4 research outputs found
A Convolutional Forward and Back-Projection Model for Fan-Beam Geometry
Iterative methods for tomographic image reconstruction have great potential
for enabling high quality imaging from low-dose projection data. The
computational burden of iterative reconstruction algorithms, however, has been
an impediment in their adoption in practical CT reconstruction problems. We
present an approach for highly efficient and accurate computation of forward
model for image reconstruction in fan-beam geometry in X-ray CT. The efficiency
of computations makes this approach suitable for large-scale optimization
algorithms with on-the-fly, memory-less, computations of the forward and
back-projection. Our experiments demonstrate the improvements in accuracy as
well as efficiency of our model, specifically for first-order box splines
(i.e., pixel-basis) compared to recently developed methods for this purpose,
namely Look-up Table-based Ray Integration (LTRI) and Separable Footprints (SF)
in 2-D.Comment: This paper was submitted to IEEE-TMI, and it's an extension of our
ISBI paper (https://ieeexplore.ieee.org/abstract/document/8759285
Total Deep Variation: A Stable Regularizer for Inverse Problems
Various problems in computer vision and medical imaging can be cast as
inverse problems. A frequent method for solving inverse problems is the
variational approach, which amounts to minimizing an energy composed of a data
fidelity term and a regularizer. Classically, handcrafted regularizers are
used, which are commonly outperformed by state-of-the-art deep learning
approaches. In this work, we combine the variational formulation of inverse
problems with deep learning by introducing the data-driven general-purpose
total deep variation regularizer. In its core, a convolutional neural network
extracts local features on multiple scales and in successive blocks. This
combination allows for a rigorous mathematical analysis including an optimal
control formulation of the training problem in a mean-field setting and a
stability analysis with respect to the initial values and the parameters of the
regularizer. In addition, we experimentally verify the robustness against
adversarial attacks and numerically derive upper bounds for the generalization
error. Finally, we achieve state-of-the-art results for numerous imaging tasks.Comment: 30 pages, 12 figures. arXiv admin note: text overlap with
arXiv:2001.0500
Total Deep Variation for Linear Inverse Problems
Diverse inverse problems in imaging can be cast as variational problems
composed of a task-specific data fidelity term and a regularization term. In
this paper, we propose a novel learnable general-purpose regularizer exploiting
recent architectural design patterns from deep learning. We cast the learning
problem as a discrete sampled optimal control problem, for which we derive the
adjoint state equations and an optimality condition. By exploiting the
variational structure of our approach, we perform a sensitivity analysis with
respect to the learned parameters obtained from different training datasets.
Moreover, we carry out a nonlinear eigenfunction analysis, which reveals
interesting properties of the learned regularizer. We show state-of-the-art
performance for classical image restoration and medical image reconstruction
problems.Comment: 21 pages, 10 figure
Convergence analysis of pixel-driven Radon and fanbeam transforms
This paper presents a novel mathematical framework for understanding
pixel-driven approaches for the parallel beam Radon transform as well as for
the fanbeam transform, showing that with the correct discretization strategy,
convergence - including rates - in the operator norm can be obtained.
These rates inform about suitable strategies for discretization of the
occurring domains/variables, and are first established for the Radon transform.
In particular, discretizing the detector in the same magnitude as the image
pixels (which is standard practice) might not be ideal and in fact,
asymptotically smaller pixels than detectors lead to convergence. Possible
adjustments to limited-angle and sparse-angle Radon transforms are discussed,
and similar convergence results are shown. In the same vein, convergence
results are readily extended to a novel pixel-driven approach to the fanbeam
transform. Numerical aspects of the discretization scheme are discussed, and it
is shown in particular that with the correct discretization strategy, the
typical high-frequency artifacts can be avoided.Comment: 33 pages, 8 figure