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.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.

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.

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.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.