Academic literature on the topic 'Lp-balls'

Abstract This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to L q ( ℝ + n + 1 , μ ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz integral of the space-time fractional heat kernel, are provided. For the extension of Lebesgue spaces, a new Lp –capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the Lp –capacity, including its dual form, the Lp –capacity of fractional parabolic balls, strong and weak type inequalities, are established.

AbstractSuppose that we have the unit Euclidean ball in ℝn and construct new bodies using three operations — linear transformations, closure in the radial metric, and multiplicative summation defined by We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in L0 that naturally extends the corresponding properties of Lp-spaces with p ≠ 0, and show that the procedure described above gives exactly the unit balls of subspaces of L0 in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in L0, and prove several facts confirming the place of L0 in the scale of Lp-spaces.

The detection, segmentation, and delineation of the targeted regions of interest in biomedical images are fundamental steps for computer-aided assessment and prescreening. In this thesis, we focus on shape-constrained active contours explicitly due to their shape-specificity, efficiency and low computational cost. In particular, they need fewer degrees-of-freedom than other active contour approaches. They are simpler to formulate and optimize. Variants of concentric shape-constrained active contour representations include circular active disc, elliptical active disc, and active oblong. The degrees of freedom include scaling, translation and rotation. The energy is specified using a contrast function and the optimization is carried out using gradient-descent/ascent technique or its faster variants; use Green’s theorem to make the optimization computationally efficient. The thesis addresses segmentation problems for two image classes: retinal fundus images in ophthalmology and ultrasound images of the carotid artery. In retinal fundus image analysis, we consider optic disc, optic cup segmentation, glaucoma diagnosis, creation of a glaucoma specific database, fovea segmentation, oxygen saturation measurement from dual-wavelength oximetry images. In the carotid artery segmentation problem, we consider both transverse-mode and longitudinal model images and develop methods for segmenting the lumen intima and the media adventitia boundary. To start with, we present a reliable and fully automated method for the segmentation and outlining of the optic disc and cup using retinal fundus images with relevant parameters for glaucoma prescreening. We calculate the cup-to-disc ratio (CDR) and rim-to-disc ratio (RDR) from the segmented optic disc and cup. We perform two-stage glaucoma classification using CDR following the International Classification of Diseases (ICD) rules and RDR value following the new disc-damage-likelihood-scale (DDLS) rule. We categorize the glaucomatous condition as normal, moderate, or severe. In addition, we incorporate a check for the pattern of decreasing rim-widths of inferior, superior, nasal, and temporal (ISNT) regions to differentiate normal fundus from a glaucomatous one. We have validated our glaucoma prescreening technique on publicly available as well as locally obtained fundus image databases. The number of retinal fundus images used for the validation of optic disc and cup segmentation is 1597 and 436, respectively, and is 436 for glaucoma severity grading. The algorithm performance is validated against expert clinician outlining and quantitative comparison is provided using Jaccard and Dice similarity measures. The tool is Java/Android/iOS-based, repeatable, easy to use, provides quantitative analysis, and takes only a few seconds per image for the diagnosis. Our solution is available as a smartphone app, “NAYANA,” which assesses severity of glaucoma by computing CDR, RDR, and ISNT parameters. The app is available in both Android and iOS versions. We have tested these versions in a hospital setup with the aforementioned set of algorithm parameters. The results showed that the system could diagnose the severity of glaucoma reliably. As part of this dissertation, we have created a new, comprehensive, and the largest glaucoma-specific retinal fundus images database (with 1500 images) containing images of both glaucomatous and normal eyes. We provide optic disc and cup manual segmentation ground-truth and a decision on glaucoma by five expert ophthalmologists. The database has been created with the help of research funding from Govt. of India’s IMPacting Research, InnovatioN and Technology (IMPRINT) - India initiative. It is a result of collaboration with five expert ophthalmologists to support comparative studies on automatic OD and OC segmentation algorithms using retinal fundus images and subsequent decision on glaucoma diagnosis. We present an automated technique for the measurement of oxygen saturation in retinal arterioles and venules using dual-wavelength retinal oximetry images. The technique is based on segmenting an optic disc centered ring-shaped region of interest and subsequent analysis of the oxygen saturation levels. The two dominant peaks in the histogram of the oxygen saturation levels correspond to arteriolar and venular oxygen saturations from which the arterio-venous saturation difference (AVSD) can be calculated. For evaluation, we have created a normative database of Asian Indian eyes containing forty four dual- wavelength retinal oximetry images. Validations against expert manual annotations exhibit high consistency across the dataset indicating that the automated technique is an accurate alternative to the manual procedure. Proceeding further, we also address the problem of fovea segmentation and develop a technique for delineation of macular regions based on the active disc formalism. We report validation results on three publicly available fundus image databases, amounting to a total of 1370 fundus images for automatic fovea localization and 370 fundus images for fovea segmentation and macular region delineation. With regard to the carotid artery segmentation problem, we develop an automated outlining technique based on circular active disc for outlining the media adventitia boundary and elliptical active disc for outlining the lumen intima boundary of the common carotid artery. We report results of experimental validation on Brno University’s Signal Processing (SP) lab database, which contains 971 transverse mode ultrasound images of the carotid artery. The database provides manual annotations, which are also circular and serve as the ground-truth/reference. The segmentation of the lumen intima layer has been accomplished with the use of elliptical active disc and the performance has been validated on the SP lab database. Comparisons with other state-of-the-art techniques are also reported. We then consider the longitudinal-mode ultrasound images of the common carotid artery and address the segmentation of the lumen intima layer. The method is hybrid in the sense that a coarse segmentation is first achieved by optimizing a locally defined contrast function of an active oblong with five degrees-of-freedom. Subsequently, fine segmentation and delineation of the carotid artery are achieved by post-processing the portion of the ultrasound image spanned by the annulus region of the optimally fitted active oblong followed by a cubic curve-fitting technique to delineate the lumen intima boundary. The algorithm has been validated on 84 common carotid artery longitudinal-mode ultrasound images provided by the SP lab, Brno university. The segmentation results of the proposed technique exhibit a good correlation with the ground-truth annotations provided by two expert radiologists. Finally, we develop a generalized formulation to derive circle/ellipse/rectangle and other shapes using active lp-balls. The concentric-shape strategy, energy function and optimization techniques follow the active disc model. By using one more parameter than required for a rectangle or an ellipse, we are able to significantly increase the degree of flexibility and cover a parametric family of shapes including rectangles and ellipses. We have demonstrated the suitability of active lp-balls for shape-specific segmentation using synthetic images. Applicability of active lp-balls for suitable biomedical applications remains to be explored. In summary, this thesis focuses on various aspects of shape-constrained active contours and demonstrates concrete applications to retinal fundus image segmentation, retinal oximetry assessment, and common carotid artery segmentation in transverse and longitudinal mode ultrasound images.
MAHE, AICTE