On The Continuous Steering of the Scale of Tight Wavelet Frames

In analogy with steerable wavelets, we present a general construction of adaptable tight wavelet frames, with an emphasis on scaling operations. In particular, the derived wavelets can be "dilated" by a procedure comparable to the operation of steering steerable wavelets. The fundamental aspects of the construction are the same: an admissible collection of Fourier multipliers is used to extend a tight wavelet frame, and the "scale" of the wavelets is adapted by scaling the multipliers. As an application, the proposed wavelets can be used to improve the frequency localization. Importantly, the localized frequency bands specified by this construction can be scaled efficiently using matrix multiplication

Steerable filtering using novel circular harmonic functions with application to edge detection

In this paper, we perform approximate steering of the elongated 2D Hermite-Gauss functions with respect to rotations and provide a compact analytical expressions for the related basis functions. A special notation introduced here considerably simplifies the derivation and unifies the cases of even and odd indices. The proposed filters are applied to edge detection. Quantitative analysis shows a performance increase of about 12.5% in terms of the Pratt’s figure of merit with respect to the well-established Gaussian gradient proposed earlier.

CLOSED FORM OF THE STEERED ELONGATED HERMITE-GAUSS WAVELETS

We provide a closed form, both in the spatial and in the frequency domain, of a family of wavelets which arise from steering elongated Hermite-Gauss filters. These wavelets have interesting mathematical properties, as they form new dyadic families of eigenfunctions of the 2D Fourier transform, and generalize the well known Laguerre-Gauss harmonics. A special notation introduced here greatly simplifies our proof and unifies the cases of even and odd orders. Applying these wavelets to edge detection increases the performance of about 12.5% with respect to standard methods, in terms of the Pratt’s figure of merit, both for noisy and noise-free input images

Discriminative Learning of Similarity and Group Equivariant Representations

One of the most fundamental problems in machine learning is to compare examples: Given a pair of objects we want to return a value which indicates degree of (dis)similarity. Similarity is often task specific, and pre-defined distances can perform poorly, leading to work in metric learning. However, being able to learn a similarity-sensitive distance function also presupposes access to a rich, discriminative representation for the objects at hand. In this dissertation we present contributions towards both ends. In the first part of the thesis, assuming good representations for the data, we present a formulation for metric learning that makes a more direct attempt to optimize for the k-NN accuracy as compared to prior work. We also present extensions of this formulation to metric learning for kNN regression, asymmetric similarity learning and discriminative learning of Hamming distance. In the second part, we consider a situation where we are on a limited computational budget i.e. optimizing over a space of possible metrics would be infeasible, but access to a label aware distance metric is still desirable. We present a simple, and computationally inexpensive approach for estimating a well motivated metric that relies only on gradient estimates, discussing theoretical and experimental results. In the final part, we address representational issues, considering group equivariant convolutional neural networks (GCNNs). Equivariance to symmetry transformations is explicitly encoded in GCNNs; a classical CNN being the simplest example. In particular, we present a SO(3)-equivariant neural network architecture for spherical data, that operates entirely in Fourier space, while also providing a formalism for the design of fully Fourier neural networks that are equivariant to the action of any continuous compact group.Comment: PhD thesi

Aeronautical engineering. A continuing bibliography with indexes, supplement 127, October 1980

A bibliography containing 431 abstracts addressing various topics in aeronautical engineering is given. The coverage includes engineering and theoretical aspects of design. construction, evaluation, testing, operation, and performance of aircraft (including aircraft engines) and associated components, equipment, and systems. It also includes research and development in aerodynamics, aeronautics, and ground support equipment for aeronautical vehicles

Learning Equivariant Representations

State-of-the-art deep learning systems often require large amounts of data and computation. For this reason, leveraging known or unknown structure of the data is paramount. Convolutional neural networks (CNNs) are successful examples of this principle, their defining characteristic being the shift-equivariance. By sliding a filter over the input, when the input shifts, the response shifts by the same amount, exploiting the structure of natural images where semantic content is independent of absolute pixel positions. This property is essential to the success of CNNs in audio, image and video recognition tasks. In this thesis, we extend equivariance to other kinds of transformations, such as rotation and scaling. We propose equivariant models for different transformations defined by groups of symmetries. The main contributions are (i) polar transformer networks, achieving equivariance to the group of similarities on the plane, (ii) equivariant multi-view networks, achieving equivariance to the group of symmetries of the icosahedron, (iii) spherical CNNs, achieving equivariance to the continuous 3D rotation group, (iv) cross-domain image embeddings, achieving equivariance to 3D rotations for 2D inputs, and (v) spin-weighted spherical CNNs, generalizing the spherical CNNs and achieving equivariance to 3D rotations for spherical vector fields. Applications include image classification, 3D shape classification and retrieval, panoramic image classification and segmentation, shape alignment and pose estimation. What these models have in common is that they leverage symmetries in the data to reduce sample and model complexity and improve generalization performance. The advantages are more significant on (but not limited to) challenging tasks where data is limited or input perturbations such as arbitrary rotations are present

Gamma ray Cerenkov telescope image analysis

The subject of this thesis is ground based gamma ray astronomy using the imaging atmospheric Cerenkov technique. The first two chapters are introductory, and describe the field of gamma ray astronomy, the generation of extensive air showers in the atmosphere and the Cerenkov radiation they induce. Chapter three describes the atmospheric Cerenkov telescope, including the development of the imaging technique for background discrimination. The characteristics of the three University of Durham atmospheric Cerenkov telescopes and the processing and calibration of their data products are outlined. Chapter four is concerned with periodic sources of gamma ray emission and includes a review of candidate sources and time series analysis techniques. An analysis of the Mark 3 telescope SMC X-1 database is presented. An upper limit of 1.2 x 10(^-11) cm(^-2) s(^-1) above a cosmic ray threshold of 1 TeV is determined for the guard ring analysis of Mark 3 data. For an analysis of medium resolution Mark 3 imaging data, the upper limit is 2 x 10(^-10) cm(^-2) s(^-1) above a cosmic ray threshold of 500 GeV. Chapter five introduces a new method for the parameterisation of Cerenkov images of extensive air showers recorded by atmospheric Cerenkov telescopes. This method, involving the optimization of a bivariate Gaussian fit to the image, is shown to be significantly better than the standard moment based parameterisation using simulated images. In Chapter six, both of these methods are employed in an attempt to enhance the signal to noise ratio for observations of the pulsar PSR 1706-44 made with the Mark 6 telescope and some evidence for steady emission is seen. The implied fluxes are (2.6 ± 0.3 ± 0.1)x 10(^-11) cm(^-2) s(^-1) above 420 GeV for the bivariate Gaussian analysis and (1.7 ± 0.4 ± 0.2)x10(^-11) cm(^-2) s(^-1) above 500 GeV for the moment analysis