Advanced algorithms for Tomography and their high throughput implementation Pierre Paleo
In the last decades, the increasing computing power enabled to use complex algorithms for tomographic reconstruction. These iterative techniques handle the reconstruction as an inverse problem. Prototyping new methods basically involves two steps : building an objective function including the geometry and the a priori knowledge ; and finding an appropriate optimization algorithm.
Introduction
Compressed Sensing ●
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Tomography : technique to obtain a 3D representation of an object Object is scanned at multiple angles : data set of radiographic images (projections)
A breakthrough in signal processing : accurately reconstruct a signal even if the Nyquist condition is not met
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Basic idea : exploit a-priori knowledge like the sparsity in some domain.
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Tomographic reconstruction amounts to an optimization problem :
Combine these projections to reconstruct the interior of the object
(
arg min x
Applications : medicine, material science, paleontology...
Computed Tomography The reconstruction process is done by dedicated algorithms. The two essential operators are the projection and the back-projection.
2
)
‖P x−d ‖2 + g(x)
for ex.
g(x) = ‖∇ x ‖1
(2)
Dictionary Learning technique : the unknown slice is a linear combination of a few elements of a basis of patches, called dictionary. This technique can yield very good reconstruction quality with few projections, especially if the patches are allowed to overlap [1]. Fig. 2 – A phantom and its basis of patches. Source : [1]
Artifacts correction
The formalism (2) can be adapted to correct artifacts arising during the reconstruction. Ring artifacts, especially, can be corrected by adapting (2) to
arg min x ,r
2
‖P x +r−d ‖ + β ‖r ‖ 2
r
+ βTV ‖∇ x ‖1
1
The parameters βTV and βr weight the image regularization and the ring correction, respectively. Fig. 1 – Illustration of Projection and Back-projection
(W. Smith, The Scientist and Engineer's Guide to Digital Signal Processing) ●
Most widely used algorithm : Filtered Back-Projection (FBP)
Optimization algorithms
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Nyquist–Shannon sampling theorem : one should have #projections ≥ # image rows
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Problem for biological samples where the radiation dose is a major concern
The increase of computational power enabled to use more complex algorithms : iterative techniques, handling the tomographic reconstruction as an inverse problem:
solve P x = d projection operator T
slice to be reconstructed
T
(1) acquired data
Neither P, P , P P can be stored as a matrix (70 000 Terabytes of memory for each of them) Equation (1) has to be solved with an iterative algorithm like Least Squares Conjugate Gradient. Other well-known algorithms are Algebraic Reconstruction Method (ART) and Simultaneous Iterative Reconstruction Technique (SIRT). step size
d
P
T
+ + -
α
PT
slice at iteration n
xn
P
2 1 F (x) = ‖P x−d‖2 2 x n+1 = x n−αn ∂ F ( x n )
backward step forward step (back-projection) (projection)
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Fig. 3 – A reconstructed slice (syntactic foam) with FBP (left) and with iterative technique including rings correction (right). Source : [2]
The prior g(x) can be non-smooth, which precludes from using gradient descent algorithms. Recent advances in convex analysis provide adapted methods for problem (2). Optimization algorithms currently used are Nesterov, Chambolle-Pock and Conjugate Subgradient. High throughput implementation ●
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A standard data set contains about 2500 projections of a 2048 × 2048 pixels area – approximately 40 GB of memory. With such amount of data, iterative algorithms have to be highly optimized : convergence rate, iteration cost, data throughput... Reconstruction routines are implemented to run on Graphics Processing Units (GPU) The parallel geometry is well adapted for parallel processing
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Direct Fourier Inversion or Projection can be used to decrease even further the iteration cost.
References [1] A Mirone, E. Brun, E. Gouillart, The PyHST2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities, Nuclear Instruments and Methods in Physics Research Section B, Volume 324, 1 April 2014 [2] P. Paleo, A. Mirone, M. Desvignes, Ring artifacts correction in compressed sensing tomographic reconstruction, Submitted to Journal of Synchrotron Radiation
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