Endometriosis is an oestrogen-dependent gynaecological condition defined as the presence of endometrial tissue outside the uterus cavity. The condition is predominantly found in women in their reproductive years, and associated with significant pelvic and abdominal chronic pain and infertility. The disease is believed to affect approximately 33% of women by a recent study. Currently, surgical intervention, often laparoscopic surgery, is the gold standard for diagnosing the disease and it remains an effective and common treatment method for all stages of endometriosis. Magnetic resonance imaging (MRI) of the patient is performed before surgery in order to locate any endometriosis lesions and to determine whether a multidisciplinary surgical team meeting is required. In this dissertation, our goal is to use image processing techniques to aid surgical planning. Specifically, we aim to improve quality of the existing images, and to automatically detect bladder endometriosis lesion in MR images as a form of bladder wall thickening. One of the main problems posed by abdominal MRI is the sparse anisotropic frequency sampling process. As a consequence, the resulting images consist of thick slices and have gaps between those slices. We have devised a method to fuse multi-view MRI consisting of axial/transverse, sagittal and coronal scans, in an attempt to restore an isotropic densely sampled frequency plane of the fused image. In addition, the proposed fusion method is steerable and is able to fuse component images in any orientation. To achieve this, we apply the Riesz transform for image decomposition and reconstruction in the frequency domain, and we propose an adaptive fusion rule to fuse multiple Riesz-components of images in different orientations. The adaptive fusion is parameterised and switches between combining frequency components via the mean and maximum rule, which is effectively a trade-off between smoothing the intrinsically noisy images while retaining the sharp delineation of features. We first validate the method using simulated images, and compare it with another fusion scheme using the discrete wavelet transform. The results show that the proposed method is better in both accuracy and computational time. Improvements of fused clinical images against unfused raw images are also illustrated. For the segmentation of the bladder wall, we investigate the level set approach. While the traditional gradient based feature detection is prone to intensity non-uniformity, we present a novel way to compute phase congruency as a reliable feature representation. In order to avoid the phase wrapping problem with inverse trigonometric functions, we devise a mathematically elegant and efficient way to combine multi-scale image features via geometric algebra. As opposed to the original phase congruency, the proposed method is more robust against noise and hence more suitable for clinical data. To address the practical issues in segmenting the bladder wall, we suggest two coupled level set frameworks to utilise information in two different MRI sequences of the same patients - the T2- and T1-weighted image. The results demonstrate a dramatic decrease in the number of failed segmentations done using a single kind of image. The resulting automated segmentations are finally validated by comparing to manual segmentations done in 2D.

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Dans cette thèse nous étudions la bornitude des transformées de Riesz associées aux opérateurs de Schrödinger avec des potentiels qui admettent des parties négatives.Cette étude a lieu dans un premier temps sur les espaces de Lebesgue Lp(RN, dx), puissur les espaces Lp(M, dx) où M est une variété Riemannienne de type homogène et dans un dernier temps sur les espaces à poids Lp(RN,wdx). Nous considérons également,sur ces espaces à poids, la bornitude du calcul fonctionnel holomorphe associé et la bornitude des puissances négatives de l’opérateur de Schrödinger. / In this thesis we study the boundedness of Riesz transforms associated to Schrödinger operators with potentials having negative parts. First we consider the boundednesson Lp(RN, dx), then on Lp(M, dx) where M is a Riemannian manifold of homogeneous type. Finally we treat the boundedness of Riesz transforms on Lp(RN,wdx). As we consider, on the weighted spaces, the boundedness of the associated holomorphicfunctional calculus and the boundedness of the negative powers of the Schrödinger operator.

Speech is a non-stationary signal and contains modulations in both spectral and temporal domains. Based on the type of modulations studied, most speech processing algorithms can be classified into short-time analysis algorithms, narrow-band analysis algorithms, or joint spectro-temporal analysis algorithms. While traditional methods of speech analysis study the modulation along either time (Short-time analysis algorithms) or frequency (Narrowband analysis) at a time. A new class of algorithms that work simultaneously along both temporal as well as spectral dimensions, called the spectro-temporal analysis algorithms, have become prominent over the past decade. Joint spectro-temporal analysis (also referred to as 2-D speech analysis) has shown promise in applications such as formant estimation, pitch estimation, speech recognition, etc. Over the past decade, 2-D speech analysis has been independently motivated from several directions. Broadly these motivations for 2-D speech models can be grouped into speech-production motivated, source-separation/machine- learning motivated and neurophysiology motivated. In this thesis, we develop 2-D speech model based on the speech production motivation. The overall organization of the thesis is as follows: We first develop the context of 2-D speech processing in Chapter one, we then proceed to develop a 2-D multicomponent AM-FM model for narrowband spectrogram patch of voiced speech and experiment with the perceptual significance of number of components needed to represent a spectrogram patch in Chapter two. In Chapter three we develop a demodulation algorithm called the inphase and the quadrature phase demodulation (IQ), compared to the state-of-the art sinusoidal demodulation, the AM obtained using this method is more robust to carrier estimation errors. The demodulation algorithm was verified on call voiced sentences taken from the TIMIT database. In chapter four we develop a demodulation algorithm based on Riesz transform, a natural extension of the Hilbert transform to higher dimensions, unlike the sinusoidal and the IQ demodulation techniques, Riesz-transform-based demodulation does not require explicit carrier estimation and is also robust to pitch discontinuous in patches. The algorithm was validated on all voiced sentences from the TIMIT database. Both IQ and Riesz-transform-based methods were found to give more accurate estimates of the 2-D AM (relates to vocal tract) and 2-D carrier (relates to source) compared with the sinusoidal modulation. In Chapter five we show application of the demodulated AM and carrier to pitch estimation and for creation of hybrid sounds. The hybrid sounds created were found to have better perceptual quality compared with their counterparts created using the linear prediction analysis. In Chapter six we summarize the work and present with possible directions of future research.