We propose a novel convolutional neural network (CNN), called \Psi DONet, designed for learning pseudodifferential operators (\Psi DOs) in the context of linear inverse problems. Our starting point is the iterative soft thresholding algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow us to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling, and convolution, which characterize our \Psi DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited-angle Xray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of \Psi DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are \Psi DOs or Fourier integral operators.

In this paper we address the problem of deconvolution of an image corrupted with Poisson noise by reformulating the restoration process as a constrained minimization of a suitable regularized data fidelity function. The minimization step is performed by means of an interior-point approach, in which the constraints are incorporated within the objective function through a barrier penalty and a forward-backward algorithm is exploited to build a minimizing sequence. The key point of our proposed scheme is that the choice of the regularization, barrier and step-size parameters defining the interior point approach is automatically performed by a deep learning strategy. Numerical tests on Poisson corrupted benchmark datasets show that our method can obtain very good performance when compared to a state-of-the-art variational deblurring strategy.

Storage conditions influence the integrity of the recoverable DNA from forensic evidence in terms of yield and quality. FTA cards are widely used in the forensic practice as their chemically-treated matrix provides protection from the moment of collection to the point of analysis with current STR typing technology. In this study we assess the recoverability and the integrity of DNA from eleven years old saliva on FTA cards using a forensic quantitative real-time polymerase chain reaction (qPCR) commercial assay. The quality after long-term storage was investigated in order to evaluate if the FTA device could assure enough stability over time, applying some internally validated quality criteria of the STR profile. Furthermore, we used a 3D interpolation model to combine the quantitative and qualitative data from qPCR to calculate the Minimum Optimal DNA Input (MODI) to add to the downstream PCR reaction based on the quantitative and qualitative data of a sample. According to our results, when saliva sample is properly transferred onto FTA cards and then correctly stored according to the manufacturer's instructions, it's possible to recover sufficient amounts of DNA for human identification even after more than a decade of storage at ambient temperature. Degradation affected the quality of results especially when the Degradation Index exceeds the value of 2.12, requiring modifications of the standard internal workflow to improve the genotyping quality. Above this value, the application of a "corrective factor" to the PCR normalization process was necessary in order to adjust the recommended manufacturer's PCR DNA input taking into account the degradation level. Our results demonstrated the importance to consider in predictive terms the parameters obtained with the real-time quantification assay, both in terms of quantity (DNA concentration) and of quality (DI, Inhibition). Informatics predictive tools including qPCR data together with the variables of storage duration and conditions should be developed in order to optimize the DNA analysis process.

We propose a novel convolutional neural network (CNN), called ΨDONet, designed for learning pseudodifferential operators (ΨDOs) in the context of linear inverse problems. Our starting point is the Iterative Soft Thresholding Algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limitedangle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling and convolution, which characterize our ΨDONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited angle X-ray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of ΨDONet on simulated data from limited angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are ΨDOs or Fourier integral operators.

In the context of deep learning, the costliest phase from a computational point of view is the full training of the learning algorithm. However, this process is to be used a significant number of times during the design of a new artificial neural network, leading therefore to extremely expensive operations. Here, we propose a low-cost strategy to predict the accuracy of the algorithm, based only on its initial behaviour. To do so, we train the network of interest up to convergence several times, modifying its characteristics at each training. The initial and final accuracies observed during this beforehand process are stored in a database. We then make use of both curve fitting and Support Vector Machines techniques, the latter being trained on the created database, to predict the accuracy of the network, given its accuracy on the primary iterations of its learning. This approach can be of particular interest when the space of the characteristics of the network is notably large or when its full training is highly time-consuming. The results we obtained are promising and encouraged us to apply this strategy to a topical issue: hyper-parameter optimisation (HO). In particular, we focused on the HO of a convolutional neural network for the classification of the databases MNIST and CIFAR-10. By using our method of prediction, and an algorithm implemented by us for a probabilistic exploration of the hyper-parameter space, we were able to find the hyper-parameter settings corresponding to the optimal accuracies already known in literature, at a quite low-cost.