In biology, radar, and image super-resolution applications, the common proxy problem of multireference alignment (MRA) can arise. The MRA problem involves estimating a one-dimensional signal from noisy and corcularly shifted observations. Expectation-maximization solves this problem by estimating distributions of shifts for each signal instead of assigning fixed shifts. Alternatively, the method of invariants/moments recovers the clean signal using third-order moments, see the definition of moments in Section 2.1. While the expectation-maximization method offers better accuracy, the invariant method is more computationally efficient. Therefore, the aim of the project is to develop signal reconstruction models using invariant features (such as the mean, the power spectrum, and the bispectrum) and neural networks to improve signal reconstruction accuracy in high noise settings relative to other invariant methods (not utilizing neural networks) for the same amount of signal observations.
The best performing invariant method found through research was the one using Frequency marching (FM) algorithm, which uses the bispectrum to estimate the signal phases. From there, the signal phases can be used together with the mean and the power spectrum to reconstruct the signal. Phase synchronization (PS) algorithm is a developed invariant method that performs similarly to FM and also uses the bispectrum to estimate the signal phases.
Three types of signal reconstruction models that used feed-forward neural network (FNN) were developed. Two of the types used a developed FNN model, named Denoising Neural Network (D-FNN), to denoise the invariant features, then one of them used the FM algorithm to reconstruct the signal, while the other used the PS algorithm. The third type was a developed FNN, named Reconstructing Neural Network (R-FNN), that reconstructed the signal directly using the invariant features as input. The models were tested on different types of signals, as described in Section 3.1. The signal types ranged from simple to complex, where the complexity of the signal type was measued by the number of unique signals it containted.
This project achieved its main objective by proving it is possible to improve signal reconstruction accuracy by using neural networks in high noise settings. The best type of the developed models was the R-FNN:s which performed better or similarly to the models using D-FNN in the tests.