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

Outline

Description of the Experimental Setup

Basic Reconstruction Algorithms

Recent Papers on TV-regularized Tomography

Limited Angle Tomography

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

Tilt Series

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.

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)

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).

Illustration of the Projection-Slice Theorem

�

�Rf (r)

F(u,v)

f(x,y)

1D-Fourier-Transform

2D-Fourier-Transform

�r

r

u

v

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).

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.

Illustration of the Backprojection Algorithm

Intermediate and final results after 1,6,12,18,24 and 30 projections

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)

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.

Illustration of the Filtered Backprojection Algorithm

Intermediate and final results after 1,6,12,18,24 and 30 projections

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

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 −→

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.

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.

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.

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 −→

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.

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.