The Structure of a General Type of Inverse Problem in Metrology

Inverse problems are ubiquitous in science. The theory and techniques of inverse problems play important roles in metrology owing to the close relation between inverse problems and indirect measurements. However, the essential connection between the concepts of inverse problems and measurement has not been deeply discussed before. This thesis is focused on a general type of inverse problem in metrology that arises naturally in indirect measurements, called the inverse problem of measurement (IPM). Based on the representational theory of measurement, a deterministic model of indirect measurements is developed, which shows that the IPM can be taken as an inference process of an indirect measurement and defined as the inference of the values of the measurand from the observations of some other quantity(s). The desired properties of solving the IPMs are listed and investigated in detail. The importance of estimating empirical relations is emphasised. Based on the desired properties, some structural properties of the IPMs are derived using category theory and order theory. Thereby, it is demonstrated that the structure of the IPMs can be characterised by a notion in order theory, called ‘Galois connection’. The deterministic model of indirect measurements is generalised to a probabilistic model by considering the effects of measurement uncertainty and intrinsic uncertainty. The propagation of uncertainty from the observed data to the values of measurands is investigated using a method of covariance matrices and a Bayesian method. The methods of estimating empirical relations with probability assigned using the solutions of IPM are discussed in two different approaches: the coverage interval approach and the random variable approach. For estimating empirical relations and determining the conformity of measurement results in indirect measurements, a strategy of estimating the empirical relations with high resolution is developed which significantly reduced the effect of measurement uncertainty; a method of estimating specification uncertainty is proposed for evaluating the intrinsic uncertainties of measurands; the impact of model resolution on the specifications of the indirectly measured quantities is discussed via a contradiction in the specifications of surface profiles

Amortised MAP Inference for Image Super-Resolution

Image super-resolution (SR) is an underdetermined inverse problem, where a large number of plausible high resolution images can explain the same downsampled image. Most current single image SR methods use empirical risk minimisation, often with a pixel-wise mean squared error (MSE) loss. However, the outputs from such methods tend to be blurry, over-smoothed and generally appear implausible. A more desirable approach would employ Maximum a Posteriori (MAP) infer- ence, preferring solutions that always have a high probability under the image prior, and thus appear more plausible. Direct MAP estimation for SR is non- trivial, as it requires us to build a model for the image prior from samples. Here we introduce new methods for amortised MAP inference whereby we calculate the MAP estimate directly using a convolutional neural network. We first introduce a novel neural network architecture that performs a projection to the affine subspace of valid SR solutions ensuring that the high resolution output of the network is always consistent with the low resolution input. Using this architecture, the amor- tised MAP inference problem reduces to minimising the cross-entropy between two distributions, similar to training generative models. We propose three methods to solve this optimisation problem: (1) Generative Adversarial Networks (GAN) (2) denoiser-guided SR which backpropagates gradient-estimates from denoising to train the network, and (3) a baseline method using a maximum-likelihood- trained image prior. Our experiments show that the GAN based approach per- forms best on real image data. Lastly, we establish a connection between GANs and amortised variational inference as in e. g. variational autoencoders

From Correlation to Causation: Estimation of Effective Connectivity from Continuous Brain Signals based on Zero-Lag Covariance

Knowing brain connectivity is of great importance both in basic research and for clinical applications. We are proposing a method to infer directed connectivity from zero-lag covariances of neuronal activity recorded at multiple sites. This allows us to identify causal relations that are reflected in neuronal population activity. To derive our strategy, we assume a generic linear model of interacting continuous variables, the components of which represent the activity of local neuronal populations. The suggested method for inferring connectivity from recorded signals exploits the fact that the covariance matrix derived from the observed activity contains information about the existence, the direction and the sign of connections. Assuming a sparsely coupled network, we disambiguate the underlying causal structure via $L^1$-minimization. In general, this method is suited to infer effective connectivity from resting state data of various types. We show that our method is applicable over a broad range of structural parameters regarding network size and connection probability of the network. We also explored parameters affecting its activity dynamics, like the eigenvalue spectrum. Also, based on the simulation of suitable Ornstein-Uhlenbeck processes to model BOLD dynamics, we show that with our method it is possible to estimate directed connectivity from zero-lag covariances derived from such signals. In this study, we consider measurement noise and unobserved nodes as additional confounding factors. Furthermore, we investigate the amount of data required for a reliable estimate. Additionally, we apply the proposed method on a fMRI dataset. The resulting network exhibits a tendency for close-by areas being connected as well as inter-hemispheric connections between corresponding areas. Also, we found that a large fraction of identified connections were inhibitory.Comment: 18 pages, 10 figure

On Dimer Models and Closed String Theories

We study some aspects of the recently discovered connection between dimer models and D-brane gauge theories. We argue that dimer models are also naturally related to closed string theories on non compact orbifolds of \BC^2 and \BC^3, via their twisted sector R charges, and show that perfect matchings in dimer models correspond to twisted sector states in the closed string theory. We also use this formalism to study the combinatorics of some unstable orbifolds of \BC^2.Comment: 1 + 25 pages, LaTeX, 11 epsf figure