A General Description of Linear Time-Frequency ... - IEEE Xplore

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 1, JANUARY 2010

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A General Description of Linear Time-Frequency Transforms and Formulation of a Fast, Invertible Transform That Samples the Continuous S-Transform Spectrum Nonredundantly Robert A. Brown, M. Louis Lauzon, and Richard Frayne

Abstract—Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and Stransforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications. Index Terms—Biomedical image processing, efficient algorithm, nonstationary analysis, S-transform, time-frequency analysis.

I. INTRODUCTION

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HE ability to describe the frequency content of a signal is a critical capability in diverse fields of science. Transforms that accomplish this task by transforming signals of one or more Manuscript received July 28, 2008; accepted July 03, 2009. First published July 31, 2009; current version published December 16, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Haldun M. Ozaktas. This work was supported by the Canadian Institutes of Health Research, by the Canada Research Chairs Program, by the Natural Sciences and Engineering Research Council of Canada, and by the Informatics Circle of Research Excellence. R. A. Brown is with the Department of Electrical and Computer Engineering, University of Calgary and with the Seaman Family MR Research Centre, Hotchkiss Brain Institute, Foothills Medical Centre, Calgary, AB, T2N 2T9 Canada (e-mail: [email protected]). M. L. Lauzon is with the Department of Radiology and the Department of Clinical Neurosciences, University of Calgary, and with the Seaman Family MR Research Centre, Hotchkiss Brain Institute, Foothills Medical Centre, Calgary, AB, T2N 2T9 Canada (e-mail: [email protected]). R. Frayne is with the Department of Radiology, Department of Clinical Neurosciences, and Department of Electrical and Computer Engineering, University of Calgary, and with the Seaman Family MR Research Centre, Hotchkiss Brain Institute, Foothills Medical Centre, Calgary, AB, T2N 2T9 Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2009.2028972

dimensions into frequency or frequency-analogue spaces have proliferated [1]–[4]. While the original Fourier transform (FT) is an extremely important signal and image analysis tool, it assumes that a signal is stationary, i.e., that the frequency content is constant at all times in a signal, or at all locations in an image. Since most interesting signals are nonstationary, a series of techniques have been developed to characterize signals with dynamic frequency content. These methodologies are the foundation of the field of time-frequency analysis. A simple approach to the problem of nonstationary signal analysis is the short-time Fourier transform (STFT) [1]. In this technique changes in frequency over time are captured by using a window function to provide time localization. However, the choice of window function represents a compromise. Narrower windows provide better time resolution but poorer frequency resolution, while wider windows provide the converse. Ideally, the window width is chosen to produce the best representation of particular features of interest in the signal, but this requires a priori knowledge. The wavelet transform (WT) [2], which has been applied to a wide variety of signal processing problems, improves on the STFT by introducing the concept of progressive resolution. The WT provides the equivalent of finer time resolution at high frequencies and finer frequency resolution at low frequencies. However, the WT does not measure frequency but only an analogue, called scale. Additionally, the WT provides either no phase information, or phase measurements which are all relative to different local references. This is in contrast to the conventional concept of phase, as provided by the FT, where all phase measurements are relative to a global reference. The S-transform (ST) [5], [6] exhibits globally referenced phase and frequency measurements similar to those of the DFT and STFT, as well as the progressive resolution of the WT. This combination of desirable features has shown promise in a wide variety of applications, including detecting abnormalities in the heart [7], identifying genetic abnormalities in brain tumors [8], analyzing electroencephalograms [9], transmitting medical images [10], characterizing the behaviour of liquid crystals [11], detecting disturbances in electrical power distribution networks [12], monitoring high altitude wind patterns [13] and detecting gravitational waves [14].

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The ST exhibits two desirable properties, particularly for biomedical signal and image analysis applications. From wavelet research, transforms that incorporate progressive resolution are known to provide better, more consistent time-frequency representations across the entire spectrum [2] and the sinusoidal basis functions used by the ST complement the physical processes underlying many biological signals. However, while very efficient methods have been developed for calculating the FT and WT, the computational demands of the ST have limited its utility and prevented more widespread usage [15]. This paper presents the general Fourier-family transform (GFT), which describes linear transforms based on the complex sinusoid, including the FT, STFT and ST. Additionally, fast forward and inverse algorithms for calculating the discrete, nonredundant form of the GFT are derived that are much more computationally efficient than existing methods for calculating the ST [5], making computation more practical. This greatly improved computational efficiency will enhance existing techniques, and allows more advanced applications than have been investigated to date. These now-possible applications include the ability to analyze longer signals and larger images, as well as transform data with three or more dimensions, e.g., volumetric data from tomographic medical imaging techniques, or seismic data. While this paper considers only the one-dimensional transform, useful for signal processing, the general formulations can be extended to higher dimensional data without unexpected increases in computational complexity, allowing analysis of images, volumes, and temporal images (e.g., MR or CT perfusion datasets). The terminology used in the literature is inconsistent between the S-transform, wavelet and Fourier transforms. In this paper “continuous” will refer to transforms expressed as integrals, and “discrete” to formulations using summation. “One-to-one” and “nonredundant” will be used to describe discrete transforms that are minimally sampled: i.e., the transform of an -point signal produces an -point spectrum. This reduces the memory requirements of the algorithm and also implies that each unique signal is transformed into a unique spectrum, and vice versa. Finally, a “fast” transform is an algorithm for calculating a discrete, one-to-one transform that eliminates any remaining computational redundancies to achieve a reduction in computational complexity. This paper considers several transforms, each of which has several variants, using the above described taxonomy. The continuous Fourier transform is familiar from analytic time-frequency analysis. The discrete Fourier transform (DFT) is also one-to-one but the fast Fourier transform (FFT) is generally computational the preferred algorithm as it offers complexity for the DFT. complexity as compared to The short-time Fourier transform (STFT) has continuous and discrete formulations. The latter may be either one-to-one or not, and fast algorithms exist to calculate the STFT efficiently. Though there is a true continuous wavelet transform, the term is often used to describe a discrete approximation. Here, this approximation is referred to as the discrete wavelet transform (DWT), which is not one-to-one. What is commonly called the discrete wavelet transform is specifically termed the one-to-one discrete wavelet transform in this paper.

Finally, the S-transform has a number of variations. The original S-transform paper [5] describes a continuous time-frequency transform that uses Gaussian windows to provide time localization. The same paper introduces the discrete (but not one-to-one) ST (DST), and the “fast” ST, which is a method to calculate the discrete ST from the frequency domain. Although this method is more efficient than the time-domain version, it is still relatively slow when compared to other time-frequency transforms. Since the time-domain DST is very rarely applied in the literature, this paper will refer to the frequency domain version of the algorithm simply as the DST. The window function of the ST was subsequently parameterized, yielding the general S-transform (GST) [16]. Stockwell has recently published the discrete orthonomal S-transform (DOST) [17]. This is a discrete (though still not one-to-one) transform that uses sinc-shaped windows (boxcar functions in the frequency domain). Since Gaussian windows are not supported, the DOST is not strictly an S-transform. The GFT introduced in this paper is a general transform that encompasses the FT, STFT, ST, GST, and DOST. Furthermore, the fast discrete GFT algorithm developed here is a fast, one-toone, discrete transform. Due to the generality of the GFT, this algorithm represents an efficient means of calculating the STFT, ST, and GST, as well as being a faster method of computing the DOST. Besides unifying these transforms in one framework, the fast GFT algorithm greatly improves on the computational efficiency of the ST, GST, and DOST. A signal that would take days to transform [15] using the frequencydomain DST algorithm can be transformed using the fast GFT in less than a second. II. THE FOURIER FAMILY OF TRANSFORMS The FT describes a signal in terms of a series of complex sinusoids, with varying amplitude and phase. The sinusoidal basis functions of the FT are purely periodic and infinite in extent, and the FT converts entirely between the time and frequency domains, with no direct temporal information remaining after the transform. As noted in the introduction, this process is ideal for stationary signals, where the frequency content does not change over time. However, in many applications, nonstationary signals are common, and frequency changes over time or space are often not only present, but are in fact the very feature that we are interested in examining. To allow examination of nonstationary signals, a number of solutions have been proposed, including the STFT and, more recently, the ST. is defined in Stockwell et al. [5] The ST of a time signal as

(1)

where and are the transform time and frequency coordinates. This equation has the same form as that of the FT, but adds a normalized-area Gaussian window for time localization. It is important to note that the window also depends on , causing the width of the window to decrease with increasing frequency.

BROWN et al.: CONTINUOUS S-TRANSFORM SPECTRUM NONREDUNDANTLY

This automatically adjusts the ST window to provide a progressive tradeoff between time and frequency resolution for each frequency, with finer frequency resolution at low frequencies and better time resolution at high frequencies. Like the FT and STFT, the ST produces a complex spectrum that includes both frequency and globally referenced phase information. The ST is very similar in form to the Morlet wavelet [5], [18]. A mathematical comparison of the Gabor, Morlet wavelet and S-transforms can be found in Zhu et al. [18] but briefly, the ST basis functions are not a proper wavelet as they do not satisfy the condition of zero mean for admissible wavelets. Additionally, the basis functions of the ST, and more generally of the GFT, are not shifted and scaled versions of a mother function, and are not self-similar. Due to this, the GFT produces a globally referenced phase spectrum where all phase measurements are relative to the origin. Though complex wavelets exist, these produce locally referenced phase measurements where the reference point changes as the wavelet is shifted. The globally referenced phase of the GFT allows the Fourier spectrum to be directly recovered from any GFT, and facilitates the use of phase analysis techniques such as cross-spectral analysis. Of particular interest in this regard, magnetic resonance (MR) imaging directly measures the complex Fourier spectrum of the object being imaged. Many MR techniques, such as phase contrast MR [19], encode velocity information in the phase. Furthermore, many artefacts in MR imaging, such as those caused by motion and magnetic field nonuniformities, manifest themselves as changes in the phase. Since phase is critically important in this modality, many filtering, reconstruction and analysis techniques are based on the FT. These techniques could potentially be improved by using the ST, since it offers the capability to perform local analysis of nonstationary signals. The ST has been generalized [16] to include arbitrary windows

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addressed in the GST paper. Finally, the ST and GST are frequently compared to the STFT in the literature as alternatives, not as a superset. Due to these problems, we suggest that the “generalized” form of the GST that encompasses the STFT and other constant window-width transforms may be more clearly identified as the general Fourier family transform (GFT) and described mathematically as (3) where sigma is defined prior to computing the transform, and may be a constant value, or depend on either or both of and . This notation eliminates the potentially misleading presence (i.e., the windows are of as an explicit parameter. If , frequency adaptive), the GFT is a generalized ST. If a constant, the GFT describes a short time Fourier transform (STFT) with a constant window width . The FT itself can be understood in this context by setting the window function to a . Other interesting transforms may be constant, constructed by defining in terms of , causing the time-frequency resolution to vary over time, or by making dependent on in a piecewise fashion. As an example of the latter, when examining radio emission, windows with a constant width could be used within a particular band, while windows of a different width are used in adjacent bands. In the GFT formalism we label the complex sinusoid, , as the transform kernel, and , as the transform window. The product of the two forms the transform basis functions. In this context, the FT, STFT, ST, and GST are examples of the GFT, differing only by properties of their respective windows. B. Discrete Fourier-Family Transforms

(2) is a general window function with unit area where for all . The definition of the general S-transform in [16] includes a list of unspecified parameters that determine the shape of the window. In this paper, since these parameters are constants, the generality they provide is included in the definition of the . window function, and we write simply The original ST is a special case of this general S-transform, is a Gaussian function. where A. The General Fourier-Family Transform According to the definition of the general S-transform, the window function depends on the frequency . This parameter might be omitted (multiplied by zero) in the definition of the window function, allowing the GST definition to encompass constant window-width transforms such as the STFT. However, this eventuality is not recognized in the GST paper [16]. Furthermore, common forms of the STFT that do not evaluate windows centered at every sampled signal point may not be invertible via the usual ST and GST inversion method, an issue that is also not

Calculation of the discrete ST, as presented in Stockwell et al. [5], or of the discrete GST, presents a considerable computational challenge [15]. Neither is an orthogonal transform [5], and a considerable amount of redundant information is contained in their spectra. Calculated from the time domain according to (1), the DST of an -point signal generates an -point spectrum. Each of those points requires a numerical approximation of the integral in (1), which is a summation over the -point signal multiplied by the basis function. Therefore, the total computational complexity of this operation is and the resulting spectrum requires storage space. The DST can be calculated somewhat more efficiently by operating on the Fourier spectrum and taking advantage of the FFT aloperations gorithm [5]. This alternative requires and has the same memory requirements, [15]. In comparison, the DFT (and some discrete, one-to-one wavelet transforms) can be calculated in place, with the transform requiring no more space than the original signal. The one-dimensional , and one-to-one DFT has a computational complexity of the FFT improves on this, with a computational complexity of . Due to the high computational complexity of the ST and its memory requirements, transforms of longer signals and even

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small images have proven to be challenging to compute and store [15]. As an example, the DST of a 512 512 pixel image, which is a typical image size encountered in medical imaging applications, can currently take more than a day to compute on current workstations and requires 1 TB of storage (double precision complex floating point), compared to the FFT, which currently takes milliseconds and requires 4 MB of storage [15]. Though parallel and distributed computing techniques for calculating the ST on multiprocessor machines and computing clusters have been developed [15], computation of the DST for large signals or images is prohibitive, severely limiting potential applications of the ST. A more efficient fast, one-to-one DST algorithm may both allow (i) improved implementations of ST-based techniques that have shown promise [7]–[14] and (ii) application of the ST in many other areas that benefit from signal processing. As discussed in Section I, transform naming conventions are inconsistent between the Fourier, wavelet and ST literature. In Section I, a transform taxonomy was defined that describes each transform variant accurately and uniquely. However, the meaning of these terms, particularly “discrete,” as used in this paper, differ from their definitions in the literature. To avoid confusion, we note here that the DST and discrete GST may be more recognizable to those familiar with wavelets as the discrete approximations of the ST and GST, respectively, analogous to what is called the continuous wavelet transform in the literature. These three transforms are all characterized by redundant sampling of the time-frequency plane, without regard to the uncertainty principle. The existing DST algorithm is not the equivalent of the literature-defined discrete wavelet transform, which does not produce a redundant spectrum. In the following sections, we will develop the discrete one-to-one GFT, which samples the time-frequency plane efficiently, and is the analog of the discrete wavelet transform. Furthermore, we will present fast algorithms for calculating this transform and the inverse. Just as the discrete one-to-one wavelet transform cannot be easily interpolated to produce the spectrum generated by the discrete approximation of the continuous wavelet transform, the discrete one-to-one GFT spectrum is not directly comparable on a pixel-by-pixel basis with the original ST spectrum. In both the GFT and wavelet cases, the continuous approximation does not contain any extra information over the one-to-one representation, but the necessary interpolation algorithm to directly convert the latter into the former is prohibitively complex.

Fig. 1. A boxcar function (A) and its Fourier transform, a sinc function (B). As defined in (5), the -domain consists of shifted copies of the FT (C) multiplied by the FT of the window functions (D). Their product (E) can be Fourier transformed along the v axis to produce the S-domain (F). This example uses Gaussian windows that scale with frequency, and so the result (F) is the discrete S-transform.

where the axis is produced by taking the Fourier transform along the axis of the S-domain. The -domain is also related to the frequency domain by (5) and (derivation provided in the Appendix), where are the FTs of the signal and the window function along the axis, respectively. The S-domain is recoverable from the -domain via (6)

III. CHARACTERIZATION OF THE GENERALIZED FOURIER FAMILY TRANSFORM In addition to the familiar signal and frequency domains, in this paper we make use of two more domains: the “(generalized) , defined by (3), and the “ -domain,” defined S-domain,” as

(4)

Fig. 1 illustrates the relevant domains for an example signal, a boxcar function [Fig. 1(A)]. The FT of the boxcar is a sinc function [Fig. 1(B)]. The -domain is constructed according to (5) by constructing a matrix where each row is the FT of the signal, shifted by one sample relative to the row above [Fig. 1(C)]. Each row of this matrix is then multiplied by the appropriate window function [windows are shown in matrix form in Fig. 1(D)], resulting in the completed -domain [Fig. 1(E)]. Finally, according to (6), the S-domain is calculated by taking the FT of each row of the -domain [Fig. 1(F)]. In


this example, the windows are adaptive, that is, they change with frequency [Fig. 1(D)], so the result is an ST. Though it has not previously been presented as such, the frequency domain DST algorithm introduced in the original ST paper [5] is calculated via the -domain: the Fourier transform of the signal is shifted and multiplied by the Fourier transform of the window, then inverse Fourier transformed to build up the S-domain. Fig. 1 shows an example of the construction of the -domain for a simple signal, and the associated S-domain. result from an -point Since the DST produces an signal, it must contain redundant information. In the S-domain, that redundancy is illuminated by the Heisenberg uncertainty principle: the more finely energy is localized in the time domain, the more uncertain is its location in the frequency domain, and vice versa. This basic property is not reflected by the S-domain sampling scheme since it produces uniand frequency resolution at all points. Howform time ever, it is not entirely clear how to perform efficient sampling in the S-domain, particularly for arbitrary windows. Though the DOST [17] acknowledged this uncertainty and uses a dyadic sampling scheme, it still produces an S-domain with more than points. Furthermore, the DOST is restricted to windows that are a boxcar function in the frequency domain, or equivalently, a sinc function in the time domain, which has several undesirable properties [20]. S-domain is more clearly The redundancy of the -domain (Fig. 1). Since revealed in the corresponding each horizontal line (along ) at a particular value is a shifted , the version of the FT of the signal multiplied by full-frequency domain, and thus the signal, is recoverable from does not any one of these horizontal lines, provided have any zero values. Similarly, each vertical line at a particular -value is also a shifted version of windowed by the . From (4) and (5), it is clear that the full Fourier known spectrum may be obtained from any -line of the S-domain by Fourier transforming the line, dividing by , and shifting accordingly. Additionally, by the Fourier projection slice theorem [21], a projection through the S-domain along the -direction produces the line of the -domain, which is also a complete Fourier spectrum. This latter procedure is equivalent to the published inverse S-transform: summing over the axis and inverse Fourier transforming the result [5]. , the GFT is an STFT. Since the windows in the When STFT are fixed, efficiently sampling the time-frequency plane is more straightforward. Depending on the window design, a one-to-one STFT is easily realized, although for practical reasons some redundancy is often incorporated into the transform. Since one-to-one STFT algorithms are already well known, the , following discussion will focus on the fast GFT where that is, where the GFT is an ST. IV. THE FAST GENERAL FOURIER-FAMILY TRANSFORM The -domain representation and the Heisenberg uncertainty principle clearly suggest two optimizations that can be made to sampling scheme: (i) bandpass filtering along and the (ii) sampling along in a way that is related to the filtering along by the uncertainty principle. For a GFT where , i.e., an STFT, (i) involves sampling the frequency axis uniformly,

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Fig. 2. The -domain for the signal in Fig. 1 (A) is bandpassed in the region axis (B). Since this example boasts progressive resolution, around the v the resulting space can be sampled along the v axis using a dyadic scale (dark horizontal lines in C). This sampling scheme selects N points, which can be Fourier transformed to produce the N -point S-domain in (D). (D) has been interpolated using a nearest neighbor algorithm to form an N N point image that is comparable to that produced by the N N point DST in Fig. 1(F).

=0

2

2

but more coarsely than if the DFT of the entire signal were performed, and is a familiar concept in STFT practice. When , as in the ST, (ii) requires an octave sampling scheme, common in efficient wavelet transforms. The -domain is weighted by the Fourier transform of the window functions. The purpose of the window is to provide localization in the time domain but, via the uncertainty principle, it also controls the degree of localization in the frequency domain. To provide localization in the frequency domain, the window must act as a band-pass filter, i.e., frequencies at a distance from the window centre will be attenuated. This is illustrated in Fig. 1(D) for the Gaussian windows of the ST. A degree of optimization can be achieved by discarding, i.e., multiplying by a boxcar, the portions of each -line that are sufficiently attenuated by the window [Fig. 2(B)]. In transforms where (the width of the window) varies, the extent of the remaining -segment will vary to match the band-pass characteristics of the window [Fig. 2(B)]. Further optimization is obtained by sampling along the axis as required by the uncertainty principle [Fig. 2(C)]. For transforms with variable , the required sam, these two optipling interval along will also vary. If mizations produce a dyadic sampling scheme [Fig. 2(C)]. The result can be Fourier transformed to produce an -point S-domain [Fig. 2(D)]. By reversing the shift applied to the frequency domain as part of the construction of the -domain (5), the relationship of this sampling scheme to the Fourier transform of the signal becomes clear [Fig. 3(B)]. Projecting the unshifted -domain onto the axis [Fig. 3(C)] shows that the operations we have former performed on the -domain have effectively windowed and partitioned the frequency domain. Obtaining the S-domain involves

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Fig. 3. The -domain consists of shifted versions of the FT of the signal. (A) shows the regions sampled by the fast GFT (dark lines). If this shifting is undone (B), the sampling scheme’s complete coverage of the frequency domain becomes more apparent. In (C), the corresponding sampled regions (dotted gray lines) and accompanying windows (solid gray lines) are shown superimposed on the FT of the signal (solid black line).

individually inverse Fourier transforming each partition, each of which provides one line of the S-domain. If the transform is to be invertible, at least one window portion must be nonzero at the location of each Fourier coefficient. If the transform is to be one-to-one, this requirement is strengthened: the windows must form an orthogonal set. The simplest example of this is where exactly one window is nonzero at the location of each Fourier coefficient. That is, the windows must not have gaps between them, nor overlap. For transforms such as the DOST or some forms of STFT, where the windows are boxcar functions, this is a straightforward requirement, but the Gaussian windows of the ST cannot be used to construct an orthogonal set. However, as with the DFT, in the discrete form the signal and its Fourier spectrum are finite: both can be regarded as an infinite signal or spectrum, multiplied by a boxcar. This implies that the ST windows are also multiplied by a boxcar. In the discrete non one-to-one ST, this boxcar is always the same width as the full spectrum, so the relationship between the bandwidth of the boxcar and that of the Gaussian window varies. An equally valid choice is to fix this relationship, so the boxcar width varies along with the Gaussian. This combination can be used to form a valid window set, and is shown in Fig. 3(C). This choice is consistent with the wavelet transform and may produce a spectrum with more consistent characteristics: from a bandwidth perspective, low frequency windows will be truncated by the same amount as those at higher frequencies so unavoidable artifacts, such as ringing and edge effects, are expected to also be present at the same level throughout the spectrum.


Fig. 3(C) shows an example of this window construction process. Gaussian windows (gray lines), are truncated using boxcar functions (dotted lines). Each resulting truncated window is then multiplied by the Fourier spectrum (black line), producing a filtered spectrum that is nonzero only where the boxcar function is nonzero. This nonzero section is then Fourier transformed to produce a line of the S-domain. Calculation of a one-to-one, discrete, fast GFT from the frequency domain is described in Algorithm 1. Algorithm 1 has (Appendix) and a computational complexity of transforms an -point signal into an -point S-domain. This performance is a great improvement over (i) the time domain and produces an -point S-domain DST, which is [15]; (ii) the frequency domain DST, which is and also produces an -point S-domain [15]; and (iii) the DOST, which is and produces a -point S-domain, . The computational complexity of the fast where GFT is of the same order as the Fast Fourier Transform, which and produces an -point result. Since is also the GFT formalism describes the STFT, GST, and DOST, the fast GFT algorithm provides a computationally and memory efficient method for calculating all three transforms. The fast GFT algorithm presented here is the minimally sampled, one-to-one transform. This configuration is efficient to compute and store, but some applications may benefit from a less optimal implementation. If the frequency domain windows are allowed to overlap, introducing some redundancy through oversampling of the axis, a smoother window shape may be achieved. Alternately, windows may be positioned closer together, resulting in oversampling along the axis. This redundancy comes at the cost of some computation time and additional storage requirements. For example, if each window is allowed to overlap its neighbors by 50%, each FFT in Algorithm 1 (step 5) will require transforming twice as many points, increasing the computation time and producing an S-domain with points. However, for constant overlap factors, such as in 2 this example, the computational complexity of the algorithm re. mains Actual computation times for the frequency domain DST, the fast GFT (Algorithm 1), and an optimized FFT library are compared in Fig. 4. Due to memory constraints, the largest signal that could be calculated with the DST algorithm had points. While the DST of this signal took over 30 s, the fast GFT and the FFT each took only a few milliseconds: 5 and 2 ms, respectively (complex 64-bit floating point math, 2.5 GHz Intel Core 2 Duo, using one core only, 4 GB RAM). Fig. 4, plotted computational comon a log-scale, also shows the plexity of the FFT and GFT, as compared to the performance of the DST. The efficiency of the fast GFT allows computation of the transform for larger signals and images. For example, Fig. 5(A) shows the first 8 s of an electrocardiogram (ECG) recording from a publicly available subset of the European ST-T Database (Physionet, http://physionet.org). In this case, ST-T refers to ST- and T- waves, which are features of an ECG and are not related to the S-transform. This sample is from the V4 lead of record e0103, a 62-year-old male subject complaining of mixed angina. The complete recording contains over one mil-


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fast GFT with adaptive Gaussian windows, and Fig. 5(C) is the GFT spectrum bilinear interpolated to present a smoother appearance. In this configuration the GFT is a fast ST. The transform took 0.77 s to calculate (complex 64-bit floating point math, 2.5-GHz Intel Core 2 Duo, using one core only, 4 GB RAM). The classic frequency domain S-transform algorithm, if it could be calculated entirely in memory, would take over three spectrum requires 16 TB of days. However, since the storage space (64-bit complex data), swapping to an array of hard drives would greatly increase the actual computation time. V. THE INVERSE FAST GENERAL FOURIER-FAMILY TRANSFORM Fig. 4. Benchmarks comparing the computation speed of the FFT (dotted black line), DST (solid black line) and fast GFT (gray line) for one-dimensional signals of various lengths. The DST and fast GFT were implemented in the interpreted language Python (Python Software Foundation; Hampton, New Hampshire), but both routines called a compiled FFT library (vecLib, Apple; Cupertino, California), which was also used to produce the FFT benchmark data. The FFT and fast GFT computation times are both ( log ) and thus show similar growth rates, while the DST computation time grows more quickly. The 2 points due to memory limitations. DST could not be calculated for

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Fig. 5. The computation speed of the fast GFT can be used to perform analysis of signals or images that are too large to transform with the DST. (A) shows the first 8 s of an electrocardiogram recording, sampled at 250 Hz. (B) is the corresponding portion of the S-transform, nearest neighbor interpolated to form an image. (C) is the same spectrum with bilinear interpolation. The negative frequencies and frequencies above 62.5 Hz are not shown. The full signal has 2 points and it took 0.77 s to transform with the fast GFT. The DST of this signal would take over three days to compute, if it could be performed entirely in workstation memory.

lion points. Fig. 5(B) is a portion of the spectrum produced by points from this signal, using the a transform of the first

The forward continuous GFT is calculated via the -domain by shifting and windowing the Fourier transform of the signal, according to (5), then inverse Fourier transforming each horizontal -line to produce the S-domain, according to (6). The inverse continuous GFT reverses these steps, using the FT of each horizontal -line in the S-domain to recover the -domain, according to (4). As discussed in the previous section, the signal can then be obtained from any horizontal line in the -domain by dividing by , shifting, and taking the inverse FT. Alternately, the complete frequency domain can be reconstructed directly from the S-domain by summing over . The fast GFT avoids redundancy by sampling in accordance with the uncertainty principle. Since the -domain is sampled using a total of -points, the corresponding S-domain will also be sampled with a total of -points and the inverse fast GFT points to reconstruct the signal. The FFT of will require all each horizontal line of the S-domain recovers a line segment it in the -domain. When each segment is divided by represents an unweighted portion of the frequency domain. As illustrated in Fig. 3, when shifted by , these segments combine to produce a full frequency domain, which can be inverse fast Fourier transformed to recover the signal, without loss. Algorithm 2 demonstrates the calculation of the inverse fast GFT in-place, making it optimally memory efficient. The fast GFT inversion process may be compared to inverting the discrete ST. The discrete ST produces an S-domain conpoints. Since this S-domain contains reduntaining dant information, several different inversion methods may be used. As originally proposed [5], the original signal is recoverable via a projection of the S-domain onto the axis. The signal can also be recovered through the FT of any one horizontal line. Since the discrete ST is not a one-to-one transform, different S-domains can produce the same signal. As noted by others, filtering the S-domain and inverting can produce a signal that is unchanged by the filtering process [22]. Using square windows, the result of transforming a signal using Algorithm 1 and inverting the transform according to Algorithm 2, differs from the original signal no more than would be expected from machine precision round-off errors during the FFTs performed as part of the algorithms. Larger errors are observed when using nonsquare windows, but this source of error is entirely due to applying and removing the windows in the frequency domain, and is no more pronounced than would be observed by performing the same windowing and unwindowing operation without the other steps of the GFT.

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VI. CONCLUSION In this paper, we have demonstrated a general time-frequency transform that encompasses the Fourier transform, the short-time Fourier transform, the S-transform, the generalized S-transform and the discrete orthonormal S-transform. We have also described an efficient method for calculating this general transform that has the same computational complexity as the fast Fourier transform and can be computed in-place. The general Fourier family transform has several advantages over other time-frequency analysis techniques. Unlike wavelet transforms, it provides globally referenced phase and DFT-like frequency measurements, which are particularly important for imaging modalities where phase consistency is required. While the DOST explicitly requires the use of windows that are boxcars in the frequency domain, the GFT allows the use of arbitrary windows. Finally, the fast GFT (Algorithm 1) provides an efficient means of calculating the discrete one-to-one GFT that is faster than the S-transform algorithms and the DOST, is memory efficient, transforming an -point signal to an -point spectrum, and also provides an accurate representation of the true frequency and time resolution at each point in the spectrum. The GFT is directly applicable to a number of applications that have been developed using the ST, and can likely be used in place of the wavelet transform in many other areas. Just as the discrete approximation of the continuous wavelet transform is still frequently used for signal analysis, there are likely to be many situations where the smoother representation produced by the original ST is preferred, despite its higher computational requirements. However, the high computational complexity of the ST has made its use impractical in many applications. For these time- or memory- sensitive applications, or those involving signals that are too large to transform with the original ST, the fast GFT algorithm can be calculated much more quickly, almost as quickly as the FFT, allowing sophisticated time-frequency analysis techniques to be more widely deployed. APPENDIX A. Derivation of (5) The -domain is defined in (4) as the Fourier transform of along the axis. That is

where indicates the 1–D Fourier transform over the variand are the Fourier transforms of the able and , and the window function , respectively. signal B. Computational Complexity of Algorithm 1 For a signal of points, Algorithm 1 consists of an point point multiply (step 4), and FFT (step 1), a shift (step 3), an an point inverse FFT (step 5). Steps 3 through 5 are performed twice, once for the positive frequencies and once for the negative frequencies (step 6). Although Algorithm 1 shows the window functions calculated inside the loop (step 2) for brevity, they may be precalculated. Of the steps in the loop, the one with the highest order is points, which can be accomplished in step 5, an FFT of . For a dyadic sampling scheme, as in Algorithm takes values in , but all 1, . If we one-to-one GFT sampling schemes have , where is the fraction of points, i.e., , let then . Using this relation we can write the order of step 5 as . Substituting we have

From above,

, so this simplifies to . Since . , which implies , the In fact, it is zero only if we have trivial case corresponding to the FFT. For so . Since the complexity of step 5 is less than log , the complexity of the loop is dominated by the -point FFT in step 1, . which gives the order of the algorithm: Algorithm 1: The in-place fast general Fourier family is an array of points. The subscript transform. on indicates its domain: is the time domain, is the frequency domain and represents the -domain samples along the axis. In this example, a dyadic frequency sampling scheme is used, but this could be adapted, as in the STFT for example. LET be an array of samples representing a signal 1. Calculate the FFT of the signal:

FOR IF LET ELSE:

in :

DO:


LET LET LET

4. Remove the window: and

2. Calculate the window function and FFT:

3. Shift the window function to be centered on the frequency band:

4. Apply the window to the frequency band:

Note: The result of this step, in the S-domain. 6. Repeat steps 3 to 5 substituting to calculate the negative frequencies. END FOR

, maps to and

Algorithm 2: The inverse in-place fast general Fourier family transform, using a dyadic sampling scheme for this example be an array of LET as produced by Algorithm 1

samples of the S-domain

in :

5. Repeat steps 2 to 4 substituting and to calculate the negative frequencies. END FOR 6. Recover the signal through the inverse FFT:

REFERENCES

5. Calculate the in-place inverse FFT of the frequency band:

FOR IF LET ELSE: LET LET LET

289

DO:

and

1. Calculate the window function and FFT:

2. Calculate the in-place FFT of the S-domain band:

3. Shift the window function to be centered on the frequency band:

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Robert A. Brown received the B.Sc. degree in computer science in 2001 and the Ph.D. degree in biomedical engineering from the University of Calgary, Calgary, AB, Canada, in 2008. He recently began a postdoctoral fellowship in the Magnetic Resonance Spectroscopy Unit at the McConnell Brain Imaging Centre of the Montreal Neurological Institute, QC, Canada. His research interests include automated and computer assisted medical image and signal analysis, high performance scientific computing, and advanced magnetic resonance (MR) image acquisition, reconstruction, and postprocessing techniques. Current specific interests include noninvasive brain tumor classification from MR imaging, longitudinal demyelination and remyelination patterns in multiple sclerosis, and practical applications of compressed sensing techniques in clinical imaging.

Richard Frayne received the B.A.Sc. degree in electrical engineering) from the University of Waterloo in 1989 and the Ph.D. degree in Medical biophysic from the University of Western Ontario in 1994. He is a Professor (with tenure) in the Departments of Radiology and Clinical Neuroscience and a member of the Hotchkiss Brain Institute at the University of Calgary, AB, Canada. He is also a Scientist at the Seaman Family MR Centre, Foothills Medical Centre, Alberta Health Sciences. In 2003 he was appointed a Canada Research Chair in Image Science. This Chair was renewed in 2008. He also holds a Senior Medical Scholar salary-support award from 2005 to 2010 from the Alberta Heritage Foundation for Medical Research (AHFMR). His research interests are in the development and application of new magnetic resonance imaging techniques in humans for the study, detection, and treatment of vascular disease. Current specific interests include imaging for stroke prevention and in acute stroke; angiography in the lower limbs, and MR-based endovascular therapy. He is also interested in advanced image reconstruction and signal processing and in the clinical vascular applications of molecular imaging and 3D ultrasound. Previously, at the University of Wisconsin (1994–1999), he helped develop time-resolved contrast-enhanced MR angiography. He has over 15 years’ experience in pursuing imaging research in a clinical research environment. He has over 95 published or in-press peer-reviewed publications, over 65 invited talks and over 300 scholarly presentations. He is very active in the training of high-quality people and has trained more than 30 undergraduate students, nine medical students, ten M.Sc., seven Ph.D., and five postdoctoral fellows. The AHFMR, Canadian Foundation for Innovation, Canadian Institutes for Health Research, Heart and Stroke Foundation of Canada, The US National Institutes for Health, the National Science and Engineering Research Council of Canada and the Whitaker Foundation, currently fund or have funded his research.

M. Louis Lauzon received the B.Sc. degree in physics from the University of Waterloo in 1991 and the Medical Biophysics M.Sc. and Ph.D. degrees from the University of Western Ontario in 1993 and 1998, respectively. He is an Assistant Research Professor in the Departments of Radiology and Clinical Neuroscience and a member of the Hotchkiss Brain Institute at the University of Calgary, AB, Canada. He is also a Scientist at the Seaman Family MR Centre, Foothills Medical Centre, Alberta Health Sciences. His research interests are in the development and application of new magnetic resonance (MR) imaging techniques and advanced signal/image processing for the study, detection, and treatment of vascular and hepatobiliary diseases. Current specific interests include body MR imaging, MR angiography in the lower limbs, fast MR imaging, time-frequency analysis, and compressed sensing.