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 considering wavelet-based approaches for synthetic image detection.

We will introduce multi-level discrete wavelet transform, which to the best of our knowledge has never been considered for this goal prior to this work. A similar approach that has been considered for the goal of synthetic image detection, is the multi-level wavelet packet transform used by Wolter et al. We will show that not only is our proposed approach more efficient and easier interpretable, it also performs better in a number of experimental settings and therefore forms a suitable addition to the toolset for the detection of synthetic images.

Moreover, we will try and generalize performance of our used classifiers to out-of-dataset samples and see that our used classifier in general does not allow for such generalization. Finally, we will discuss the challenges of this work and offer interesting directions for further research.

In this thesis we shall consider a generalization on Pólya Processes as have been described by Chung et al. [7]. Given finitely many bins, containing an initial configuration of balls, additional balls arrive one at a time. For each new ball, a new bin is created with probability 푝, or with probability 1 − 푝 this new ball shall be placed in an existing bin such that the probability of this ball ending in a specific bin, is proportional to 푓(푚) where 푚 is the number of balls currently in that bin and 푓 is some feedback function. We shall show that for 푝 = 0, which will be defined as Finite Pólya Processes, the behaviour of the process can be classified into one of three mutual exclusive regimes: Monopolistic Regime, Eventual Leadership Regime or Almost-Balanced Regime. This behaviour solely depends on the convergence of the following sums: Ση≥1 푓(푛)–1 and Ση≥1 푓(푛)-2. We shall explore the limiting distribution of fractions of balls in bins when 푓(푥) = 푥, which is a known result for classical multi-coloured Pólya Urn problems.Using a similar method, we find a limiting distribution for Finite Pólya Processes with general positive linear feedback functions, which has not previously been researched. We then consider the case where 푝 > 0, which are defined as Infinite Pólya Processes and restrict the feedback function to be of the form 푓(푥) = 푥γ where 훾 ∈ ℝ. We shall show that if 훾 > 1, almost surely one bin will dominate or a new bin will be created. We shall show that for 훾 = 1 a preferential attachment scheme arises. We consider 훾 < 1 under the assumptions that some limits exist and show that the fraction of bins having 푚 balls shrinks exponentially as a function of 푚. Finally, we reflect on our results and discuss interesting future research subjects.