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The Chirplet Transform:
A Generalization of Gabor's Logon Transform
Steve Mann and Simon Haykin
Communications Research Laboratory, McMaster University, Hamilton Ontario, L8S 4K1
email: manns@McMaster. CA
Abstract We propose a novel transform, an expansion of an arbitrary function onto a basis of multiscale chirps (swept frequency wave packets). We apply this new transform to a practical problem in marine radar: the detection of floating objects by their "acceleration signature" (the "chirpyness" of their radar backscatter), and obtain results far better than those previously obtained by other current Doppler radar methods. Each of the chirplets essentially models the underlying physics of motion of a floating object. Because it so closely captures the essence of the physical phenomena, the transform is near optimal for the problem of detecting floating objects. Gabor's "Logon" Paradigm ^{1}Although the logon representation is not a "frame" [1], and is therefore not numerically stable, we proceed in the development of this line of thought, and will later address the issue of numerical stability by overrepresenting the expansion. 
envelope. This notion is known formally in the physics literature as the WeylHiesenberg group. (The "sliding window Fourier transform" belongs to the WeylHiesenberg group since we can think of the bases as being modulated versions of one parent window.) Gabor emphasised the use of a Gaussian window, since it minimises the uncertainty product tf (provided that these support measures are quantified in terms of the root mean square deviations from their mean epochs[2]). The timefrequency logon diagrams, depicted in figures 1 and 2 show how the Gabor function bases cover the timefrequency space, and how we can trade frequency resolution for improved temporal resolution. Gabor originally used rectangles to designate each of these elementary signals. If each of these rectangles were a pixel, and its brightness was set in accordance with the appropriate coefficient in the signal expansion (for example by replacing the rectangle with a dither pattern), the logon diagram would be a density plot (image) of the TF distribution. 
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1.1 The wavelet transform within the logon paradigm 2 "Wavelets" and Wavelets 
3 Chirps 4 The Chirplet at a Single Scale ^{2} We have greatly simplified our analysis here: what we really want is an approximation to an ellipse in TF space; the bases are actually found by optimization techniques, based on certain constraints. 
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but with one small modification. Since we desire unit L2 norm, we instead let 
it easier to form an orthonormal set, but here we have deliberately introduced a specific form of asymmetry and will exploit this structure in what follows.
4.1The Nyquist boundary problem 
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down to 1/2). This chirp will lie on the b = 0 axis, as far to the right as possible (or as far to the left as possible). But then a chirp which has the same value of a, but a nonzero intercept (for example one going from fractional frequency 1/4 to 3/4), will violate the Nyquist limit and give rise to aliasing in frequency space. 5 MultiScale Chirping "Wavelets" (GLT basis functions) 
(zero acceleration) in terms of Doppler. Our previously described "bowtie" space is also a special case of this generalization, where the scale is fixed constant.
If we are only looking at the magnitude or power distribution of the GLT, we can ignore the last parameter because all the GLT bases are complex and lie in the Hardy space (their imaginary parts are equal to the Hilbert transforms of their real parts). Because these functions are analytic, the contribution due to one of the bases is very insensitive to the relative phase. Chirps correspond to slanted logons which can fill the timefrequency space completely. We can expand any arbitrary signal onto a basis of upchirps, downchirps, or a mixed basis of upchirps, downchirps, and ordinary Gabor functions which all show up as nearly elliptical contours (having equal areas) in timefrequency space^{3 } but with different slants (orientations) and aspect ratios. By tilting the logons, we can simultaneously increase the support in both domains, without losing optimality in cojoint resolution, along a tilted set of axes ("s/" for slant), where t_{sl}f_{sl} is still 1/2. 6 Application of the GLT to Detection of Floating Objects in Ocean Based Radar ^{3}The approach of Slepian[4], who used functions which are truly rectangular in TF space, can be extended to chirps. Our "Slepian chirplet" then represents an ideal parallelogram in TF space. We may then apply Thomson's method of multiple windows[5] to our chirplet paradigm. 
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6.1 Fourier based Doppler processing 6.2 Underlying physics of motion 6.3 Projections of the chirplet as indicators of Doppler evolution of floating objects 6.3.1The singlescale chirplet snapshot revisited ^{4}The well known Pierson Moskowitz (PM) spectrum [8] describes sea surface heights as a function of time and position. The temporal PM spectrum is very pesky, indicating strong periodicity A lowpass filtered (due to growler inertia) version of the PM spectrum would describe the spectral characteristics of a time series formed by the height of the floating object. 
floating object, (compare figures 9 and 10), but in an even more pronounced way. Furthermore, we have an explicit measure of the target's "chirpyness". In other words, we can see the acceleration signature of the target, and use this information to fit to a physical model. Such physical constraints are useful for verification and also for target tracking. We will refer to this particular choice of two variable parameters as a singlescale chirplet snapshot. The term singlescale refers to the fact that all the bases have the same duration, or physical support (in this case one second). The word snapshot refers to the fact that we are glancing at the target at a particular "instant" (over a short interval of time) and are not tracking the temporal evolution of the acceleration signature.

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the chirplet) of the original time series. In figure 11 (a) we show this slice. If, however, all we want is one free rameter, we are far better off to take the slice through the "bowtie" center. In figure 11 (b) we see that the "Average Instantaneous Frequency" spectrum (AIF spectrum) is much sharper than the Fourier spectrum. 7 Classification using the GLT 7.1 GLT snapshot features

distribution to the data Although the logon is no longer Gaussian in the GLT space, the Gaussian fit allows us to quantify the spread by the determinant of the covariance matrix, S. These values correspond to the secondorder moments of the distribution.
7.2 Classification Results 
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We have used Fishers Linear Discriminant (FLD)[10] in preference to Principal Components Analysis (PCA) which only looks at the variance of the features, and requires scaling in accordance with an assumed a priori knowledge of the feature importances. 7.3 The NeymanPearson Paradigm 7.4 Weighted kNearest Neighbours 
from the corresponding exemplars. If, for example, the distance from the input feature vector to the "winner" was just a bit less than the distance to the "runner up", the first and second weights would be nearly equal. If, on the other hand, the winner was much closer than all the others, its class would be weighted very highly compared to the classes of all the others. 8 Skeletonisation and Time Evolution of the GLT Snapshot
evolution of the GLT snapshot itself. The images in figures 9 and 10 were just "snapshots" of the GLT evaluated at a fixed temporal center. If we move the center epoch of the bases through the data, we can see, basically, a single prominent elementary chirp, moving around on an elliptical locus. (We developed software to display a "movie" or animation of the successive GLT snapshots in sequence on the computer screen. 64 of these snapshots appear in figure 12.) We have coined the term "hypermatrix" for this structure, which we say has 64 "pages", and the rows and 
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columns of each page make up the image. In figure 13 the loci of this movement are shown, for two different sets of 64 snapshots, projected onto the temporal center axis. (By projection, we mean the average, summing over all pages of the hypermatrix, to reduce three "index dimensions" to two.)
If we visualise this path in three dimensions, we have a helix. A slice through this hypermatrix, along the plane f_{beg} = f_{end} results in the timefrequency squiggle we saw in figure 7. 9 Current Research 
process on the screen as either constant area ellipses moving and twisting to fit the TF distribution, or as bowties moving around in the chirplet space. References [2] D. Gabor. Theory of communication. J. Inst. Elec. Eng., 93:429456, 1946. [3] Steve Mann and Simon Haykin. An Adaptive Wavelet Like Transform. SPIE, 36th Annual International Symposium on Optical and Optoelectronic Applied Science and Engineering, 2126 July 1991. [4] D. Slepian, H. Pollack, and H. Landau. Prolate spheroidal wave functions, fourier analysis and uncertainty. Bell System Technical Journal, 1961. [5] D. Thomson. Spectrum estimation and harmonic analysis. Proc IEEE, Sept. 1982. [6] B.W. Currie, S. Haykin, and C Krasnor. Timevarying spectra for dual polarized radar returns from targets in an ocean environment. IEEE Conference proceedings RADAR90, Washington D.C., May 1990. [7] S Haykin, C Krasnor, Tim Nohara, B Currie, and D Hamburger. Coherent dual polarized radar studies of an ocean environment. IEEE Transactions on Geosciences and Remote Sensing, Feb 1991. [8] Gary Mastin, Peter Watterberg, and John Mareda. Fourier synthesis of ocean scenes. IEEE CG&A, 1987. [9] Michael Kass, Andrew Witkin, and Demetri Terzopoulos. Snakes: Active Contour Models . Academic Publishers, 1987. [10] Richard Duda and Peter Hart. Pattern Recognition and Scene Analysis. John Wiley and Sons, 1973. [11] Stephane Mallat. Compact image representation from multiscale edges. Submitted to 3rd International Conference on Computer Vision, 1990. [12] Tomaso Poggio and Federico Girosi. Networks for approximation and learning. Proc. IEEE, 1990. Sept. [13] D Lowe. Joint representations in quantum mechanics and signal processing theory: why a probability function of time and frequency is disallowed. Royal Signals and Radar Establishment, 1986. report 4017. 
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