Carlos Milovic^{1,2}, Jose Miguel Pinto^{1,2}, Julio Acosta-Cabronero^{3}, Pablo Irarrazaval^{1,2}, and Cristian Tejos^{1,2}

^{1}Department of Electrical Engineering, Pontificia Universidad Catolica de Chile, Santiago, Chile, ^{2}Biomedical Imaging Center, Pontificia Universidad Catolica de Chile, Santiago, Chile, ^{3}German Center for Neurodegenerative Deceases (DZNE), Magdeburg, Germany

### Synopsis

**An
strategy for parameter selection in TGV regularized reconstruction problems
is presented, with applications to deconvolutions and QSM. This
allows fine-tuning of parameters in an efficient way and the use of
predictors that are correlated to optimized results in terms of MSRE.
This allows users to automatize or accelerate the parameter
selection, critical in expensive problems such as QSM and reduce the
error in the reconstruction.**### Purpose

A common approach for
image reconstruction is the optimization of a functional with a
regularization term. Tikhonov, Total Variation and Sparse
Representations (Compressed Sensing) are typical regularizers. Total
Generalized Variation1 (TGV) is a relatively new
regularizer that addresses most of the common problems of the
mentioned approaches, yielding piece-wise smooth images, without
staircase artifacts or blurry edges. TGV involves the selection of
three parameters: α0,
α1 and λ.
Whereas α0 and α1
regulate an anisotropic diffusion process, λ weights a data fidelity
term. The ratio α0/α1is
normally set between 0.5 and 2.0, so only α1
and λ are free parameters1,2,3,4. Reconstruction results
tend to be highly sensitive to the chosen α1
and λ values and there is no consensus on how to set them;
therefore, selecting those parameters is done heuristically in a slow
and tedious process. Although α1
and λ are somehow linked, how they affect the quality of the
reconstructions remains unclear. Additionally, there are some TGV
applications such as Quantitative Susceptibility Mapping (QSM) and
Undersampled Reconstructions where traditional strategies to set
reconstruction parameters (e.g. L-curve) may fail or cannot be
applied. We present a strategy for parameter selection in TGV-based
reconstruction problems. Our method delivers quasi-optimal
reconstruction parameters with a relatively small number of sampled
points, and can be applied to a variety of expensive reconstruction
problems including QSM.

### Methods

We discovered that the
Mean Square Errors (MSE) of TGV reconstructions show a common
pattern, for a variety of applications including deconvolution,
undersampled reconstruction and QSM. Using TGV we reconstructed
several synthetic images from corrupted samples (blurred and additive
Gaussian noise with different SNR). Each image was reconstructed with
22x22 combinations of α1
and λ values. Plotting the MSE between the ground truth and the
reconstructed image as a function of (-log(α1),-log(λ))
we found that the minimal error was always at the dark 45º diagonal
(Fig. 1). For different images the 45º diagonal changed its location
but did not rotate.
The optimal (α1,
λ) values can be approximately found a priori, i.e. without knowing
the ground truth. The first step is to locate the dark 45º diagonal.
This can be done reconstructing images with a few (α1,
λ) pairs (mapping a -45º diagonal) and choosing: (Metric 1) the one
that maximizes the $$$L_2$$$-norm
of the updated diffusion image of the TGV process for the case of
deconvolution and undersampled reconstructions
(Fig2)
or (Metric 2) the one that minimizes the $$$L_2$$$-norm of the
difference between the diffusion image (v) and data fidelity (f) of the TGV
process for the case of QSM reconstructions (Fig.3).
The second step is to
locate the optimum within the identified 45º diagonal. This can be
done reconstructing images with a few (α1,
λ) pairs (mapping the identified 45º diagonal) and choosing the one
according to Metric 2.

### Results

We tested our algorithm
for three TGV-based applications: deconvolution (noisy blurred
synthetic bi-gradient image, Fig 2), undersampled reconstruction
(not-shown), and QSM (reconstruction of analytic
phantom, Fig. 3).

For deconvolution, we obtained an error 6% larger than the optimal, with slightly diffeferent parameters. For QSM our methods obtained a larger error (20%), however the resulting image showed almost no visual difference. Defferences for the obtained parameters were also small, in this case due to a discrepancy generated in the first step of our search.

### Discussion

Our results indicate
that our strategy is valid for setting TGV-based reconstruction
parameters. Compared with an exhaustive search that typically needs
of the order of $$$N^2$$$ reconstructed points, our method needs of
the order of $$$2N$$$ points since the search is within two
orthogonal lines that can be approximately identified, thus achieving
an important reduction of processing time. Although the parameter
search is not perfect, the reconstructed error is relatively small
when compared with the optimal found from ground truth experiments.
Additionally, the search could be processed completely blind, without
any visual inspection as is usually needed for exhaustive parameter
search processes. Finally, our proposed metrics produced relatively
convex spaces of search, thus multiscale approaches might be used to
refine or fine-tune our search procedure opening up a wide range of
possibilities to optimize the algorithm.

### Conclusion

Our proposed strategy
delivers quasi-optimal parameters for TGV-based reconstructions. It
significantly reduces the processing time compared with exhaustive
searches maintaining good quality reconstruction results. Less trial
and error is vital in expensive processes such as QSM problems, where
the optimization process may require 20 to 200 minutes for a single
reconstruction depending on the number of voxels and computer speed.

### Acknowledgements

Anillo
ACT1416### References

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