An Analysis of Stockwell Transforms, with Applications to Image Processing

Time-frequency analysis is a powerful tool for signal analysis and processing. The Fourier transform and wavelet transforms are used extensively as is the Short-Time Fourier Transform (or Gabor transform). In 1996 the Stockwell transform was introduced to maintain the phase of the Fourier transform, while also providing the progressive resolution of the wavelet transform. The discrete orthonormal Stockwell transform is a more efficient, less redundant transform with the same properties. There has been little work on mathematical properties of the Stockwell transform, particularly how it behaves under operations such as translation and modulation. Previous results do discuss a resolution of the identity, as well as some of the function spaces that may be associated with it [2]. We extend the resolution of the identity results, and behaviour under translation, modulation, convolution and differentiation. boundedness and continuity properties are also developed, but the function spaces associated with the transform are unrelated to the focus of this thesis. There has been some work on image processing using the Stockwell transform and discrete orthonormal Stockwell transform. The tests were quite preliminary. In this thesis, we explore some of the mathematics of the Stockwell transform, examining properties, and applying it to various continuous examples. The discrete orthonormal Stockwell transform is compared directly with Newland’s harmonic wavelet transform, and we extend the definition to include varitions, as well as develop the discrete cosine based Stockwell transform. All of these discrete transforms are tested against current methods for image compression

Window-Dependent Bases for Efficient Representations of the Stockwell Transform

Since its appearing in 1996, the Stockwell transform (S-transform) has been applied to medical imaging, geophysics and signal processing in general. In this paper, we prove that the system of functions (so-called DOST basis) is indeed an orthonormal basis of L^2([0,1]), which is time-frequency localized, in the sense of Donoho-Stark Theorem (1989). Our approach provides a unified setting in which to study the Stockwell transform (associated to different admissible windows) and its orthogonal decomposition. Finally, we introduce a fast -- O(N log N) -- algorithm to compute the Stockwell coefficients for an admissible window. Our algorithm extends the one proposed by Y. Wang and J. Orchard (2009).Comment: 27 pages, 6 figure

Recognition of Microseismic and Blasting Signals in Mines Based on Convolutional Neural Network and Stockwell Transform

The microseismic monitoring signals which need to be determined in mines include those caused by both rock bursts and by blasting. The blasting signals must be separated from the microseismic signals in order to extract the information needed for the correct location of the source and for determining the blast mechanism. The use of a convolutional neural network (CNN) is a viable approach to extract these blast characteristic parameters automatically and to achieve the accuracy needed in the signal recognition. The Stockwell Transform (or S-Transform) has excellent two-dimensional time-frequency characteristics and thus to obtain the microseismic signal and blasting vibration signal separately, the microseismic signal has been converted in this work into a two-dimensional image format by use of the S-Transform, following which it is recognized by using the CNN. The sample data given in this paper are used for model training, where the training sample is an image containing three RGB color channels. The training time can be decreased by means of reducing the picture size and thus reducing the number of training steps used. The optimal combination of parameters can then be obtained after continuously updating the training parameters. When the image size is 180 × 140 pixels, it has been shown that the test accuracy can reach 96.15% and that it is feasible to classify separately the blasting signal and the microseismic signal based on using the S-Transform and the CNN model architecture, where the training parameters were designed by synthesizing LeNet-5 and AlexNet

Seismic Image Analysis Using Local Spectra

This report considers a problem in seismic imaging, as presented by researchers from Calgary Scientific Inc. The essence of the problem was to understand how the S-transform could be used to create better seismic images, that would be useful in identifying possible hydrocarbon reservoirs in the earth. The important first step was to understand what aspect of the imaging problem we were being asked to study. However, since we would not be working directly with raw seismic data, traditional seismic techniques would not be required. Rather, we would be working with a two dimensional image, either a migrated image, a common mid-point (CMP) stack, or a common depth point (CDP) stack. In all cases, the images display the subsurface of the earth with geological structures evident in various layers. For a given image the local spectrum is computed at each point. The various peaks in the spectrum are used to classify each pixel in the original seismic image resulting in an enhanced and hopefully more useful seismic pseudosection. Thus, the objective of this project was to improve the identification of layers and other geological structures apparent in the two dimensional image (a seismic section, or CDP gather) by classifying and coloring image pixels into groups based on their local spectral attributes

All the Groups of Signal Analysis from the (1+1)-affine Galilei Group

We study the relationship between the (1+1)-affine Galilei group and four groups of interest in signal analysis and image processing, viz., the wavelet or the affine group of the line, the Weyl-Heisenberg, the shearlet and the Stockwell groups. We show how all these groups can be obtained either directly as subgroups, or as subgroups of central extensions of the affine Galilei group. We also study this at the level of unitary representations of the groups on Hilbert spaces.Comment: 28 pages, 1 figur

Special Affine Stockwell Transform Theory, Uncertainty Principles and Applications

In this paper, we study the convolution structure in the special affine Fourier transform domain to combine the advantages of the well known special affine Fourier and Stockwell transforms into a novel integral transform coined as special affine Stockwell transform and investigate the associated constant Q property in the joint time frequency domain. The preliminary analysis encompasses the derivation of the fundamental properties, Rayleighs energy theorem, inversion formula and range theorem. Besides, we also derive a direct relationship between the recently introduced special affine scaled Wigner distribution and the proposed SAST. Further, we establish Heisenbergs uncertainty principle, logarithmic uncertainty principle and Nazarovs uncertainty principle associated with the proposed SAST. Towards the culmination of this paper, some potential applications with simulation are presented.Comment: arXiv admin note: text overlap with arXiv:2010.01972 by other author