R. Wilson, A.D. Calway and E.R.S. Pearson, "A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis", IEEE Trans. Inform. Theory, 38, pp. 674-690, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....occur in the retinal image as it does not attempt to combine information across scales. Also, it does not distinguish between true junctions and points of high curvature. 9 6 Conclusions and Further work This work uses and extends the ideas previously presents by Davies, Wilson and Calway [19] [22]. Its main contribution is that we apply a superposition of Gaussian models and use a synthesised magnitude to reconstruct the data. This allows us to derive a likelihood, P ( Y jGK ) to select a model G k , which models a junction with K = 1; 2; 3 branches. By using an explicit Gaussian ....

R. Wilson, A. D. Calway, and E. R. S. Pearson, \A generalized wavelet transform for fourier analysis: the multiresolution fourier transform and its application to image and audio signal analysis," IEEE Tran. IT, Special Issue on Wavelet Representations, 1992.

....in work of Mann and Haykin [22] and Baraniuk and Jones [5] especially as the source of chirplets. We expand the use of the term warblet from [22] where it is used to mean a particular function and its orbit under the action of the AWH group) to include the action of the UT group also. In [33], Wilson et al. investigate applications of the transform associated to the AWH group and its matrix coefficients (which they call the multiresolution Fourier transform) to problems in the analysis of images and audio signals. Appearance of the UT group in signal analysis wavelet theory is more ....

....the AWH and UT representations are not square integrable in the usual sense, using generalized notions of square integrability one may still produce desired frames. Particular examples of (continuous) frame operators appear in work of Torresani [29] Segman and Schempp[24] and Wilson et al. [33]. We shall, however, establish more general criteria in the context of both the AWH and UT groups for constructing continuous and discrete frames for L 2 (R) and L p (R) The generalized notions of square integrability and frames used in establishing these criteria have been developed in a ....

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R. Wilson, A.D. Calway and E.R.S. Pearson, A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis, IEEE Trans. Inform. Theory 38 (1992), 674--712.

....with noise robustness, for example, arise because the analysis region does not contain sufficient data to uniquely constrain the problem. One approach which has been advocated as a means of dealing with the above problem in various areas of image analysis is the use of multiresolution techniques [146, 14, 149, 41]. A multiresolution approach enables the choice of analysis window to adapt to the underlying data set so that estimation accuracy can be traded off against spatial resolution 1.6 A Multiresolution Affine Approach 14 across multiple resolutions. An analogy can be made with surface approximation ....

....regions. Calway et al. 27] developed a frequency domain approach based on a correlation technique, which can be shown to be equivalent to full search block matching. Their method is also implemented within an efficient multiresolution framework, the Multiresolution Fourier Transform (MFT) [146]. Correlation in the spatial domain is prohibitively expensive but, if implemented via the frequency domain, the computational load becomes manageable. Young and Kingsbury use an approach reminiscent of the correlationtechnique mentioned above, but limited to a single fixed resolution [152] ....

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R. Wilson, A. D. Calway, and E. R. S. Pearson. A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analyis. IEEE Transactions on Information Theory, 38(2), March 1992.

....of leading to a worse spatial localization. This splitting is however still strongly orientation dependent. Another solution to the coarse frequency sampling has been proposed by Wilson et al. in their Multiresolution Fourier Transform , which incorporates Fourier analysis in a wavelet scheme [11]. An alternative frequency splitting scheme has been studied by Kacker et al. 12] to improve orientational selectivity. C. Continuous wavelet decomposition Finer frequency selectivity can be obtained from (1) when dropping the constraints of a complete orthogonal decomposition, at the expense ....

R. Wilson, A.D. Calway, and E.R.S. Pearson, "A generalized wavelet transform for fourier analysis: The multiresolution fourier transform and its application to image and audio signal analysis," IEEE Trans. Info. Theory, vol. 38, pp. 674--690, 1992.

....with this is found in the work of Kacker et al. [17] who claim classification improvements to the level of Gabor transform results by using an additional splitting in a pyramidal wavelet scheme. A different but related solution to the coarse frequency sampling has been proposed by Wilson et al. [36] in their multiresolution Fourier Transform, which incorporates Fourier analysis in a wavelet scheme. C. Wavelet versus Gabor The Gabor decomposition approach has been successful in texture analysis for many years [16] 21] The newer wavelet approaches also perform well, but fail to yield ....

R. Wilson, A. D. Calway, and E. R. S. Pearson. A generalized wavelet transform for fourier analysis: The multiresolution fourier transform and its apllication to image and audio signal analysis. IEEE Trans. Info. Theory, 38:674-- 690, 1992.

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R. Wilson, A.D. Calway and E.R.S. Pearson, "A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis", IEEE Trans. Inform. Theory, 38, pp. 674-690, 1992.

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R. Wilson, A. D. Calway, and E. R. S. Pearson. A Generalized Wavelet Transform for Fourier Analysis: the Multiresolution Fourier Transform and its Application to Image and Audio Signal Analysis. IEEE Trans. IT, Special Issue on Wavelet Representations, 38(2):674--690, 1992.

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R.G. Wilson, A.D. Calway, and E.R.S. Pearson. A generalized wavelet transform for Fourier analysis: The multiresolution Fourier transform and its applications to image and audio signal analysis. IEEE Transancation on Information Theory, 38(2):674--690, March 1992.

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R.G. Wilson, A.D. Calway, and E.R.S. Pearson. A generalized wavelet transform for Fourier analysis: The multiresolution Fourier transform and its applications to image and audio signal analysis. IEEE Transancation on Information Theory, 38(2):674--690, March 1992.

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R. Wilson, A. D. Calway & E. R. S. Pearson. "A Generalized Wavelet Transform for Fourier Analysis: the Multiresolution Fourier Transform and its Application to Image and Audio Signal Analysis." IEEE Trans. IT, Special Issue on Wavelet Representations 38(2), pp. 674--690, 1992.

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R. Wilson, A. D. Calway, and E. R. S. Pearson. A Generalized Wavelet Transform for Fourier Analysis: the Multiresolution Fourier Transform and its Application to Image and Audio Signal Analysis. IEEE Trans. IT, Special Issue on Wavelet Representations, 38(2):674--690, 1992.

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R. Wilson, A. D. Calway, and E. R. S. Pearson. A Generalized Wavelet Transform for Fourier Analysis: the Multiresolution Fourier Transform and its Application to Image and Audio Signal Analysis. IEEE Trans. IT, Special Issue on Wavelet Representations, 38(2):674-690, 1992.

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R. Wilson, A. D. Calway, and E. R. S. Pearson. A Generalized Wavelet Transform for Fourier Analysis: the Multiresolution Fourier Transform and its Application to Image and Audio Signal Analysis. IEEE Trans. IT, Special Issue on Wavelet Representations, 38(2):674--690, 1992.

.... x for clarity) A t 1 = X x fw g( t ) X x g( t )g( t ) 3) t 1 = X x xf w g( t ) X x fw g( t ) 4) C t 1 = 2 X x ( x t ) x t ) T fw g( t ) X x fw g( t ) 5) The initial estimate 0 is obtained by using the multiresolution Fourier Transform (MFT) [7]. The windowed Fourier transform of fw ( x) fw ( u) is also a Gaussian with the spectral energy distribution dependent on the type of feature. The principal components of the moment of inertia tensor I of the spectral energy: I = 1 B n X u u u T j fw ( u)j 2 (6) where B is ....

R. Wilson, A. D. Calway & E. R. S. Pearson. "A Generalized Wavelet Transform for Fourier Analysis: the Multiresolution Fourier Transform and its Application to Image and Audio Signal Analysis." IEEE Trans. IT, Special Issue on Wavelet Representations 38(2), pp. 674--690, 1992.

.... have been described: techniques such as the active contour models, which rely on energy minimisation at pixel level [17] global methods, such as the Hough transform (HT) 18] or intermediate approaches, such as the hierarchical HT [19] and Multiresolution Fourier Transform (MFT) based methods [20]. It is clearly attractive to combine region processing with boundary detection to overcome their respective shortcomings. Haddon and Boyce extended relaxation techniques to combine edge and region information, using co occurrence matrices [21] In MRF modelling, edges are incorporated by the ....

....using an explicit boundary representation in the present work. Nonetheless, the region process has contributed to the boundary identification, giving a result which compares favourably with Canny edge detection (e.g. figures 13(e) and (f) and more sophisticated methods reported elsewhere (e.g. [20]) 5 Conclusions A new model and algorithm for image segmentation have been described and been shown to be effective in describing and estimating regions and their boundaries with high accuracy from noisy data. In applications where region shape is important, it has a number of advantages over ....

R. Wilson, A. D. Calway, and E. R. S. Pearson. A Generalized Wavelet Transform for Fourier Analysis: the Multiresolution Fourier Transform and its Application to Image and Audio Signal Analysis. IEEE Trans. IT, Special Issue on Wavelet Representations, 1992.

.... to translations and dilations, two of the most important sources of variation in images [4] Not every multiresolution representation is equally e ective in this regard: schemes using dyadic decimation, such as pyramids and orthonormal wavelets, sacri ce translation invariance to reduce redundancy [6, 25, 12]. While this is desirable in image compression, it is not ideal for problems in computer vision, where compactness is less important than utility in a range of problems from segmentation to motion analysis. In those cases, it is the combination of statistical inference, usually from incomplete ....

R. Wilson, A. Calway, and E.R.S. Pearson. A Generalized Wavelet Transform for Fourier Analysis: The Multiresolution Fourier Transform and Its Application to Image and Audio Signal Analysis. IEEE Trans. Information Theory, 38(2):674{ 690, March 1992.

....testing strategies have become more subtle, so have the algorithms of feature extraction. From the work of Roberts [19] on luminance edge detection in 1965, this has evolved to the sophisticated techniques used today, such as regularization [15] Hough transforms [20, 21] multiresolution analysis [22, 23, 24], tensors [25] and tangent bundles [16] aimed at providing ever more effective ways of detecting edges and grouping them into potential boundary contours. Of course, in many systems, there is information available from shading, texture, motion and stereopsis (e.g. 7] but this is essentially a ....

R. Wilson, A.D. Calway and E.R.S. Pearson,"A Generalized Wavelet Transform for Fourier Analysis: the Multiresolution Fourier Transform and its Application to Image and Audio Analysis",IEEE Trans. Inform. Th., 38, pp.674-690, Mar. 1992.

....Figure 6: MFT 2 D structure 4 Implementation 4.1 Multiresolution Fourier Transform Central to the above analysis is the availability of local frequency estimates over multiple scales. These can be obtained from the MFT of the texture image. The MFT is a generalized form of wavelet transform [1] and [2] The MFT is to have scale oe as part of a signal representation in phase space ( so that MFT is able to perform local Fourier analysis at various scales. The general structure of the MFT is shown in figure 6. This section briefly introduces the properties of MFT. 4.1.1 Continuous ....

R. Wilson A. Calway, E.R.S. Pearson. A Generalized Wavelet Transform for Fourier Analysis: The Multiresolution Fourier Transform and Its Application to Image and Audio Signal Analysis. IEEE Trans. I.T., 38(2), March 1992.

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Wilson R, Calway A.D, Pearson E.R.S, 1992, "A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis", IEEE Trans. Inf Th., 38, 2, 674-690.

....frequency methods in the analysis of images, and, in the case of the latter, a realization that such methods are best employed over a range of spatial scales. More recently, these approaches have been taken a step further by the introduction of the multiresolution Fourier transform (MFT) 9] [13]. This is a generalised form of WT in which the properties of the two above mentioned approaches are combined to yield a general multiresolution framework. It achieves this by combining a number of single resolution representations defined at different scales into one hierarchical entity; yielding ....

....of embedded WTs each defined over a range of spatial and spatial frequency resolutions. The advantage of this approach is that it combines the completeness of the local spectra provided by a STFT, with the multiresolution properties underlying the traditional form of WT. As was demonstrated in [13], this generalisation enables considerable extension of the applicability of such representations and provides a potentially more versatile framework in which to base suitable image models. An additional characteristic which is of interest in image processing is that of orientation selectivity. ....

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R.Wilson, A.D.Calway, and E.R.S.Pearson, "A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis", in submission.

.... representation, but in the present work this is the multiresolution Fourier transform (MFT) a generalisation of the wavelet and windowed Fourier (Gabor) transforms in which scale and frequency are independently varied, allowing local Fourier analysis to be carried out over a range of scales [11]. This means that the scale of analysis can be adapted to the data, a generalisation of the more familiar scale space concept [12] From the Fourier representation at a given scale, it is possible to identify the affine co ordinate transform which warps a given block of texture into an ....

....scale will be unknown a priori, ie the size of the ideal prototype will vary according to the structure within a given texture, the spectra will need to be available over a range of scales. As indicated earlier, these are obtained from the MFT, which, for the discrete image x( j ) is given by [11] x( i (n) j (n) n) X k w n ( k Gamma i (n) x( k ) exp[ Gamma j (n) k ] 16) and hence corresponds to the DFT of the region defined by the window w n ( j ) centred on the point i (n) where the spatial extent of the former varies with the scale ....

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R. Wilson, A.D. Calway, E.R.S. Pearson,"A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis", IEEE Trans. Inf. Th., IT-38, pp. 674-90, 1992.

....and k in frames m and m 1 respectively, at scale n (cf. equation 23) To implement the above algorithm we need local spectral estimates over different sized regions. An ideal tool for this is the Multiresolution Fourier Transform (MFT) extensively described by Calway [24] and Wilson et al. [25]. The MFT of a discrete 2 D image x( i ) at scale oe n , frequency j and spatial position i is defined as x( i ; j ; oe n ) X k w n ( k Gamma i )x( k )e Gamma k Delta j (20) Estimation Motion Estimation Motion Motion Estimation Figure 3: ....

R. Wilson, A. D. Calway, and E. R. S. Pearson, "A Generalized Wavelet Transform for Fourier Analysis: The Multiresolution Fourier Transform and its Application to Image and Audio Signal Analyis," IEEE Transactions on Information Theory, vol. 38, Mar. 1992.

....transformations of the signal. The purpose of this paper is to consider the potential of one particular form of representation to fulfil this role. The Multiresolution Fourier Transform (MFT) is a generalised form of wavelet transform designed for performing Fourier analysis at multiple scales [8]. It enables the symmetry and invariance properties of Fourier representations to be utilized at scales appropriate to the signal, a property which has been exploited in a number of applications [8] and as shown below, also recommend it for use when dealing with signals which are symmetric under ....

.... a generalised form of wavelet transform designed for performing Fourier analysis at multiple scales [8] It enables the symmetry and invariance properties of Fourier representations to be utilized at scales appropriate to the signal, a property which has been exploited in a number of applications [8], and as shown below, also recommend it for use when dealing with signals which are symmetric under the 2 D affine group of transformations. After defining an affine version of the MFT, an algorithm for estimating the affine coordinate transformation relating two image regions based on the ....

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R.Wilson, A.D.Calway and E.R.S.Pearson, "A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis", IEEE Trans. Information Theory, 38(2), pp. 674-690, 1992.

....called the Multiresolution Fourier Transform (MFT) an overcomplete image representation which combines the wavelet and windowed Fourier transforms in a unified framework. The resulting representation is an abstraction of the multiscale representation apparently used in the mammalian visual cortex [5, 6], which greatly simplifies the modelling of image features and the estimation of model parameters from global structure local structure Figure 1: Multiresolution doubly affine image model.Global structure is expressed by relations between neighbouring quadtree leaf nodes. Local model represents ....

....by a node. image data. This paper presents a summary of the basic properties of the MFT and the associated modelling techniques, which will illustrate its effectiveness in capturing the image information which is important to viewers. Fuller descriptions of the work can be found in the references [7, 6, 10]. 2 IMAGE MODELLING BASED ON THE MFT In order to model image features such as boundary contours and textures, two aspects of structure have to be captured: local structure relating the luminance at one pixel to that of its neighbours and global structure, which typically reflects the large scale ....

R. Wilson, A.D. Calway and E.R.S. Pearson,"A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis ",IEEE Trans. Inform. Theory, 38, pp. 674- 690, 1992.

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#36# R. Wilson, A. D. Calway, and E. R. S. Pearson. A generalized wavelet transform for fourier analysis: The multiresolution fourier transform and its apllication to image and audio signal analysis. IEEE Trans. Info. Theory, 38:674#

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