Reconstruction methods for sparse-data tomography

Transcription

1 Reconstruction methods for sparse-data tomography Part B: filtered back-projection Samuli Siltanen Department of Mathematics and Statistics University of Helsinki, Finland INDAM intensive period Computational Methods for Inverse Problems in Imaging Como, Italy, June 26 28, 2018

2 Lotus root tomography YouTube search: lotus tomography Video: thanks to Tatiana Bubba, Andreas Hauptmann and Juho Rimpeläinen

3 Reconstruction of a function from its line integrals was first invented by Johann Radon in 1917 f (P) = 1 π 0 df p (q) q Johann Radon ( )

4 Outline X-ray attenuation as line integral The Radon transform and its inverse Illustration of Filtered Back-Projection (FBP)

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7 Wilhelm Conrad Röntgen invented X-rays and was awarded the first Nobel Prize in Physics in 1901

8 X-ray intensity attenuates inside matter, here shown with a homogeneous block

9 Formula for X-ray attenuation along a line inside homogeneous matter An X-ray with intensity I 0 enters a homogeneous physical body. I 0 I 1 } {{ } s The intensity I 1 of the X-ray when it exits the material is I 1 = I 0 e µs, where s is the length of the path of the X-ray inside the body and µ > 0 is X-ray attenuation coefficient.

10 X-ray intensity attenuates inside matter, here shown with two homogeneous blocks

11 A digital X-ray detector counts how many photons arrive at each pixel photon count 1000 X-ray source 1000 Detector

12 Adding material between the source and detector reveals the exponential X-ray attenuation law photon count

13 We take logarithm of the photon counts to compensate for the exponential attenuation law 1000 photon count 1000 log

14 Final calibration step is to subtract the logarithms from the empty space value (here 6.9) photon count log line integral

15 Formula for X-ray attenuation along a line: Beer-Lambert law Let f : [a, b] R be a nonnegative function modelling X-ray attenuation along a line inside a physical body. Beer-Lambert law connects the initial and final intensities: We can also write it in the form I 1 = I 0 e b a f (x)dx. log(i 1 /I 0 ) = b a f (x)dx, where I 0 is known from calibration and I 1 from measurement.

16 Outline X-ray attenuation as line integral The Radon transform and its inverse Illustration of Filtered Back-Projection (FBP)

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18 Let us start with a crucial calculation To reconstruct f at a point x, the most obvious data related to f (x) are the integrals over lines passing through x. Let us sum them all together, call the result Tf (x) and see what we get by introducing polar coordinates: Tf (x) = = = = π 0 2π 0 R 2 where stands for convolution. 0 f (x + tθ)dtdθ f (x + y) dy R 2 y f (y) x y dy = (f (y) 1 y )(x), f (x + tθ) tdtdθ t

19 The Calderón operator Λ is the inverse of T Recall that Fourier transform converts convolution to multiplication (ĝ h = ĝĥ) and that 1 y (ξ) = 1 ξ. Furthermore, define the Calderón operator Λ by ( ) Λf (x) := F 1 ξ ˆf (ξ) = 1 (2π) 2 e ix ξ ξ ˆf (ξ)dξ, R 2 where F 1 is the inverse Fourier transform. Note that Λ can be thought of as a high-pass filter. Now we see that Tf (ξ) = ˆf (ξ) ξ, and thus ΛTf = f.

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22 Radon transform organizes sets of line integrals Rf (θ, s) = s f (x)dx x θ=s θ

23 Definition of the Radon transform Let f (x) = f (x 1, x 2 ) be the X-ray attenuation coefficient. The classical model for tomographic data is the Radon transform Rf (θ, s) = f (x)dx = x θ=s f (sθ + y)dy, y θ θ S 1, s R, where S 1 is the unit circle, θ is the orthogonal complement of the unit vector θ and x θ denotes vector inner product. Note that f is defined on R 2 and Rf is defined on S 1 R 1. Also: Rf (θ, s) = Rf ( θ, s). θ [0, 2π].

24 Radon transform as sinogram

25 Adjoint of the Radon transform R: the back-projection operator R Use Fubini s theorem and change of variables x = sθ + y with y θ and dx = dsdy and s = θ x to get Rf (θ, s) g(θ, s) dsdθ S 1 R 1 ( ) = f (sθ + y)dy g(θ, s) dsdθ S 1 R 1 θ ( ) = f (x) g(θ, x θ) dx dθ S 1 R 2 ( ) = f (x) g(θ, x θ)dθ dx R 2 S 1 =: f (x) R g(x) dx R 2 Both f and R g are defined on R 2. Note that Tf = R Rf.

26 We arrive at the modern way of writing the filtered back-projection reconstruction formula f = ΛR Rf ( f (P) = 1 π 0 ) df p (q) q Johann Radon ( )

27 Outline X-ray attenuation as line integral The Radon transform and its inverse Illustration of Filtered Back-Projection (FBP)

28 Simple example of tomographic imaging with a double-disc target

29 The inverse problem of tomography is to recover the unknown target from the measured X-ray data

30 Since we know the projection directions we can back-project the data into the image

31 Since we know the projection directions we can back-project the data into the image

32 Since we know the projection directions we can back-project the data into the image

33 Since we know the projection directions we can back-project the data into the image

34 Since we know the projection directions we can back-project the data into the image

35 Since we know the projection directions we can back-project the data into the image

36 Since we know the projection directions we can back-project the data into the image

37 Back-projection becomes more useful by summing up the images

38 Summing all the back-projections results in a blurred reconstruction

39 Summing all the back-projections results in a blurred reconstruction

40 Summing all the back-projections results in a blurred reconstruction

41 Summing all the back-projections results in a blurred reconstruction

42 Summing all the back-projections results in a blurred reconstruction

43 Summing all the back-projections results in a blurred reconstruction

44 Summing all the back-projections results in a blurred reconstruction

45 Here we use more directions, so the reconstruction quality is higher

46 Final reconstruction involves filtering on top of the back-projection ξ fb(ξ) fb(ξ) Multiplication with ice-cream cone FFT IFFT

47 These books are recommended for learning the mathematics of practical X-ray tomography 1983 Deans: The Radon Transform and Some of Its Applications 1986 Natterer: The mathematics of computerized tomography 1988 Kak & Slaney: Principles of computerized tomographic imaging 1996 Engl, Hanke & Neubauer: Regularization of inverse problems 1998 Hansen: Rank-deficient and discrete ill-posed problems 2001 Natterer & Wübbeling: Mathematical Methods in Image Reconstruction 2008 Buzug: Computed Tomography: From Photon Statistics to Modern Cone-Beam CT 2008 Epstein: Introduction to the mathematics of medical imaging 2010 Hansen: Discrete inverse problems 2012 Mueller & S: Linear and Nonlinear Inverse Problems with Practical Applications 2014 Kuchment: The Radon Transform and Medical Imaging

48 Thank you for your attention!