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		The Multiresolution Fourier Transform
		-------------------------------------

A generalized form of Wavelet Transform which enables adaptive
analysis of the `time-frequency' plane over multiple scales has been
developed by the Signal Processing Group at the University of Warwick,
England. The MFT has been applied successfully to segmentation
problems in both image and audio signal analysis. Details of these and
properties of the transform can be found in the following:


A Generalized Wavelet Transform for Fourier Analysis : the
Multiresolution Fourier Transform and its Application to Image and
Audio Signal Analysis

R. Wilson, A.D. Calway, and E.R.S. Pearson.

IEEE Trans. Information Theory, Special Issue on Wavelet Transforms
and Multiresolution Signal Analysis, vol. 38, 2, 1992.

Abstract

A wavelet transform specifically designed for Fourier analysis at
multiple scales is described and shown capable of providing a { em
local  representation which is particularly well suited to
segmentation problems. It is shown that by an appropriate choice of
analysis window and sampling intervals, it is possible to obtain a
Fourier representation which can be computed efficiently and overcomes
the limitations of using a fixed scale of window, yet by virtue of its
symmetry properties allows simple estimation of such fundamental
signal parameters as instantaneous frequency and onset time/position.
The transform is applied to the segmentation of both image and audio
signals, demonstrating its power to deal with signal events which are
localised in either time/space or frequency.  Feature extraction and
segmentation are tackled through the introduction of a class of
multiresolution Markov models, whose parameters represent the signal
events underlying the segmentation. In the case of images, this
provides a unified and computationally efficient approach to boundary
curve segmentation; in audio analysis, it provides an effective way of
note segmentation, giving accurate estimates of onset time and pitch
in polyphonic musical signals.


Additional information can also be found in:

R. Wilson and A.D. Calway, ``A general multiresolution signal
descriptor and its application to image analysis'', Proc.
EUSIPCO-88, pp. 663-666, Grenoble, 1988.

A. Calway and R. Wilson, ``A unified approach to feature extraction
based on an invertible linear transform'',  Proc.  3rd Int. Conf. 
Image process.   & its Appl. , pp.  651-655, 1989.

A.D. Calway,  The Multiresolution Fourier Transform: A General
Purpose Tool for Image Analysis , Ph.D.  Thesis, Warwick Univ., 1989.

A.D. Calway and R. Wilson, ``Curve extraction in images using the
multiresolution Fourier transform'',  Proc. IEEE ICASSP-90 , pp.
2129-2132, Albuquerque, 1990.

E.R.S. Pearson and R. Wilson, ``Musical event detection from audio
signals within a multiresolution framework'', Proc. ICMC-90,
Glasgow,1990.

E.R.S. Pearson, The Multiresolution Fourier Transform and its
Application to the Analysis of Polyphonic Music , Ph.D. Thesis,
Warwick Univ., 1991.

A.R. Davies and R. Wilson, ``Curve and corner extraction using the
multiresolution Fourier transform'',  Proc.  IEE Conf.  on Image
Process.  and its Appl. , Maastricht, Holland, 1992.

R. Wilson, A.D. Calway, E.R.S. Pearson and A.R. Davies. ``An
introduction to the multiresolution Fourier transform and its
applications'', Res.  Report RR204, Dept.  of Computer Science,
University of Warwick, 1992.

Tao-I Hsu, A.D. Calway and R.Wilson, ``Analysis of structured texture
using the multiresolution Fourier transform'', Res.  Report RR226,
Dept.  of Computer Science, University of Warwick, 1992.

A.D. Calway, H.Knutsson, and R. Wilson. ``Multiresolution estimation
of 2-d disparity using a frequency domain approach'', Proc.  British
Machine Vision Conf., Leeds, September 1992.

For copies of the above and further information contact:

Dr Roland Wilson  
Department of Computer Science  
University of Warwick  
Coventry CV4 7AL  
England

email: rgw@dcs.warwick.ac.uk