HK

23 records found

## Authored

We propose alternatives to Bayesian prior distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well-defined infinite ...

We consider the statistical non-linear inverse problem of recovering the absorption term f > 0 in the heat equation {∂tu-12Δu+fu=0onO×(0,T)u=gon∂ O×(0,T)u(·,0)=u0onO, where O ϵ ℝd is a bounded domain, T < ∞ is a fixed time, and g, u 0 are given sufficiently smooth functi ...

How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to students and the general public using diffe ...
We consider the statistical inverse problem of recovering an unknown function f from a linear measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction of f corre ...
In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded in the image space of the forward operator. We introduce a Banach space setting that allows to define a reasonable notion of solutions for more general ...

The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable U (x, w) is Mδ (y, ...

Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function u(x) is m(x) = Au(x) + δε(x) where δ > 0 ...

## Total Least Squares

### Comparing Least Squares Methods for Signal Reconstruction

A common problem in wireless communication is the existence of multipath propagation. This means that a transmitted signal is received multiple times because of reflections caused by the environment. We present two ways of modeling multipath propagation of an acoustic underwater ...

## X-Ray Tomography

### An Inverse Problem

This thesis aims at introducing the reader to the mathematical concepts behind the imaging technique X-ray tomography, commonly known as a CT scan. It includes the derivation of the filtered and unfiltered backprojection reconstruction methods for noise-free data. It was conclude ...

## Using Benford's Law for wavelet coefficients to differentiate images

### Het toepassen van Benford's Law op wavelet coëfficiënten om afbeeldingen te onderscheiden

This thesis explores the application of Benford's Law to wavelet coefficients derived from the Discrete Wavelet Transform (DWT) of images, aiming to provide a novel method for image differentiation. The study focuses on the DWT, specifically utilizing Haar and Daubechies wavelets ...
Background Dynamic SPECT scanning provides a non-invasive way to image the time-dependent distribution of radio-labelled tracers inside living tissue. Beside human medicine, dynamic SPECT also finds its applications in pre-clinical research on small animals. In pre- ...
In this thesis we develop a Bayesian approach to graph contrastive learning and propose a new uncertainty measure based on the disagreement in likelihood due to different positive samples. Moreover, we extend contrastive learning to simplicial complexes and show that it can be u ...
Integrated circuits are vital in the modern world. Testing these circuits is often a months long process involving measurements at multiple times during long stress tests. In this work, final measurements from such tests are predicted based on early measurements, potentially redu ...
With the rise of zero-shot synthetic image generation models, such as Stability.ai's Stable Diffusion, OpenAI's DALLE or Google's Imagen, the need for powerful tools to detect synthetic generated images has never been higher. In this thesis we contribute to this goal by consideri ...
Probabilistic numerics methods are a novel approach to quantifying the approximation errors in numerical computations as probabilistic uncertainties. A recent method that was developed is the Bayesian Finite Element Method, which aims to determine the discretization errors along ...

## Exploring the potential of wavelets

### In the field of image processing

This research consists of two applications of image processing, namely, image compression and image denoising. Image compression aims to reduce the size of an image without losing too many features. This is often used to store a large number of images such as fingerprints. Denois ...
In anti-cancer therapy, ntiangiogenic treatments are applied and take effect on the vascularization of tissue. To evaluate the efficacy of treatments, we adopt two methods to solve the physiological pharmacokinetic model’s parameter estimation problem, providing discrete, partial ...
When a second tumor arises in the contralateral breast in a patient with a previous or synchronous breast cancer, it is of clinical importance to determine if this tumor is a new unrelated tumor or a metastasis, i.e. clone, of the primary tumor. A new, unrelated tumor may be trea ...
The random graph is a mathematical model simulating common daily cases, such as ranking and social networks. Generally, the connection between different users in the network is established through preference, and this phenomenon leads to a power-law behaviour of the degree sequen ...
In this report, our goal is to find a way to get some information such as the mean out of high dimensional densities. If we want to calculate the mean we need to calculate integrals, which are difficult to do for high dimensional densities. We cannot use the analytical or classic ...