Extensions of the theory of sampling signals with finite rate of Innovation, performance analysis and an application to single image super-resolution

Abstract

Sampling is the reduction of a continuous-time signal to a discrete sequence. The classical sampling theorem limits the signals that can be perfectly reconstructed to bandlimited signals. In 2002, the theory of finite rate of innovations (FRI) emerged and broadened classical sampling paradigm to classes of signals with finite number of parameters per unit of time, which includes certain classes of non-bandlimited signals. In this thesis we analyse the performance of the FRI reconstruction algorithm and present extensions of the FRI theory. We also extend the FRI theory for the application of image upsampling. First, we explain the breakdown phenomenon in FRI reconstruction by subspace swap and work out at which noise level FRI reconstruction algorithm is guaranteed to achieve the optimal performance given by the Cramer-Rao bound. Our prediction of the breakdown PSNR is directly related to the distance between adjacent Diracs, sampling rate and the order of the sampling kernel and its accuracy is verified by simulations. Next, we propose an algorithm that can estimate the rate of innovation of the input signals and this extends the current FRI framework to a universal one that works with arbitrarily unknown rate of innovation. Moreover, we improve the current identification scheme of “parametrically sparse” systems, i.e. systems that are fully specified by small number of parameters. Inspired by the denoising technique used for FRI signals, we propose the modified Cadzow denoising algorithm which leads to robust system identification. We also show the possibility of perfectly identifying the input signal and the system simultaneously and we also propose reliable algorithm for simultaneous identification of both in the presence of noise. Lastly, by noting that lines of images can be modelled as piecewise smooth signals, we propose a novel image upsampling scheme based on our proposed method for reconstructing piecewise smooth signals which fuses the FRI method with the classical linear reconstruction method. We further improve our upsampled image by learning from the errors of our upsampled results at lower resolution levels. The proposed algorithm outperforms the state-of-the-art algorithms.