves and measurement is acoustic waves. Geometry: The sensors are assumed to be along a circle at points P φ. y axis φ O Object α Pφ C(ρ, φ) x axis Figure: Measurement Setup. Tabish (IIIT-H) July 1, / 39

62 Imaging setup Type: Photoacoustic type sensors where source of excitement is EM waves and measurement is acoustic waves. Geometry: The sensors are assumed to be along a circle at points P φ. Sensor Structure: Each sensor is assumed to have a limited conical view equal to α y axis Pφ α C(ρ, φ) φ O x axis Object Figure: Measurement Setup. Tabish (IIIT-H) July 1, / 39

63 Imaging setup Type: Photoacoustic type sensors where source of excitement is EM waves and measurement is acoustic waves. Geometry: The sensors are assumed to be along a circle at points P φ. Sensor Structure: Each sensor is assumed to have a limited conical view equal to α y axis Pφ ρ α R r C(ρ, φ) θ φ O x axis Object Figure: Measurement Setup. Tabish (IIIT-H) July 1, / 39

64 Mathematial Model y axis We define Circular arc Radon(CAR) Transform g α of a function f as follows ρ α Pφ g α (ρ, φ) = f (r, θ) ds (1) O r θ φ R C(ρ, φ) x axis A α(ρ,φ) Object where is α is the view angle and s is the arc length measure. Figure: Measurement Setup. Tabish (IIIT-H) July 1, / 39

65 Mathematial Model y axis We define Circular arc Radon(CAR) Transform g α of a function f as follows ρ α Pφ g α (ρ, φ) = Object {}}{ f (r, θ) ds (1) O r θ φ R C(ρ, φ) x axis A α(ρ,φ) Object where is α is the view angle and s is the arc length measure. Figure: Measurement Setup. Tabish (IIIT-H) July 1, / 39

66 Mathematial Model y axis We define Circular arc Radon(CAR) Transform g α of a function f as follows ρ α Pφ g α (ρ, φ) = Object {}}{ f (r, θ) ds (1) O r θ φ R C(ρ, φ) x axis A α(ρ,φ) Object where is α is the view angle and s is the arc length measure. Figure: Measurement Setup. Tabish (IIIT-H) July 1, / 39

67 Mathematial Model y axis We define Circular arc Radon(CAR) Transform g α of a function f as follows ρ α Pφ g α (ρ, φ) }{{} Measured Data = A α(ρ,φ) Object {}}{ f (r, θ) ds (1) r θ φ O Object R C(ρ, φ) x axis where is α is the view angle and s is the arc length measure. Figure: Measurement Setup. Tabish (IIIT-H) July 1, / 39

68 CAR Transform: Back projection based inversion An approximate inversion of the transform may be done using an algorithm based on Backprojection, such that Tabish (IIIT-H) July 1, / 39

69 CAR Transform: Back projection based inversion An approximate inversion of the transform may be done using an algorithm based on Backprojection, such that f (x, y) = 2π 0 g(ρ, (x cos φ) 2 + (y sin φ) 2 )dφ Tabish (IIIT-H) July 1, / 39

70 CAR Transform: Back projection based inversion An approximate inversion of the transform may be done using an algorithm based on Backprojection, such that f (x, y) = 2π 0 g(ρ, (x cos φ) 2 + (y sin φ) 2 )dφ (a) Original Phantom (b) α = 5 (c) α = 17 (d) α = 21 Examples of image reconstructions using a näive Backprojection Algorithm Tabish (IIIT-H) July 1, / 39

71 CAR Transform: Back projection based inversion (a) Original Phantom (b) α = 5 (c) α = 17 (d) α = 21 Examples of image reconstructions using a näive Backprojection Algorithm The BP based algotithm is an approximate inversion and leads to lot of artifacts as well as blurring. Tabish (IIIT-H) July 1, / 39

72 CAR Transform: Back projection based inversion (a) Original Phantom (b) α = 5 (c) α = 17 (d) α = 21 Examples of image reconstructions using a näive Backprojection Algorithm The BP based algotithm is an approximate inversion and leads to lot of artifacts as well as blurring. Due the form of transform, it is non-trivial to derive the exact form of the filter. Tabish (IIIT-H) July 1, / 39

73 CAR Transform: Back projection based inversion (a) Original Phantom (b) α = 5 (c) α = 17 (d) α = 21 Examples of image reconstructions using a näive Backprojection Algorithm The BP based algotithm is an approximate inversion and leads to lot of artifacts as well as blurring. Due the form of transform, it is non-trivial to derive the exact form of the filter. To improve the quality of reconstruction, we adopt a Fourier series based solution. Tabish (IIIT-H) July 1, / 39

74 CAR Transform: Fourier Series based analysis g α (ρ, φ) = A α(ρ,φ) f (r, θ) ds Since both f, g are 2π periodic in angular variable, we may expand them into their Fourier series such that, Tabish (IIIT-H) July 1, / 39

75 CAR Transform: Fourier Series based analysis Since both f, g are 2π periodic in angular variable, we may expand them into their Fourier series such that, then, gn α (ρ) e inφ = f n (r)e inθ dθ. n= g α (ρ, φ) = A α(ρ,φ) f (r, θ) ds n= A α(ρ,φ) Tabish (IIIT-H) July 1, / 39

76 CAR Transform: Fourier Series based analysis On Simplifying and equating the Fourier coefficients, the equation reduces to ρ gn α K n (ρ, u) (ρ) = F n (u)du ρ u where and R R 2 +ρ 2 2ρR cos α F n (u) = f n (R u) [ ] 2ρ(R u)t (R u) 2 +R 2 ρ 2 n 2R(R u) K n (ρ, u) =. (2) (u + ρ)(2r + ρ u)(2r ρ u) where, T n (x) = cos(n cos 1 (x)) Tabish (IIIT-H) July 1, / 39

77 CAR Transform: Fourier Series based analysis On Simplifying and equating the Fourier coefficients, the equation reduces to ρ gn α K n (ρ, u) (ρ) = F n (u)du ρ u where and R R 2 +ρ 2 2ρR cos α F n (u) = f n (R u) [ ] 2ρ(R u)t (R u) 2 +R 2 ρ 2 n 2R(R u) K n (ρ, u) =. (2) (u + ρ)(2r + ρ u)(2r ρ u) where, T n (x) = cos(n cos 1 (x)) Tabish (IIIT-H) July 1, / 39

78 CAR Transform: Fourier Series based analysis On Simplifying and equating the Fourier coefficients, the equation reduces to ρ gn α K n (ρ, u) (ρ) = F n (u)du ρ u where and R R 2 +ρ 2 2ρR cos α F n (u) = f n (R u) [ ] 2ρ(R u)t (R u) 2 +R 2 ρ 2 n 2R(R u) K n (ρ, u) =. (2) (u + ρ)(2r + ρ u)(2r ρ u) where, T n (x) = cos(n cos 1 (x)) Tabish (IIIT-H) July 1, / 39

79 CAR Transform: Integral equation g α n (ρ) = ρ R R 2 +ρ 2 2ρR cos α K n (ρ, u) ρ u F n (u)du (3) The equation is a non-standard Volterra integral equation of first kind with a weakly singular kernel. Tabish (IIIT-H) July 1, / 39

80 CAR Transform: Integral equation g α n (ρ) = ρ R R 2 +ρ 2 2ρR cos α Singural Kernel {}}{ K n (ρ, u) ρ u F n (u)du (3) The equation is a non-standard Volterra integral equation of first kind with a weakly singular kernel. Tabish (IIIT-H) July 1, / 39

81 CAR Transform: Integral equation Functions g α n (ρ) = ρ Singural Kernel {}}{ K n (ρ, u) ρ u F n (u)du (3) R R 2 +ρ 2 2ρR cos α The equation is a non-standard Volterra integral equation of first kind with a weakly singular kernel. Tabish (IIIT-H) July 1, / 39

82 CAR Transform: Integral equation Functions g α n (ρ) = ρ Singural Kernel {}}{ K n (ρ, u) ρ u F n (u)du (3) R R 2 +ρ 2 2ρR cos α The equation is a non-standard Volterra integral equation of first kind with a weakly singular kernel. The exact (closed form) solution of such an equation is not known. Tabish (IIIT-H) July 1, / 39

83 CAR Transform: Integral equation Functions g α n (ρ) = ρ Singural Kernel {}}{ K n (ρ, u) ρ u F n (u)du (3) R R 2 +ρ 2 2ρR cos α The equation is a non-standard Volterra integral equation of first kind with a weakly singular kernel. The exact (closed form) solution of such an equation is not known. A direct numerical solution of the equation does not require closed form solution. Tabish (IIIT-H) July 1, / 39

84 Discrete CAR Transform g α n (ρ) = ρ R R 2 +ρ 2 2ρR cos α K n(ρ, u) ρ u F n(u)du. k gn α (ρ k ) = ρ q q=1ρ q 1 F n (u)k n (ρ, u) du. ρ u Tabish (IIIT-H) July 1, / 39

85 Discrete CAR Transform k gn α (ρ k ) = ρ q q=1ρ q 1 F n (u)k n (ρ, u) du. ρ u Approximating the integrand as a linear function over each interval [ρ q 1, ρ q], and integrating we get g n (ρ k ) = k h b kq K n (ρ k, ρ q )F n (ρ q ) q=l Tabish (IIIT-H) July 1, / 39

86 Discrete CAR Transform k gn α (ρ k ) = ρ q q=1ρ q 1 F n (u)k n (ρ, u) du. ρ u Approximating the integrand as a linear function over each interval [ρ q 1, ρ q], and integrating we get where g n (ρ k ) = k h b kq K n (ρ k, ρ q )F n (ρ q ) q=l 4 3 {(k q + 1) (k q) (k q) 1 2 ( ) q = l b kq = 4 (k q + 1) 3 2 2(k q) (k q 1) 3 2 q = l + 1,...k q = k. 3 ( ) and l = max 0, R R 2 + ρ 2 k 2ρ kr cos α where x is the greatest integer less than equal to x. Tabish (IIIT-H) July 1, / 39

87 Discrete CAR Transform g n(ρ k ) = k h b kq K n(ρ k, ρ q)f n(ρ q) q=l The previous equation can be written in the matrix from as gn α = B n F n (4) Tabish (IIIT-H) July 1, / 39

88 Discrete CAR Transform g n(ρ k ) = k h b kq K n(ρ k, ρ q)f n(ρ q) q=l The previous equation can be written in the matrix from as gn α = B n F n (4) where gn α = g α n (ρ 0)... g α n (ρ M 1) Fn = F n(ρ 0 )... F n(ρ M 1 ). Tabish (IIIT-H) July 1, / 39

89 Discrete CAR Transform g n(ρ k ) = k h b kq K n(ρ k, ρ q)f n(ρ q) q=l The previous equation can be written in the matrix from as gn α = B n F n (4) where gn α = g α n (ρ 0)... g α n (ρ M 1) Fn = F n(ρ 0 )... F n(ρ M 1 ). Matrix B n lower triangular matrix which is a piecewise linear, discrete approximation of the integral in Equation (3). Tabish (IIIT-H) July 1, / 39

90 Discrete CAR Transform The previous equation can be written in the matrix from as g α n = B n F n (4) where gn α = g α n (ρ 0)... g α n (ρ M 1) Fn = F n(ρ 0 )... F n(ρ M 1 ). Matrix B n lower triangular matrix which is a piecewise linear, discrete approximation of the integral in Equation (3). Diagonal entries of B n, b ii = 4 3 h 0, hence the matrix is invertible. Tabish (IIIT-H) July 1, / 39

91 Numerical inversion of CAR Transform. g α n = B n F n Matrix B n lower triangular matrix which is a piecewise linear, discrete approximation of the integral in Equation (3). Diagonal entries of B n, b ii = 4 3 h 0, hence the matrix is invertible. The matrix is B n has a high condition number (O(10 15 )), hence direct inversion is unstable. Tabish (IIIT-H) July 1, / 39

92 Numerical inversion of CAR Transform. g α n = B n F n Matrix B n lower triangular matrix which is a piecewise linear, discrete approximation of the integral in Equation (3). Diagonal entries of B n, b ii = 4 3 h 0, hence the matrix is invertible. The matrix is B n has a high condition number (O(10 15 )), hence direct inversion is unstable. We use a Truncated SVD based r-rank inverse (r < M) such that, F n B 1 n,r g α n Tabish (IIIT-H) July 1, / 39

93 Experiments and Results: Effect of Rank original phantom (f ) used in experiments. Tabish (IIIT-H) July 1, / 39

94 Experiments and Results: Effect of Rank Full rank inversion is expected to be unstable. (a) r = n/6 (b) r = n/2 (c) r = 9n/10 (d) r = n Effect of rank r of matrix B n,r on the reconstruction quality. n = 300 Tabish (IIIT-H) July 1, / 39

95 Experiments and Results: Effect of Rank Full rank inversion is expected to be unstable. If the rank r is set to be too low reconstructed image is expected to have ringing artifacts. (a) r = n/6 (b) r = n/2 (c) r = 9n/10 (d) r = n Effect of rank r of matrix B n,r on the reconstruction quality. n = 300 Tabish (IIIT-H) July 1, / 39

96 Experiments and Results: Choosing Rank original phantom (f ) used in experiments. Tabish (IIIT-H) July 1, / 39

97 Experiments and Results: Choosing Rank (a) r = n/3 (b) r = n/2 (c) r = 9n/10 (d) r = n original phantom (f ) used in experiments. Effect of rank r of matrix B n,r on the reconstruction quality. n = 300 Tabish (IIIT-H) July 1, / 39

98 Experiments and Results: Choosing Rank (a) r = n/3 (b) r = n/2 (c) r = 9n/10 (d) r = n Plot of Mean Square Error as a function of rank r. Effect of rank r of matrix B n,r on the reconstruction quality. n = 300 Tabish (IIIT-H) July 1, / 39

99 Experiments and Results: Choosing Rank (a) r = n/3 (b) r = n/2 original phantom (f ) used in experiments where region to (c) r = 9n/10 (d) r = n be zoomed is shown in red. Effect of rank r of matrix B n,r on the reconstruction quality. n = 300 Tabish (IIIT-H) July 1, / 39

100 Experiments and Results: Choosing Rank (a) r = n/3 (b) r = n/2 Based on our experiments, as a rule of thumb, dropping highest 10% of singular values gives a fairly stable reconstruction. (c) r = 9n/10 (d) r = n Effect of rank r of matrix B n,r on the reconstruction quality. n = 300 Tabish (IIIT-H) July 1, / 39

101 Reconstruction In Limited View Scenario How to remove the artifacts which arise in the Circular arc Radon transform due to the limited view? Tabish (IIIT-H) July 1, / 39

102 Reconstruction of Singularities under Radon Transforms. Figure: Image with visualization of projection value along direction shown. Tabish (IIIT-H) July 1, / 39

103 Reconstruction of Singularities under Radon Transforms. Figure: Image with visualization of projection value along direction shown. Tabish (IIIT-H) July 1, / 39

104 Reconstruction of Singularities under Radon Transforms. Figure: Image with visualization of projection value along direction shown. Tabish (IIIT-H) July 1, / 39

105 Reconstruction of Singularities under Radon Transforms. Figure: Image with visualization of projection value along direction shown. Tabish (IIIT-H) July 1, / 39

106 Reconstruction of Singularities under Radon Transforms. Let C be the set of curves, along which we measure projections. Then for an edge to be visible there must be at least one element in the interior of set C, tangential to the edge Figure: Image with visualization of projection value along direction shown. Tabish (IIIT-H) July 1, / 39

107 Effect of limited view Due to limited view not all edges are visible, in the sense of meeting tangency criterion w.r.t set C. Tabish (IIIT-H) July 1, / 39

108 Effect of limited view Due to limited view not all edges are visible, in the sense of meeting tangency criterion w.r.t set C. The end points of the arc lie inside the object, which leads to curves C having discontinuities at end points. Tabish (IIIT-H) July 1, / 39

109 Effect of limited view Due to limited view not all edges are visible, in the sense of meeting tangency criterion w.r.t set C. The end points of the arc lie inside the object, which leads to curves C having discontinuities at end points. The presence of these sharp discontinuities in data set C and limited view will lead to streak and circular artifacts. Tabish (IIIT-H) July 1, / 39

110 Effect of limited view y axis Due to limited view not all edges are visible, in the sense of meeting tangency criterion w.r.t set C. The end points of the arc lie inside the object, which leads to curves C having discontinuities at end points. The presence of these sharp discontinuities in data set C and limited view will lead to streak and circular artifacts. Object Figure: A sharp circular artifact is x axis observed due to discontinuity in angular, as well as radial direction Tabish (IIIT-H) July 1, / 39

111 Reducing artifacts in reconstructed images To reduce the artifacts in the reconstructed images we smooth out the discontinuities of the elements of C Tabish (IIIT-H) July 1, / 39

112 Reducing artifacts in reconstructed images To reduce the artifacts in the reconstructed images we smooth out the discontinuities of the elements of C This is achieved by gracefully decaying arcs to zero at the edges. Tabish (IIIT-H) July 1, / 39

113 Reducing artifacts in reconstructed images To reduce the artifacts in the reconstructed images we smooth out the discontinuities of the elements of C This is achieved by gracefully decaying arcs to zero at the edges. Algorithmically, this achieved by weighing rows of B n by a factor of the form e (i h)2 σ 2 ; visualized below. Tabish (IIIT-H) July 1, / 39

114 Reducing artifacts in reconstructed images To reduce the artifacts in the reconstructed images we smooth out the discontinuities of the elements of C This is achieved by gracefully decaying arcs to zero at the edges. Algorithmically, this achieved by weighing rows of B n by a factor of the form e (i h)2 σ 2 ; visualized below. Visualization of structure of unmodified original matrix B n. Visualization of structure of modified matrix B n for artifact suppression. Tabish (IIIT-H) July 1, / 39

115 Experiments and Results: Artifact Reduction. (a) α = 21 Tabish (IIIT-H) (b) α = 31 (c) α = 46 (d) α = 76 July 1, / 39

116 Experiments and Results: Artifact Reduction. (a) α = 21 (b) α = 31 (c) α = 46 (d) α = 76 (e) α = 21 (f) α = 31 (g) α = 46 (h) α = 76 Reconstructed images corresponding to different α before (row 1), and after artifact suppression (row 2). Tabish (IIIT-H) July 1, / 39

117 Experiments and Results: Artifact Reduction y axis ρ α Pφ φ R x axis Object Figure: Setup with support outside the acquisition circle Tabish (IIIT-H) July 1, / 39

118 y axis O r θ φ Object ρ R α Pφ C(ρ, φ) x axis Experiments and Results: Artifact Reduction g α n (ρ) = R 2 +ρ 2 +2ρR cos α R ρ K n(ρ,u) ρ u F n (u)du Tabish (IIIT-H) July 1, / 39

119 y axis O y axis r θ Object φ φ ρ Pφ R Object ρ R α α Pφ C(ρ, φ) x axis x axis Experiments and Results: Artifact Reduction g α n (ρ) = R 2 +ρ 2 +2ρR cos α R ρ K n(ρ,u) ρ u F n (u)du g α n (ρ) = ρ K n(ρ,u) ρ u F n (u)du R R 2 +ρ 2 2ρR cos α Tabish (IIIT-H) July 1, / 39

120 Experiments and Results: Artifact Reduction Figure: Phantom with support outside the acquisition circle. Tabish (IIIT-H) July 1, / 39

121 Figure: Reconstructed images corresponding to different α before (row 1), and after artifact suppression Tabish (row (IIIT-H) 2), for the support outside case. July 1, / 39 Experiments and Results: Artifact Reduction (a) α = 21 (b) α = 31 (c) α = 46 (d) α = 76

122 Figure: Reconstructed images corresponding to different α before (row 1), and after artifact suppression Tabish (row (IIIT-H) 2), for the support outside case. July 1, / 39 Experiments and Results: Artifact Reduction (a) α = 21 (b) α = 31 (c) α = 46 (d) α = 76 (e) α = 21 (f) α = 31 (g) α = 46 (h) α = 76

123 Summary & Future Work. We Proposed a method of numerical inversion of circular arc Radon transform, a limited view generalization of circular Radon transform. Tabish (IIIT-H) July 1, / 39

124 Summary & Future Work. We Proposed a method of numerical inversion of circular arc Radon transform, a limited view generalization of circular Radon transform. We also proposed a strategy to reduce the artifacts which arise in the image due to limited view. Tabish (IIIT-H) July 1, / 39

125 Summary & Future Work. Provide a rigorous mathematical justification of the artifacts. Tabish (IIIT-H) July 1, / 39

126 Summary & Future Work. Provide a rigorous mathematical justification of the artifacts. Derive a closed form solution of the Volterra integral equation arising in the transform. Tabish (IIIT-H) July 1, / 39

127 Related Publications PET Image Reconstruction And Denoising On Hexagonal Lattices. Syed T. A. and Sivaswamy J. International Conference on Image Processing(ICIP) 2015, Quebec city. Numerical inversion of circular arc Radon transform Syed T. A., Krishnan V. P. and Sivaswamy J. (Under review). Tabish (IIIT-H) July 1, / 39

128 Thank You Tabish (IIIT-H) July 1, / 39