Blind Image Deconvolution using Approximate Greatest Common Divisor and Approximate Polynomial Factorisation

Images play a significant and important role in diverse areas of everyday modern life. Examples of the areas where the use of images is routine include medicine, forensic investigations, engineering applications and astronomical science. The procedures and methods that depend on image processing would benefit considerably from images that are free of blur. Most images are unfortunately affected by noise and blur that result from the practical limitations of image sourcing systems. The blurring and noise effects render the image less useful. An efficient method for image restoration is hence important for many applications. Restoration of true images from blurred images is the inverse of the naturally occurring problem of true image convolution through a blurring function. The deconvolution of images from blurred images is a non-trivial task. One challenge is that the computation of the mathematical function that represents the blurring process, which is known as the point spread function (PSF), is an ill-posed problem, i.e. an infinite number of solutions are possible for given inexact data. The blind image deconvolution (BID) problem is the central subject of this thesis. There are a number of approaches for solving the BID problem, including statistical methods and linear algebraic methods. The approach adopted in this research study for solving this problem falls within the class of linear algebraic methods. Polynomial linear algebra offers a way of computing the PSF size and its components without requiring any prior knowledge about the true image and the blurring PSF. This research study has developed a BID method for image restoration based on the approximate greatest common divisor (AGCD) algorithms, specifically, the approximate polynomial factorization (APF) algorithm of two polynomials. The developed method uses the Sylvester resultant matrix algorithm in the computation of the AGCD and the QR decomposition for computing the degree of the AGCD. It is shown that the AGCD is equal to the PSF and the deblurred image can be computed from the coprime polynomials. In practice, the PSF can be spatially variant or invariant. PSF spatial invariance means that the blurred image pixels are the convolution of the true image pixels and the same PSF. Some of the PSF bivariate functions, in particular, separable functions, can be further simplified as the multiplication of two univariate polynomials. This research study is focused on the invariant separable and non-separable PSF cases. The performance of state-of-the-art image restoration methods varies in terms of computational speed and accuracy. In addition, most of these methods require prior knowledge about the true image and the blurring function, which in a significant number of applications is an impractical requirement. The development of image restoration methods that require no prior knowledge about the true image and the blurring functions is hence desirable. Previous attempts at developing BID methods resulted in methods that have a robust performance against noise perturbations; however, their good performance is limited to blurring functions of small size. In addition, even for blurring functions of small size, these methods require the size of the blurring functions to be known and an estimate of the noise level to be present in the blurred image. The developed method has better performance than all the other state-of-the-art methods, in particular, it determines the correct size and coefficients of the PSF and then uses it to recover the original image. It does not require any prior knowledge about the PSF, which is a prerequisite for all the other methods

Image Restoration

This book represents a sample of recent contributions of researchers all around the world in the field of image restoration. The book consists of 15 chapters organized in three main sections (Theory, Applications, Interdisciplinarity). Topics cover some different aspects of the theory of image restoration, but this book is also an occasion to highlight some new topics of research related to the emergence of some original imaging devices. From this arise some real challenging problems related to image reconstruction/restoration that open the way to some new fundamental scientific questions closely related with the world we interact with

Bispectrum Inversion with Application to Multireference Alignment

We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called multireference alignment (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data which is important at low signal-to-noise ratio when the number of observations must be large

Heart rates estimation using rPPG methods in challenging imaging conditions

Abstract. The cardiovascular system plays a crucial role in maintaining the body’s equilibrium by regulating blood flow and oxygen supply to different organs and tissues. While contact-based techniques like electrocardiography and photoplethysmography are commonly used in healthcare and clinical monitoring, they are not practical for everyday use due to their skin contact requirements. Therefore, non-contact alternatives like remote photoplethysmography (rPPG) have gained significant attention in recent years. However, extracting accurate heart rate information from rPPG signals under challenging imaging conditions, such as image degradation and occlusion, remains a significant challenge. Therefore, this thesis aims to investigate the effectiveness of rPPG methods in extracting heart rate information from rPPG signals in these imaging conditions. It evaluates the effectiveness of both traditional rPPG approaches and rPPG pre-trained deep learning models in the presence of real-world image transformations, such as occlusion of the faces by sunglasses or facemasks, as well as image degradation caused by noise artifacts and motion blur. The study also explores various image restoration techniques to enhance the performance of the selected rPPG methods and experiments with various fine-tuning methods of the best-performing pre-trained model. The research was conducted on three databases, namely UBFC-rPPG, UCLA-rPPG, and UBFC-Phys, and includes comprehensive experiments. The results of this study offer valuable insights into the efficacy of rPPG in practical scenarios and its potential as a non-contact alternative to traditional cardiovascular monitoring techniques