The Bayes Lepski's Method and Credible Bands through Volume of Tubular Neighborhoods

For a general class of priors based on random series basis expansion, we develop the Bayes Lepski's method to estimate unknown regression function. In this approach, the series truncation point is determined based on a stopping rule that balances the posterior mean bias and the posterior standard deviation. Equipped with this mechanism, we present a method to construct adaptive Bayesian credible bands, where this statistical task is reformulated into a problem in geometry, and the band's radius is computed based on finding the volume of certain tubular neighborhood embedded on a unit sphere. We consider two special cases involving B-splines and wavelets, and discuss some interesting consequences such as the uncertainty principle and self-similarity. Lastly, we show how to program the Bayes Lepski stopping rule on a computer, and numerical simulations in conjunction with our theoretical investigations concur that this is a promising Bayesian uncertainty quantification procedure.Comment: 42 pages, 2 figures, 1 tabl

On solving operator equations by Galerkin\u27s method with Gabor frame

This paper deals with solving boundary value problems by Galerkin\u27s method in which we use Gabor frames as trial and test functions‎. ‎We show that‎, ‎the preconditioned stiffness matrix resulted by discretization is compressible and its sparsity‎ ‎pattern involves a bounded polyhedron structure‎. ‎Moreover‎, ‎we introduce an adaptive Richardson iterative method to‎ ‎solve the resulting system and we also show that by choosing a suitable smoothing parameter‎, ‎the method would be convergent‎

Multiresolution analysis as an approach for tool path planning in NC machining

Wavelets permit multiresolution analysis of curves and surfaces. A complex curve can be decomposed using wavelet theory into lower resolution curves. The low-resolution (coarse) curves are similar to rough-cuts and high-resolution (fine) curves to finish-cuts in numerical controlled (NC) machining.;In this project, we investigate the applicability of multiresolution analysis using B-spline wavelets to NC machining of contoured 2D objects. High-resolution curves are used close to the object boundary similar to conventional offsetting, while lower resolution curves, straight lines and circular arcs are used farther away from the object boundary.;Experimental results indicate that wavelet-based multiresolution tool path planning improves machining efficiency. Tool path length is reduced, sharp corners are smoothed out thereby reducing uncut areas and larger tools can be selected for rough-cuts

Parametric shape processing in biomedical imaging

In this thesis, we present a coherent and consistent approach for the estimation of shape and shape attributes from noisy images. As compared to the traditional sequential approach, our scheme is centered on a shape model which drives the feature extraction, shape optimization, and the attribute evaluation modules. In the first section, we deal with the detection of image features that guide the shape-extraction process. We propose a general approach for the design of 2-D feature detectors from a class of steerable functions, based on the optimization of a Canny-like criterion. As compared to previous computational designs, our approach is truly 2-D and yields more orientation selective detectors. We then address the estimation of the global shape from an image. Specifically, we propose to use cubic-spline-based parametric active contour models to solve two shape-extraction problems: (i) the segmentation of closed objects and (ii) the 3-D reconstruction of DNA filaments from their stereo cryo-electron micrographs. We present several enhancements of existing snake algorithms for segmentation. For the detection of 3-D DNA filaments from their orthogonal projections, we introduce the concept of projection-steerable matched filtering. We then use a 3-D snake algorithm to reconstruct the shape. Next, we analyze the efficiency of curve representations using refinable basis functions for the description of shape boundaries. We derive an exact expression for the error when we approximate a periodic signal in a scaling-function basis. Finally, we present a method for the exact computation of the area moments of such shapes

Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review

The paper characterizes classes of functions for which deep learning can be exponentially better than shallow learning. Deep convolutional networks are a special case of these conditions, though weight sharing is not the main reason for their exponential advantage

Adaptive Scattered Data Fitting with Tensor Product Spline-Wavelets

The core of the work we present here is an algorithm that constructs a least squares approximation to a given set of unorganized points. The approximation is expressed as a linear combination of particular B-spline wavelets. It implies a multiresolution setting which constructs a hierarchy of approximations to the data with increasing level of detail, proceeding from coarsest to finest scales. It allows for an efficient selection of the degrees of freedom of the problem and avoids the introduction of an artificial uniform grid. In fact, an analysis of the data can be done at each of the scales of the hierarchy, which can be used to select adaptively a set of wavelets that can represent economically the characteristics of the cloud of points in the next level of detail. The data adaption of our method is twofold, as it takes into account both horizontal distribution and vertical irregularities of data. This strategy can lead to a striking reduction of the problem complexity. Furthermore, among the possible ways to achieve a multiscale formulation, the wavelet approach shows additional advantages, based on good conditioning properties and level-wise orthogonality. We exploit these features to enhance the efficiency of iterative solution methods for the system of normal equations of the problem. The combination of multiresolution adaptivity with the numerical properties of the wavelet basis gives rise to an algorithm well suited to cope with problems requiring fast solution methods. We illustrate this by means of numerical experiments that compare the performance of the method on various data sets working with different multi-resolution bases. Afterwards, we use the equivalence relation between wavelets and Besov spaces to formulate the problem of data fitting with regularization. We find that the multiscale formulation allows for a flexible and efficient treatment of some aspects of this problem. Moreover, we study the problem known as robust fitting, in which the data is assumed to be corrupted by wrong measurements or outliers. We compare classical methods based on re-weighting of residuals to our setting in which the wavelet representation of the data computed by our algorithm is used to locate the outliers. As a final application that couples two of the main applications of wavelets (data analysis and operator equations), we propose the use of this least squares data fitting method to evaluate the non-linear term in the wavelet-Galerkin formulation of non-linear PDE problems. At the end of this thesis we discuss efficient implementation issues, with a special interest in the interplay between solution methods and data structures

A New Segmentation Algorithm for Prostate Boundary Detection in 2D Ultrasound Images

Prostate segmentation is a required step in determining the volume of a prostate, which is very important in the diagnosis and the treatment of prostate cancer. In the past, radiologists manually segment the two-dimensional cross-sectional ultrasound images. Typically, it is necessary for them to outline at least a hundred of cross-sectional images in order to get an accurate estimate of the prostate's volume. This approach is very time-consuming. To be more efficient in accomplishing this task, an automated procedure has to be developed. However, because of the quality of the ultrasound image, it is very difficult to develop a computerized method for defining boundary of an object in an ultrasound image. The goal of this thesis is to find an automated segmentation algorithm for detecting the boundary of the prostate in ultrasound images. As the first step in this endeavour, a semi-automatic segmentation method is designed. This method is only semi-automatic because it requires the user to enter four initialization points, which are the data required in defining the initial contour. The discrete dynamic contour (DDC) algorithm is then used to automatically update the contour. The DDC model is made up of a set of connected vertices. When provided with an energy field that describes the features of the ultrasound image, the model automatically adjusts the vertices of the contour to attain a maximum energy. In the proposed algorithm, Mallat's dyadic wavelet transform is used to determine the energy field. Using the dyadic wavelet transform, approximate coefficients and detailed coefficients at different scales can be generated. In particular, the two sets of detailed coefficients represent the gradient of the smoothed ultrasound image. Since the gradient modulus is high at the locations where edge features appear, it is assigned to be the energy field used to drive the DDC model. The ultimate goal of this work is to develop a fully-automatic segmentation algorithm. Since only the initialization stage requires human supervision in the proposed semi-automatic initialization algorithm, the task of developing a fully-automatic segmentation algorithm is reduced to designing a fully-automatic initialization process. Such a process is introduced in this thesis. In this work, the contours defined by the semi-automatic and the fully-automatic segmentation algorithm are compared with the boundary outlined by an expert observer. Tested using 8 sample images, the mean absolute difference between the semi-automatically defined and the manually outlined boundary is less than 2. 5 pixels, and that between the fully-automatically defined and the manually outlined boundary is less than 4 pixels. Automated segmentation tools that achieve this level of accuracy would be very useful in assisting radiologists to accomplish the task of segmenting prostate boundary much more efficiently