Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure

Automatic Segmentation and Classification of Red and White Blood cells in Thin Blood Smear Slides

In this work we develop a system for automatic detection and classification of cytological images which plays an increasing important role in medical diagnosis. A primary aim of this work is the accurate segmentation of cytological images of blood smears and subsequent feature extraction, along with studying related classification problems such as the identification and counting of peripheral blood smear particles, and classification of white blood cell into types five. Our proposed approach benefits from powerful image processing techniques to perform complete blood count (CBC) without human intervention. The general framework in this blood smear analysis research is as follows. Firstly, a digital blood smear image is de-noised using optimized Bayesian non-local means filter to design a dependable cell counting system that may be used under different image capture conditions. Then an edge preservation technique with Kuwahara filter is used to recover degraded and blurred white blood cell boundaries in blood smear images while reducing the residual negative effect of noise in images. After denoising and edge enhancement, the next step is binarization using combination of Otsu and Niblack to separate the cells and stained background. Cells separation and counting is achieved by granulometry, advanced active contours without edges, and morphological operators with watershed algorithm. Following this is the recognition of different types of white blood cells (WBCs), and also red blood cells (RBCs) segmentation. Using three main types of features: shape, intensity, and texture invariant features in combination with a variety of classifiers is next step. The following features are used in this work: intensity histogram features, invariant moments, the relative area, co-occurrence and run-length matrices, dual tree complex wavelet transform features, Haralick and Tamura features. Next, different statistical approaches involving correlation, distribution and redundancy are used to measure of the dependency between a set of features and to select feature variables on the white blood cell classification. A global sensitivity analysis with random sampling-high dimensional model representation (RS-HDMR) which can deal with independent and dependent input feature variables is used to assess dominate discriminatory power and the reliability of feature which leads to an efficient feature selection. These feature selection results are compared in experiments with branch and bound method and with sequential forward selection (SFS), respectively. This work examines support vector machine (SVM) and Convolutional Neural Networks (LeNet5) in connection with white blood cell classification. Finally, white blood cell classification system is validated in experiments conducted on cytological images of normal poor quality blood smears. These experimental results are also assessed with ground truth manually obtained from medical experts

Rotation of 2D orthogonal polynomials

© 2017 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license: http://creativecommons.org/licenses/by-nc-nd/4.0/ This author accepted manuscript is made available following 24 month embargo from date of publication (Dec 2017) in accordance with the publisher’s archiving policyOrientation-independent object recognition mostly relies on rotation invariants. Invariants from moments orthogonal on a square have favorable numerical properties but they are difficult to construct. The paper presents sufficient and necessary conditions, that must be fulfilled by 2D separable orthogonal polynomials, for being transformed under rotation in the same way as are the monomials. If these conditions have been met, the rotation property propagates from polynomials to moments and allows a straightforward derivation of rotation invariants. We show that only orthogonal polynomials belonging to a specific class exhibit this property. We call them Hermite-like polynomials

Mining for cosmological information: Simulation-based methods for Redshift Space Distortions and Galaxy Clustering

The standard model of cosmology describes the complex large scale structure of the Universe through less than 10 free parameters. However, concordance with observations requires that about 95\% of the energy content of the universe is invisible to us. Most of this energy is postulated to be in the form of a cosmological constant, $\Lambda$, which drives the observed accelerated expansion of the Universe. Its nature is, however, unknown. This mystery forces cosmologists to look for inconsistencies between theory and data, searching for clues. But finding statistically significant contradictions requires extremely accurate measurements of the composition of the Universe, which are at present limited by our inability to extract all the information contained in the data, rather than being limited by the data itself. In this Thesis, we study how we can overcome these limitations by i) modelling how galaxies cluster on small scales with simulation-based methods, where perturbation theory fails to provide accurate predictions, and ii) developing summary statistics of the density field that are capable of extracting more information than the commonly used two-point functions. In the first half, we show how the real to redshift space mapping can be modelled accurately by going beyond the Gaussian approximation for the pairwise velocity distribution. We then show that simulation-based models can accurately predict the full shape of galaxy clustering in real space, increasing the constraining power on some of the cosmological parameters by a factor of 2 compared to perturbation theory methods. In the second half, we measure the information content of density dependent clustering. We show that it can improve the constraints on all cosmological parameters by factors between 3 and 8 over the two-point function. In particular, exploiting the environment dependence can constrain the mass of neutrinos by a factor of 8$ better than the two-point correlation function alone. We hope that the techniques described in this thesis will contribute to extracting all the cosmological information contained in ongoing and upcoming galaxy surveys, and provide insight into the nature of the accelerated expansion of the universe

Analysing spatial point patterns in digital pathology: immune cells in high-grade serous ovarian carcinomas

Multiplex immunofluorescence (mIF) imaging technology facilitates the study of the tumour microenvironment in cancer patients. Due to the capabilities of this emerging bioimaging technique, it is possible to statistically analyse, for example, the co-varying location and functions of multiple different types of immune cells. Complex spatial relationships between different immune cells have been shown to correlate with patient outcomes and may reveal new pathways for targeted immunotherapy treatments. This tutorial reviews methods and procedures relating to spatial point patterns for complex data analysis. We consider tissue cells as a realisation of a spatial point process for each patient. We focus on proper functional descriptors for each observation and techniques that allow us to obtain information about inter-patient variation. Ovarian cancer is the deadliest gynaecological malignancy and can resist chemotherapy treatment effective in cancers. We use a dataset of high-grade serous ovarian cancer samples from 51 patients. We examine the immune cell composition (T cells, B cells, macrophages) within tumours and additional information such as cell classification (tumour or stroma) and other patient clinical characteristics. Our analyses, supported by reproducible software, apply to other digital pathology datasets

Generative modelling: addressing open problems in model misspecification and differential privacy

Generative modelling has become a popular application of artificial intelligence. Model performance can, however, be impacted negatively when the generative model is misspecified, or when the generative model estimator is modified to adhere to a privacy notion such as differential privacy. In this thesis, we approach generative modelling under model misspecification and differential privacy by presenting four different works. We first present related work on generative modelling. Subsequently, we delve into the reasons that necessitate an examination of generative modelling under the challenges of model misspecification and differential privacy. As an initial contribution, we consider generative modelling for density estimation. One way to approach model misspecification is to relax model assumptions. We show that this can also help in nonparametric models. In particular, we study a recently proposed nonparametric quasi-Bayesian density estimator and identify its strong model assumptions as a reason for poor performance in finite data sets. We propose an autoregressive extension relaxing model assumptions to allow for a-priori feature dependencies. Next, we consider generative modelling for missingness imputation. After categorising current deep generative imputation approaches into the classes of nonignorable missingness models as introduced by Rubin [1976], we extend the formulation of variational autoencoders to factorise according to a nonignorable missingness model class that has not been studied in the deep generative modelling literature before. These explicitly model the missingness mechanisms to prevent model misspecification when missingness is not at random. Then, we focus the attention of this thesis on improving synthetic data generation under differential privacy. For this purpose, we propose differentially private importance sampling of differentially private synthetic data samples. We observe that importance sampling helps more, the better the generative model is. We next focus on increasing data generation quality by considering differentially private diffusion models. We identify training strategies that significantly improve the performance of DP image generators. We conclude the dissertation with a discussion, including contributions and limitations of the presented work, and propose potential directions for future work

Quantitative volumetric study of brain in chronic striatolenticular stroke

Perforating branches of the middle cerebral artery, namely the striato-lenticular arteries provide the majority of blood supply for the striatum and posterior limb of the internal capsules. Occlusions of these arteries cause a small stroke but have a devastating effect on patients’ functions. Previous studies showed that the anterior two thirds of the internal capsule is occupied by the prefrontal tracts with the posterior one third by connection to/from sensorimotor, temporal and posterior parietal cortices. In this study, we aimed to examine the long-term effect of infarction in the striato-capsular region on cerebral cortex thickness and also its association with stroke volume and different functional tests. We hypothesized that because of extensive connections of striatum and internal capsule with the cerebral cortex, infarction of this area results in an extensive cortical thickness degeneration which could in turn cause low fictional measurement scores. High resolution T1 weighted MRI was obtained from 21 patients with ischemic stroke in the striatum/posterior limb of the internal capsule region. Subjects were carefully selected from a pool of 140 stroke cases recruited for the Northstar Stroke Project. 63 healthy volunteers (30 male), matched for age and gender were also chosen to form the control group from the OASIS database. Patients and normal subjects were right handed except for 3 patients who have the stroke in the left side of the brain. Patients were defined as left-sided stroke and right-sided stroke depending on the side of the stroke in brain. MRI scans were done 6 months to 2 years after the stroke. To measure cortical thickness, we used Freesurfer software. Vertexwise group comparison was carried out using General Linear Models (GLM). With the Significance level set at 0.05. Population maps of stroke lesions showed that the majority of strokes were located in the striatum and posterior internal capsule. Cortical thickness reduction was greater in the ipsilateral hemisphere. Vertex-wise group comparison between leftsided stroke patients and controls group showed significant reduction in the cortical II thickness in the dorsal and medial prefrontal, premotor, posterior parietal, precuneus, and temporal cortex which survived after correction for multiple comparison using false discovery rate at Freesurfer. Similar comparison for rightsided stroke showed a similar pattern of cortical thinning, however the extent of cortical thinning was much less than in that of the left-sided stroke patients but the ROI analysis showed the main effect of side was significant (f (1, 19) =6.909, p=0.017), which showed that the left hemisphere stroke side group had a thicker cortex (mean=2.463, sd= 0.020) on average compare to the right hemisphere stroke side (mean=2.372, sd= 0.028). Primary motor cortex was surprisingly spared in both stroke groups. In addition, volume of the corpus callosum increased significantly in the stroke group. The differences between motor cortex (M1) thickness in left-hemispheric stroke patients versus controls (t=1.24, n=14, p>0.05) and right-hemispheric stroke patients versus controls (t=-0.511, n=7, p>0.05) were not significant. There was a negative correlation between the volume of the stroke lesions and the affected M1 thickness. There was no correlation between the stroke volume and functional tests in patients and also no correlation between the motor cortex thickness and functional tests in patients. Regarding normal subjects, comparison between two sides of the brain showed that the both hemispheres are symmetrical. In addition, correlation between age and cortical thickness showed a negative significant correlation (1-tailed, p<0.0007, manual correction for multiple comparisons) in M1, superior frontal, lingual cortex at both side of the brain and also negative significant correlation in superior temporal cortex and isthmus cingulated cortex on the left side of brain and supramarginal cortex on the right side of brain but there was no significant difference in cortical thickness between males and females. The finding from this study suggests that the size of the lesion can be a predictor of further M1 cortex reduction. The correlation of M1 thickness with stroke volume showed that secondary cortical degeneration may be mainly depends on the size of neuronal loss in strital-capsular stroke. From normal subject study it can be concluded that generally cortical thickness will decrease with ageing but gender does not have an effect on the cortical thickness. III Furthermore, the lack of behavioural correlation with M1 thickness and stroke volume and also the non significant M1 cortex reduction versus control group may suggest that the long-term functional disability after capsular-striatal stroke may not be entirely dependent on primary motor cortex and secondary motor cortex and primary somatosensory cortex could have an important role as well. These results may help to understand why relatively small subcortical infarcts often cause severe disability that is relatively resistant to recovery in the long term