Wavelet Model Research Articles

Wavelet Modelling of High Resolution Radar Imaging and Clinical Magnetic Resonance Tomography

The speed with which clinical magnetic resonance imaging (MRI) systems spread throughout the world was phenomenal. Coherent wavelets allow for a unified model of the multichannel perfect reconstruction analysis-synthesis filter bank of high resolution radar imaging and MRI. The geometric quantization construction of matched bank filters depends upon the Kepplerian spatiotemporal strategy which succeeds in the synchronous and stroboscopic summation over phase histories in local frequency encoding channels. The Kepplerian planetary clockwork of quantum holography is implemented in symplectic affine planes by Fourier analysis of the Heisenberg nilpotent Lie group G, and the associated reconstructing symbolic calculus on the selected energetic stratum of the unitary dual G or the quantized calculus of the C*-algebra of G. The neural network performed by quantum holograms allows for localization of cortical activations of the human brain.

A wavelet model for vocalic speech coarticulation

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On the limit of sampled signal extrapolation with a wavelet signal model∗

The extrapolation of sampled signals from a given interval using a wavelet model with various sampling rates is examined in this research. We present sufficient conditions on signals and wavelet bases so that the discrete-time extrapolated signal converges to its continuous-time counterpart when the sampling rate goes to infinity. Thus, this work provides a practical procedure to implement continuous-time signal extrapolation, which is important in wideband radar and sonar signal processing, with a discrete one via carefully choosing the sampling rate and the wavelet basis. A numerical example is given to illustrate our theoretical result.

A computational vision approach to image registration

A computational vision approach is presented for the estimation of 2-D translation, rotation, and scale from two partially overlapping images. The approach results in a fast method that produces good results even when large rotation and translation have occurred between the two frames and the images are devoid of significant features. An illuminant direction estimation method is first used to obtain an initial estimation of camera rotation. A small number of feature points are then located, using a Gabor wavelet model for detecting local curvature discontinuities. An initial estimate of scale and translation is obtained by pairwise matching of the feature points detected from both frames. Finally, hierarchical feature matching is performed to obtain an accurate estimate of translation, rotation and scale. A method for error analysis of matching results is also presented. Experiments with synthetic and real images show that this algorithm yields accurate results when the scale of the images differ by up to 10%, the overlap between the two frames is as small as 23%, and the camera rotation between the two frames is significant. Experimental results and applications are presented.

Segmentation and stress in the rhythmic structure of music and speech: A wavelet model

The rhythm of a piece of music or a poem can be described by hierarchical tree structure. A new application of wavelet analyses to rhythm in music and speech is described whose result is a hierarchical structure [N. P. McAngus Todd, J. Acoust. Soc. Am. 92, 2380 (A) (1992)]. This structure has two complementary components that are obtained from the projection of loci, in the frequency-time plane, of zero crossings of the slope of the energy flux surface in the energy-time plane: (a) a segmentationstructure, corresponding to positive second derivatives at the zero crossings; (b) a stressstructure, corresponding to negative second derivatives at the zero crossings. The value of the stress is given by the second derivative, which has a ‘‘Mexican hat’’ impulse response. These structures resemble the grouping/metrical structures of the theory of F. Lerdahl and R. Jackendoff [AGenerativeTheoryofTonalMusic (MIT, Cambridge, MA, 1983)]. Example analyses of musical and speech performance are demonstrated for this method and evidence is provided that the resultant structures have a reasonable perceptual basis. The possibility of using this technique for the synthesis of rhythm using a cascade structure is discussed.

Proposal of a wavelet model for vocalic speech coarticulation

A recent development in the theory of wavelets is wavelet system characterization [R. K. Young, WaveletTheoryanditsApplications (Kluwer, Boston, 1993)]. In wavelet system theory, the wavelet transform can be used to describe the time-frequency behavior of a transmission channel, by virtue of its ability to describe the time-frequency content of the system’s input and output signals. The present research proposes a wavelet-system model of speech production; wherein, the system is the process of transformation from one speechstate to another. The transmission channel generates an effected utterance from a control utterance. A speech production effect (such as voice quality, or the influence of the speaker) is thereby characterized in a time-frequency manner. The effect of CVC coarticulation is investigated in particular, using samples of real speech. In this case, an isolated vowel utterance (the control) is contrasted with an utterance of the same vowel imbedded within a /C–C/ context (the effected). The channel thus describes (in terms of wavelets) the CVC coarticulation. [Work supported by Appl. Res. Lab.]

Application of wavelet transforms to ultrasonic medical imaging

In this paper, wavelet transform is applied to study ultrasonic medical imaging. Nondiffracting source ultrasonic computed tomography and diffraction ultrasonic tomography are considered. The multiresolution property of wavelet transform is used. Wavelet transform is also applied to study nonstationary ultrasonic medical images such as blood flow in heart which windowed Fourier transform (short-time Fourier transform or Gabor transforms) is not convenient to use. For the nondiffracting (straight ray) sources, the Fourier slice theorem and parallel and fanbeam reconstruction algorithm are modified for wavelet transforms. For diffracting sources, and Fourier diffraction theorem is modified for wavelet transforms. The wavelet approach to multiresolution decomposition is studied. The multiresolution wavelet model is reviewed which shows that the difference of information between two successive resolutions can be computed by decomposing the signal in a wavelet orthonormal basis. This is an improved multiresolution pyramid algorithm. The equivalence between multiresolution approximations and wavelet basis is used to derive wavelet bases. The concept of multiresolution which gives different information of an image at two different resolutions is particularly useful to texture discrimination in medical imaging. A preliminary simulation result is given.

Comparative analysis of wavelet models and vision

The authors compare visual stimulus decomposition with the wavelet multiresolution model. The retinal image is analyzed by a multifrequency channel decomposition and then the information is transmitted to the visual cortex. They describe a multiresolution decomposition of the stimulus into a wavelet base. The chosen stimulus is a continuous luminance variation between two fields of respective luminances L1 and L2. The variation is described by the function: B (b,x)=1/2 (tanh (x/b)+1). The wavelet analysis of the function B shows that the wavelet transform decreases (absolute values) with area width. With a sharp area, the transmitted information is more important than that corresponding to a wide area.

THE DETERMINATION OF THE SEISMIC QUALITY FACTOR Q FROM VSP DATA: A COMPARISON OF DIFFERENT COMPUTATIONAL METHODS1

AbstractTen methods for the computation of attenuation have been investigated, namely: amplitude decay, analytical signal, wavelet modelling, phase modelling, frequency modelling, rise‐time, pulse amplitude, matching technique, spectral modelling and spectral ratio. In particular, we have studied the reliability of each of these methods in estimating correct values of Q using three synthetic VSP seismograms for plane P‐waves with different noise contents.The investigations proved that no single method is generally superior. Rather, some methods are more suitable than others in specific situations depending on recording, noise or geology. The analytical signal method has been demonstrated to be superior if true amplitude recordings are available. Otherwise spectral modelling or, in the ‘ noise‐free’ case the spectral ratio method, is optimal.Finally, two field VSPs in sediments are investigated. Only in the case of the highest quality VSP can significant information be deduced from the computed attenuation.

Comparison of seven methods for the computation of Q

Seven methods for the evaluation of the seismic quality factor Q have been investigated: (1) wavelet modelling, (2) spectral modelling, (3) spectral ratio method, (4) matching technique, (5) amplitude decay method, (6) risetime method, and (7) the determination of Q using the analytical signal. These methods were tested using three theoretical VSP seismograms which are based on the same velocity and Q depth model. Whereas in the noise-free case the method of the analytical signal (7) proved to be superior to all other methods, the modelling techniques (1) and (2) seem to have advantages when minor noise is present. In the case of intensive noise it is hard to obtain any reliable estimate of attenuation.

Multifrequency channel decompositions of images and wavelet models

The author reviews recent multichannel models developed in psychophysiology, computer vision, and image processing. In psychophysiology, multichannel models have been particularly successful in explaining some low-level processing in the visual cortex. The expansion of a function into several frequency channels provides a representation which is intermediate between a spatial and a Fourier representation. The author describes the mathematical properties of such decompositions and introduces the wavelet transform. He reviews the classical multiresolution pyramidal transforms developed in computer vision and shows how they relate to the decomposition of an image into a wavelet orthonormal basis. He discusses the properties of the zero crossings of multifrequency channels. Zero-crossing representations are particularly well adapted for pattern recognition in computer vision. >

Optimal Identification of Non-minimum Phase ARMA (P, Q) Seismic Wavelet

This paper is dealing with the application of parametric methods to the classical seismic problem. A new maximum-likelihood algorithm is proposed, which gives a fast and globally optimal identification of non-minimum phase seismic wavelets. As ARMA(p, q) model is used, we can identify the MA part beyond the limit of the AH order so that the wavelet model can more easily represent a geophysical seismic wavelet. First, we find a minimum phase wavelet to estimate the AA part. Then, we search for the globally optimal MA parameters by using the zero-discrimination method the resulting algorithm is 2p times faster than Mendel's method. Moreover the solution is globally optimal and can avoid suboptimal solution that may result from other methods. Identification in synthetic data demonstrates the effectiveness and efficiency of the algorithm.

State-space Methods of Deconvolution for Geophysical Data Processing

Seismic data processing is of great economic importance and it is not surprising that many different procedures have been proposed, particularly since the development of modern digital computers. An important innovation in seismic analysis was proposed by Mendel (1977), who was the first to utilise state-space methodology in performing “minimum-variance deconvolution”. This novel approach seems to work well in the seismic context but appears to have some limitations when applied in related areas of geophysical data processingIn this paper, we describe a new approach to the deconvolution problem, based on optimal state-space smoothing procedures, which has been developed for a new geophysical prospecting system. It proves useful if the multiple signals from the system are combined by principal component analysis (PCA) to yield two orthogonal signals which carry most of the information that is useful for the deconvolution analysis. Recursive smoothing algorithms are then applied to the analysis of these orthogonal components with a non-minirnum phase wavelet model, which is required to define the algorithm, obtained by optimal instrumental variable methods of estimation. The Expectation and Maximization (EM) algorithm is utilised to estimate the noise process variances. In effect, the smoothing algorithm acts as an off-line, lag-free wave shaping filter, which attenuates the high frequency noise components in the deconvolved trace. The results of the new method are compared with those obtained by Mendel's minimum-variance deconvolution procedure.

Minimum-Variance and Maximum-Likelihood Deconvolution for Noncausal Channel Models

This paper extends the previous works of Mendel and his students on the subject of deconvolution from causal channel (wavelet) models to noncausal channel models. Noncausal wavelets occur, for example, in seismic data processing when a land vibrator is used to excite the Earth. Minimum-variance and maximum-likelihood deconvolution algorithms are developed herein for symmetrical and/or nonsymmetrical time-invariant wavelets that are excited by stationary and/or nonstationary white noise inputs. Minimum-variance deconvolution algorithms for a noncausal wavelet turn out to be quite different than those for a causal wavelet; however, maximum-likelihood deconvolution algorithms for a noncausal wavelet, which involve event detection and amplitude restoration, are essentially the same as those for a causal wavelet. Examples are provided that illustrate the performance of the different deconvolution algorithms.

COMPARISON OF METHODS TO DETERMINE Q IN SHALLOW MARINE SEDIMENTS FROM VERTICAL REFLECTION SEISMOGRAMS*

AbstractFour methods for the determination of Q in marine sediments are compared: two traditional methods, i.e. the risetime and the spectral ratio method, and two newly established ones, the spectrum modeling and the wavelet modeling method. In the latter one Q and the reflection time T are determined simultaneously, which gives a much better accuracy for T than reading it from the seismogram. The risetime and the spectral ratio methods are used for obtaining Q directly from the data. The principle of the modeling methods is to calculate the effect of absorption and dispersion on a reference wavelet or its spectrum for various values of Q, and the best fit between the observed and the calculated data leads to the optimum result. Numerical tests on synthetic data show that a precision of more than 25% for data containing noise or superposed arrivals can hardly be achieved; in any case, wavelet modeling is the superior method. Application to data from a vertical reflection profile in the Baltic Sea yields Q in the range of 15–100 for different layers, which is to be expected in the sedimentary environment of this area.The computations were performed in the Computer Center of Kiel University. The authors thank R. Meissner for his comments on the manuscript.

Minimum entropy deconvolution and simplicity: A noniterative algorithm

Minimum entropy deconvolution (MED) is a technique developed by Wiggins (1978) with the purpose of separating the components of a signal, as the convolution model of a smooth wavelet with a series of impulses. The advantage of this method, as compared with traditional methods, is that it obviates strong hypotheses over the components, which require only the simplicity of the output. The degree of simplicity is measured with the Varimax norm for factor analysis. An iterative algorithm for computation of the filter is derived from this norm, having as an outstanding characteristic its stability in presence of noise. Geometrical analysis of the Varimax norm suggests the definition of a new criterion for simplicity: the D norm. In case of multiple inputs, the D norm is obtained through modification of the kurtosis norm. One of the most outstanding characteristics of the new criterion, by comparison with the Varimax norm, is that a noniterative algorithm for computation of the deconvolution filter can be derived from the D norm. This is significant because the standard MED algorithm frequently requires in each iteration the inversion of an autocorrelation matrix whose order is the length of the filter, while the new algorithm derived from the D norm requires the inversion of a single matrix. On the other hand, results of numerical tests, performed jointly with Graciela A. Canziani, show that the new algorithm produces outputs of greater simplicity than those produced by the traditional MED algorithm. These considerations imply that the D criterion yields a new computational method for minimum entropy deconvolution. A section of numerical examples is included, where the results of an extensive simulation study with synthetic data are analyzed. The numerical computations show in all cases a remarkable improvement resulting from use of the D norm. The properties of stability in the presence of noise are preserved as shown in the examples. In the case of a single input, the relation between the D norm and the spiking filter is analyzed (Appendix B).

Advances In Surface Analysis Technology

The needs of modem surface metrology assessment cover not only the extraction of roughness, waviness, but also identification of surface texture or the multiscalar properties of a surface topography. In answer to this, the newly developing wavelet theory has been introduced into surface characterisation in 1994. Wavelet analysis employs time-frequency windows and offers the relevant time-frequency analysis, as a result, it can divide surface topography into different frequency components, and then study each component with the multiresolution. This paper provides a survey of the most recent work in the field of surface metrology by using wavelet theory. The major wavelet models used in surface metrology will be introduced and the basic theory, algorithm and the properties of these models will also be discussed.