tv-regularization in tomography part 1: introduction and...
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TV-Regularization in TomographyPart 1: Introduction and Overview
Philipp Lamby
Interdisciplinary Mathematics Institute,University of South Carolina
Peter Binev, Wolfgang Dahmen, Ronald DeVore,Philipp Lamby, Andreas Platen, Dan Savu, Robert Sharpley
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Outline
Description of the Experimental Setup
Basic Reconstruction Algorithms
Recent Papers on TV-regularized Tomography
Limited Angle Tomography
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Experimental Setup
I The aim is to identify the position, size and shape of heavyatom clusters in a carrier material from a tilt series of STEMmicrographs.
I The electron gun scans theobject rastering along aCartesian grid in thexy -plane.
I The specimen is tiltedaround an axis parallel tothe y -axis.
I We are confined to alimited tilt range (±600)
I The mechanics is notprecise.
y
x
z
Sk
k
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Tilt Series
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Program
Work Steps
1. Register the micrographs.
2. Invert the Radon transform in every slice y = const.
3. Stack the slices to get a 3D reconstruction.
Notes on Step 2
I Problem is a typical limited angle tomography from parallelprojections.
I The intensity values measure the Z 2-distribution along therays taken by the electron beams.
I We will make the assumption that the distribution in theslices can be well represented by piecewise constant functions.
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The Radon Transform (2D)
Let θ ∈ [0, 2π], r ∈ R. Set eθ = (cos(θ), sin(θ)). Define
L(θ, r) :={x ∈ R2 : 〈eθ, x〉 = r
}.
Definition
Rf (θ, r) :=f̂ (θ, r) := f̂θ(r) :=
∫L(θ,r)
f (x)dm(x)
=
∫ ∞−∞
f (r cos θ − t sin θ, r sin θ + t cos θ) dt
Symmetry: f̂ (θ, r) = f̂ (θ + π,−r)
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The Projection-Slice Theorem
Let s ∈ R and θ ∈ [0, 2π] and denote with
f̃ (k) :=
∫Rn
f (x)e−ix ·kdx , k ∈ Rn.
the Fourier-transform of a n-variate function f . Then
f̃ (seθ) = ˜̂fθ(s).
Proof: f̃ (seθ) =
∫ ∞−∞
(∫x ·eθ=r
f (x)e−isx ·eθ dm(x)
)dr
=
∫ ∞−∞
(∫x ·eθ=r
f (x)dm(x)
)e−isr dr
=
∫ ∞−∞
f̂ (θ, r)e−isr dr = ˜̂fθ(s).
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Illustration of the Projection-Slice Theorem
�
�Rf (r)
F(u,v)
f(x,y)
1D-Fourier-Transform
2D-Fourier-Transform
�r
r
u
v
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Fourier-Methods
I The PST motivates the following algorithm:I Measure the projections f̂θi , i = 1, . . . , n
I Compute their Fourier transforms ˜̂fθi .I Interpolate the Fourier data onto a Cartesian grid.I Take the inverse 2D FFT to obtain f (x , y).
I This algorithm has not been very popular in the past, becauseI The interpolation step leads to inaccuracies,I The algorithm requires complete data.
I But there are new developments:I Equally-Sloped Tomography (later).
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Backprojection
I The Radon transform integrates the values of f along a line.
I Its dual operator averages the value of all the line integralsthrough a given point:
2Bf̂ (x) = ˇ̂f (x) =1
2π
∫ 2π
0f̂ (θ, xeθ) dθ
I Backprojection is the adjoint and “almost” an inverse of theRadon transform:
ˇ̂f (x) =1
r∗ f (x)
I ϕ̌ is an average of plane waves.
I In practice one replaces the integral with a trapezoidal sum.
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Illustration of the Backprojection Algorithm
Intermediate and final results after 1,6,12,18,24 and 30 projections
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Filtered Backprojection
Using the inverse Fourier transform, the PST and introducing polarcoordinates one gets:
f (x) =1
2π
∫R2
f̃ (k)e ixkdk
=1
2π
∫ 2π
0
∫ ∞0
f̃ (reθ)e ixreθ r dr dθ
=1
2π
∫ π
0
∫ ∞−∞
˜̂fθ(r) |r | e irxeθ dr dθ
=1
2π
∫ π
0F−1[˜̂fθ(r) |r |](xeθ) dθ
=: B[Q(θ, r)](x)
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Filtered Backprojection Algorithm
For k = 1, . . . , nθI Measure the Projection f̂θk (r).
I Compute its 1D-Fourier Transform
I Multiply it with |r |I Take the inverse 1D-Fourier Transform to get Q(θk , r)
I Backproject Q(θk , r) to the image domain
Notes
I In practice one has to sample the projections.
I This requires a modification of the filter, because theprojections are not strictly band-limited.
I This leads to convolution-backprojection methods.
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Illustration of the Filtered Backprojection Algorithm
Intermediate and final results after 1,6,12,18,24 and 30 projections
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TV-Regularization in Image Processing
TV-norm of an N × N Grayscale Image
cont. : ‖f ‖TV =∫‖∇f ‖ dx
isotrop : ‖f ‖TV =∑N
i ,j=2
√(fi ,j − fi−1,j)2 + (fi ,j − fi ,j−1)2
anisotrop : ‖f ‖TVa =∑N
i ,j=2 |fi ,j − fi−1,j |+ |fi ,j − fi ,j−1|
ROF-Image Denoising model
Given a noisy image v , find
u∗ = arg minu
1
2‖u − v‖22 + µ‖u‖TV
I Removes noise and fine scale artifacts
I Perserves sharp edges
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The Candes-Romberg-Tao Experiment
I Idea: Take an N × N-image g and compute its discrete 2DFourier transform Fg(k) for k ∈ K ⊂ {−N/2, . . . ,N/2− 1}2.Then find (∗ = `2 or ∗ = TV )
f ∗ = arg min ‖f ‖∗, s.t. F f (k) = Fg(k), k ∈ K .
I Observation: Many piecewise constant functions can bereconstructed “exactly” from only a small number ofcoefficients.
Reconstruction
from 22 “ra-
dial” lines.
←− `2
TV −→
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Remarks
Discrete Fourier Transform
F f (k) = F f (kx , ky ) =N−1∑i=0
N−1∑j=0
f (i , j)e−2πi(kx i+ky j)/N
FFT
I Computes the points on a Cartesian grid in frequency space.
Notes
I The CRT experiment does not reflect a practical situation.
I It would be prohibitively expensive to compute the Fouriertransform for points on a polar grid.
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Equally-Sloped Tomography
AuthorsA. Averbuch, R.R. Coifman, D.L. Donoho, M.Israeli, Y.Shkolnisky,I.Sedelnikov (2008).
Idea
I Define an pseudo-polar grid which allows for a fast,algebraically exact Fourier transform.
I There is a close connection to shearlets.
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The Mao-Fahimian-Osher-Miao Paper (2010)
I Given the Fourier transformation on a pseudo-polar grid, onecan compute the DFT of other points on the radial lines by1D trigonometric interpolation.
I Measure the projections for equally sloped directions andcompute their 1D-Fourier transforms.
I Let S be the operator that computes from the values on thepp-grid the values of the DFT at the “measured” frequencysamples f .
I Then find
u∗ = arg minu‖u‖TV s.t. SFpp(u) = f .
I Solve this optimization problem with Split-Bregman iterations.
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Limited Angle Tomography
I Often, projections are not available for all directions.
I Frequency information is missing.
I Fourier methods and backprojection fail.
Reconstructionfrom 22 linesin 0o − 120o
←− FBP
CRT −→
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Limited Angle Tomography
QuestionIs this just a problem of these particular algorithms, or is there adeeper reason for these artifacts?
Some kind of an answer
I No finite number of projections determines the phantomuniquely, (i.e. the reconstruction problem is ill-posed)
I Any infinite number of projections determines the phantomuniquely. (even a thin wedge of data in the Fourier domain!)
I But the Radon data in the limited tilt range does not givestable information about singularities parallel to missingdirections.
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Algorithms for the Limited Angle Problem
Iterated Backprojection
I Compute an initial guess for the reconstruction.I Until convergence, do
I Compute the missing data from the current solution.I Backproject using the measured and the computed data.I Enforce constraints in the reconstruction.
Sinogram Restoration
I Estimate the missing data (usually by series expansion)
I Then backproject.
Regularized ART
I Our choice - see next talk.