L. Mansinha

The S‐transform with windows of arbitrary and varying shape

The S-transform is an invertible time-frequency spectral localization technique which combines elements of wavelet transforms and short-time Fourier transforms. In previous usage, the frequency dependence of the analyzing window of the S-transform has been through horizontal and vertical dilations of a basic functional form, usually a Gaussian. In this paper, we present a generalized S-transform in which two prescribed functions of frequency control the scale and the shape of the analyzing window, and apply it to determining P-wave arrival time in a noisy seismogram. The S-transform is also used as a time-frequency filter; this helps in determining the sign of the P arrival.

Pattern analysis with two-dimensional spectral localisation: Applications of two-dimensional S transforms

Abstract An image is a function, f(x, y) , of the independent space variables x and y . The global Fourier spectrum of the image is a complex function F(k x , k y ) of the wave numbers k x and k y . The global spectrum may be viewed as a construct of the spectra of an arbitrary number of segments of f(x, y) , leading to the concept of a local spectrum at every point of f(x, y) . The two-dimensional S transform is introduced here as a method of computation of the local spectrum at every point of an image. In addition to the variables x and y , the 2-D S transform retains the variables k x and k y , being a complex function of four variables. Visualisation of a function of four variables is difficult. We skirt around this by removing one degree of freedom, through examination of ‘slices’. Each slice of the 2-D S transform would then be a complex function of three variables, with separate amplitude and phase components. By ranging through judiciously chosen slice locations the entire S transform can be examined. Images with strictly periodic patterns are best analysed with a global Fourier spectrum. On the other hand, the 2-D S transform would be more useful in spectral characterisation of aperiodic or random patterns.

Local S Spectrum Analysis of 1-D and 2-D Data

Abstract The local changes of the spectrum with time are often more interesting than the spectrum of the whole time series. For example, there is an apparent drift in the nominal 28 day fluctuations of sunspot numbers over the period of the sunspot cycle, averaging ∼ 11.1 years. This time-local change in spectrum is due to a combination of Sporers Law and the differential rotation of the sun. Similarly, the space-local variations in the 2-D spectrum on an image conveys visual information on textures, boundaries and shapes. In this paper we use the recently developed S -transform to analyse two segments of the Wolf Sunspot series, a seismogram, and a synthetic 2-D image as examples of applications of the S -transform for time-local and space-local spectral analysis.

Segmentation of petrographic images by integrating edge detection and region growing

Abstract A novel approach to segmenting petrographic images is proposed in this paper. A series of edge operators with various sizes of masks are first defined. By considering a larger neighborhood, the effects of noise or surface irregularities on edges are reduced. Color edges in an image are obtained by combining the edge operators and a color edge detection algorithm. A seeded region-growing algorithm is then used to segment the image based on the color edge information and the distances between edge-pixels and non-edge pixels. Seed regions are created automatically. These regions grow simultaneously. After all pixels in the image are labeled, the boundaries shared by two regions are checked. If a boundary is weak enough, it is eliminated and the corresponding two regions are merged. In the ultimate segmented map, each region whose size is large enough corresponds to a mineral grain in the image. This approach has been implemented in C++ under the Linux environment. Three sets of petrographic images were used to test the method.

Local quaternion Fourier transform and color image texture analysis

Color images can be treated as two-dimensional quaternion functions. For analysis of quaternion images, a joint space-wavenumber localized quaternion S transform (QS) is presented in this study for a simultaneous determination of the local color image spectra. The QS transform uses a two-dimensional Gaussian localizing window that scales with wavenumbers. Rotation invariance, invertibility and computational aspects of the QS transform are discussed. The power map of the QS transform is presented here as a powerful tool in color image texture and pattern analysis. Examples are presented.

Generation of aquifer heterogeneity maps using two-dimensional spectral texture segmentation techniques

Numerical models that solve the governing equations for subsurface fluid flow and transport require detailed quantitative maps of spatially variable hydraulic properties. Recently, there has been great interest in methods that can map the spatial variability of hydraulic properties such as porosity and hydraulic conductivity (permeability). Presently, only limited data on natural permeability spatial structure are available. These data are often based on extensive discrete sampling in outcrops or boreholes. Then methods are used to interpolate between data values to map aquifer heterogeneity. Interpolation methods often mask critical local or intermediate scale heterogeneities. As sediment texture is directly correlated with many hydraulic properties we developed two new texture segmentation algorithms based on a space-local two-dimensional wavenumber spectral method known as the S-Transform. Existing texture segmentation algorithms could not delineate the subtle and continuous texture variations that exist in natural sediments. The S-Transform algorithms successfully delineated geologic structures and grain size patterns in photographs of outcrops in a glacial fluvial deposit; thus, no interpolation methods were required to produce continuous two-dimensional maps of texture facies. The S-Transform method is robust and is insensitive to changes in light intensity, and moisture variations. This makes the algorithm particularly applicable to natural sedimentary outcrops. The effectiveness of our methods are tested by correlating measured relative grain sizes in the images with actual grain size measurements taken from the sedimentary outcrops.

Time Localised Band Filtering Using Modified S-Transform

A noisy time series, with both signal and noise varying in frequency and in time, presents special challenges for improving the signal to noise ratio. A modified S-transform time-frequency representation is used to filter a synthetic time series in a two step filtering process. The filter method appears robust within a wide range of background noise levels.

The trinion Fourier transform of color images

Any color may be represented in terms of three components (RGB or HSL) or four components (CMYK). For the four component color representation the use of quaternions, with one real and three imaginary components, is natural. By setting one component to zero, quaternions have been used in RGB or HSL representation of colors and color images. In this paper a new quantity, trinion, with one real and two imaginary components, is introduced and its use in color image representation is examined. The goal is to see if significant efficiencies in representation, analysis and computation involving three component color images accrue with the use of trinions. Two versions of the trinion Fourier transform (TFT) are introduced and it is shown that using TFT is preferable for combined analysis of three component color images rather than separate monochromatic analysis of each component and use of quaternions. Joint space-wavenumber localized trinion S (TS) transform with a two-dimensional Gaussian window function that scales with wavenumbers is also presented. Invertibility, rotation invariance, and computational aspects of the TS transform are discussed.

Earthquakes and the spectrum of the Brussels Superconducting Gravimeter data for 1982–1986

Abstract Spectral analysis of the Brussels Superconducting Gravimeter data from 1982 to 1986 by earlier workers revealed a decaying spectral peak at 13.89 h following the December 30, 1983, Hindu Kush earthquake. Similar associations were presented for several other large earthquakes. The presence of the observed 13.89-h spectral peak was ascribed to possible excitation of inertial waves in the fluid outer core of the Earth. In this paper, the same data set is analysed with a moving window spectral method that shows the variation of the power in this and other spectral bands. Also, confidence intervals for the spectral estimates are computed. No unambiguous relation with the earthquake sequence used previously is found, and no statistically significant non-tidal spectral peak is found in the band between 12 and 24 h. However, the moving window spectral method applied to the 4.8- and 8-h spectral bands shows a closer association with the earthquake sequence. Again, although confidence interval calculations show some statistically significant spectral features in these bands, there is evidence that at least some of these features are generated by non-linear tidal interactions or possibly over-correction of the raw data for tidal signals. It is speculated that if indeed earthquake-induced, long-period core waves cause a detectable gravity signal, then the spectrum is perhaps richer in the short-period bands.

Time-frequency localization with the Hartley S-transform

The Hartley transform, like the Fourier transform, is used for determining the spectrum of a complete time series. However, problems arise if the time series has time-dependent spectral content, since the Hartley kernel has no time localization. To this end, a short-time Hartley transform has been proposed and defined in analogy with the short-time Fourier transform. The frequency invariance of the window used in the short-time Hartley transform, though, leads to problems similar to those encountered in the short-time Fourier transform; namely, poor time resolution at high frequencies, and artifacts at low frequencies. In the Fourier case, these problems can be addressed through use of the S-transform, whose window scales with frequency to accommodate the scaling of the Fourier sinusoid. We apply the same principles to define the Hartley S-transform, using a scalable window.