LOCAL MULTISCALE FREQUENCY AND BANDWIDTH ESTIMATION Hans Knutsson Carl-Fredri Westin Gösta Granlund Department of Electrical Engineering, Computer Vision Laboratory Linöping University, S-58 83 Linöping, Sweden email: nutte@isy, westin@isy.liu.se, gosta@isy.liu.se Abstract This paper describes a robust algorithm for estimation of local signal frequency and bandwidth. The method is based on combining local estimates of instantaneous frequency over a large number of scales. The filters used are a set of lognormal quadrature wavelets. A novel feature is that an estimate of local frequency bandwidth can be obtained. The bandwidth can be used to produce a measure of certainty for the estimated frequency. The algorithm is applicable to multidimensional data and examples of the performance of the method are demonstrated for one-dimensional and two-dimensional signals. Introduction The concept of frequency is a mathematically well defined entity. Any stationary signal can be represented as a weighted sum of sine and cosine functions having particular amplitudes, phases and frequencies. For nonstationary signals, however, this is not the case and a description in term of sines and cosines impossible. This fact has, since most real life signals are non stationary, led to a number of attempts to find methods which would allow non stationary signals to be analysed in a frequency-lie manner. Instantaneous frequency One such attempt was the introduction of the notion of instantaneous frequency. Carson and Fry [CF37] and Van der Pol [vdp46] defined instantaneous frequency and applied it to frequency modulated signals. The instantaneous frequency is commonly defined as the rate of change in phase of the analytical signal. The analytic signal is a complex signal uniquely defined by a real signal as: f A = f if Hi where f Hi denotes the Hilbert transform of f. A historical review of the concept of instantaneous frequency can be found in [Boa9a, Boa9b]. It is noteworthy that - despite it s name -, instantaneous frequency is a global entity in that local alterations of the signal will effect the instantaneous frequency everywhere, due to the infinite ernel associated with the Hilbert transform. Local Fourier transforms A perhaps more common approach is to design methods to obtain local frequency estimates. In 946 Gabor [Gab46] proposed a combined representation of time and frequency. This representation is a Fourier transform modified by inserting a Gaussian window in order to better describe local signal properties. His motivation for choosing modulated Gaussian functions were that they are maximally compact in time and frequency. This new transform, commonly referred to as the Gabor transform, is in one dimension defined by e x x σ fx e iux dx, where u is the frequency variable. The complex modulated Gaussian basis functions are frequently termed Gabor filters, g u,σ x = e iux e x σ 3 The main criticism of the Gabor transform is that although Gabor functions are well localized they have infinite support - forcing truncation in all practical implementations. A natural extension to Gabors modification is the insertion of a general window function instead of the Gaussian one. This defines a windowed Fourier transform, Gx, u = wx x fx e iux dx 4 As above, this transform can be seen as a transform having different complex modulated windows as basis functions. The spectrogram provides a common example of windowed Fourier analysis resulting in a combined space-frequency representation. This representation has long been used for analyzing D-dimensional time varying signals, see e.g. [AR77]. The wavelet concept has over the last few years gained considerable attention. A wavelet basis is produced by translation and dilatations of a mother wavelet. By using dilation of the wavelet for controlling its frequency characteristic, a constant relative bandwidth is obtained. The use of such filter sets, based on Gabor filters, was first suggested to the computer
vision community by Granlund in 978[Gra78]. A feature of constant relative bandwidth filters is that the spatial localization becomes proportional to the local signal wavelength. It is also appropriate to note that constant bandwidth filters are consistent with current multi frequency channel theories in psychophysics. A transform based on a set of scaled Gabor functions was introduced by Morlet in 988. The set of wavelets used for frequency estimation in this paper can be described as Gabor functions on a logarithmic scale and was introduced by Knutsson in 98 [Knu8]. A lognorm space-frequency representation In this paper local frequency can be interpreted in terms of a single frequency as well as in terms of local spectrum. The notion of instantaneous frequency is used for narrowband analysis over a number of scales. The estimates are weighted and summed to produce a wide range frequency estimate.. The lognormal filter function The lognormal filter Qu is a spherically separable quadrature filter with a radial frequency function which is Gaussian on a logarithmic scale. and Qu = R i ρd û 5 R i ρ = e CBln ρ/ 6 where: C B = 4 B ln ρ = u is the norm of the frequency vector is the centre frequency B is the 6 db relative bandwidth in octaves, i.e. B is given by B = ln ρu ln 7 ρ l with ρ l and ρ u being the ρ values for which R i ρ =.5. The directional function of the quadrature filters have the following form: { D û =û ˆn if u ˆn > 8 D û = otherwise where ˆn is the filter directing vector. i.e. Dû varies as cos ϕ, where ϕ is the angle difference between u and the filter direction ˆn. For simple signals, having an orientation given by ˆx, the output magnitude from a quadrature filter in direction is given by = d i ˆx ˆn, 9 where d i is independent of the filter orientation and will be determined by the radial distribution of the signal spectrum Gρ and the radial filter function R i ρ. An isotropic estimate of signal strength which is simultaneously local in the spatial and the frequency domain can be obtained by summing the magnitude of the outputs from a number of quadrature filters in different orientations. = = d i ˆx ˆn Expanding ˆx in the filter orienting vectors ˆn yields = d i ˆn x lˆn l = d i x lˆn l ˆn, l l where x l are the coefficients of ˆx. If the filter orienting vectors ˆn constitute an orthonormal basis of the same dimensionality as ˆx, Equation reduces to = d i ˆx = d i This shows that an isotropic estimate of signal strength, that is local both spatially and in the frequency domain, is obtained by summing the magnitudes of the outputs of orthogonally oriented quadrature filters. The number of filters required is the same as the dimensionality of the signal.. Frequency estimation using lognormal quadrature filters By combining the outputs from two or more sets of filters which differ only in centre frequency, it is possible to produce a local frequency estimate. For simple single frequency neighbourhoods, the contribution in the Fourier domain will be concentrated at a point at a distance ρ from the origin. The isotropic magnitude of a sum of filters is then given by Equation 6, = d i = Ae CBln ρ/ 3 where A is the local signal amplitude. The ratio between the outputs from two filters differing only in their centre frequencies and ρ j is ρ/ρ j q j = e CBln 4 e CBln ρ/ Simplification of this equation leads to a simple power relation [Knu8], q j ρ CBln ρ j/ = 5 ρi ρ j
It is convenient at this point to introduce two constants. The first, α, depends on the relative bandwidth and the ratio of the centre frequencies of the filters and the second j, is the geometric mean of the two centre frequencies. [ ρj ] α = C B ln 6 j = ρ j 7 The ratio between to lognormal filter outputs can then be written α q j ρ =. 8 j Solving for ρ yields: α 9 qj ρ = j A particularly simple situation occurs if α =. Using Equation 6 then gives the following relation between the ratio of the filter frequencies and the relative bandwidth ρ j = B /8 A centre frequency ratio of is a natural choice and has been shown to wor well, for α to be one, this choice requires a relative filter bandwidth B of. π 3π/4 π/ π/4 3π/4 ρ R 4ρ 3.5 R 3ρ ρ π/4 π/ 3π/4 π 3.5 π π/ π/4 ρ R 7ρ 6.5 R 6ρ π/4 ρ π/ 3π/4 π 6.5 Figure : The local radial frequency ρ can be estimated as the ratio between two lognormal functions times +.5, the geometric mean of the two centre frequencies, if α =, Equation 9..3 Narrow band instantaneous frequency It is not hard to show that the magnitude of the ratio between two lognormal quadrature filters can be interpreted as a narrow band estimation of instantaneous frequency. To start, note that a quadrature filter suppresses all negative frequencies and produces a filtered analytic signal where i indicates filter center frequency. The instantaneous frequency ω i for is given by: ω i = x arg This can be rewritten as: x ω i =Im[ ] It is well nown that differentiation in the spatial domain corresponds to multiplication by i and the frequency variable in the Fourier domain, i e: x iuf Ai 3 where F Ai is the Fourier transform of. Then, for α =, i+ = x 4 and the instantaneous frequency is given by: ω i =re[ + ] 5.4 Wide range frequency estimation Using two band pass filters to estimate local frequency will wor well if the signal spectrum falls within the range of the filters. However, a wide range local frequency estimate can easily be obtained by weighted summation over a number of different filter pairs. Assume N quadrature filters differing only in their centre frequencies. Let the relative bandwidth of the filters be and the centre frequencies be given by: = i ρ 6 Then a wide range frequency estimate can be obtained as N N q ρ = [ c i ] c i+ i ρ +.5, 7 i i= i= where: +.5 = + = ρ i+.5 c i are the weighting coefficients. For single frequency neigbourhoods all filter pairs will estimate the same frequency, see Figure, and any choice of weighting coefficients c i will yield the same result. For other situations, however, the choice of weighting coefficients is important. To obtain reliable estimates the weighting coefficients should reflect the amount of signal energy that a given filter pair encounters. A choice of c i that satisfies this requirement at the 3
5 5 5 3 35 4 45 5.8.9 3 Experimental results.6.4. -. -.4 -.6 -.8 -.8.7.6.5.4.3.. 5 5 5 3 35 4 45 5 Figure : Left: A test signal constructed by sinusoids of a stepwise increasing frequency π/64, π/6, π/4. Right: Estimated local frequency and certainty dashed The first example contains a non-stationary signal, i.e. a signal with space variant spectrum. It is constructed from three sinusoids having frequencies π/64, π/6 and π/4, see Figure. Note that the certainties of the estimates are high in the constant frequency regions and low in the regions where transitions between two frequencies occur..5.5.8.6.4. same time as it leads to a particularly simple estimation algorithm is: c i = 8 With this choice Equation 7 becomes: N N ρ = ρ [ ] i= i= i+.5 + 9 i.e. a wide range frequency estimate is obtained as the ratio between to sums of lognormal filter outputs. This estimation algorithm was used in the experiments below with eight filters, with a relative filter bandwidth B of, one octave apart. -.5 - -.5-5 5 5 3 35 4 45 5.95.9.85.8.75.7.65.6.55.5 5 5 5 3 35 4 45 5.9.8.7.6.5.4.3.. -. -.4 -.6 -.8-5 5 5 3 35 4 45 5.4..8.6.4..9.8.7.6.5.4.3.. 5 5 5 3 35 4 45 5 5 5 5 3 35 4 45 5 5 5 5 3 35 4 45 5.5 Estimation of spectrum variance The local frequency may be considered the average of the frequencies present at a given spatial position. It is useful to have a measure defining the deviation from this value. Such a measure may be interpreted as instantaneous bandwidth [Boa9a] and will be directly related to the reliability of the mean frequency estimate, ρ. The broader the bandwidth the more uncertain the local frequency estimate. An estimation of the local signal spectrum variance can be obtained by summing the squared differences between the wide range estimation of the mean frequency, ρ, and the estimates obtained by the individual filter pairs, i.e. : N N σ = [ qi ] i= i= q i [ i+.5 + ρ] 3 A certainty measure, c, based on the local spectrum variance can be defined: c = +σ 3 This is the certainty measure used in the examples below. Figure 3: Left Top: A test signal containing one constant and one linearly increasing frequency component. Middle: Estimated local frequency before and after normalized averaging the smoother one is after the averaging. Bottom: Estimated certainty and the the certainty after the normalized averaging step dashed. Right Top: A test signal containing constant mean frequency but with exponential increasing bandwidth. Middle: Estimated local frequency before and after normalized convolution the smoother on is after the averaging. Bottom: Estimated certainty and the certainty after the normalized averaging step the smoother one. The test signal in the second example, see Figure 3 left, is constructed as a sum of two components, one constant frequency component, f = π/4, plus a chirp signal which has linearly increasing frequency from f = π/8 tof =3π/8. This means that the single frequency model is not valid. In this case the algorithm estimates the mean of the two frequency components. Due to interference between the frequencies, the mean frequency is difficult to measure in some spatial positions. In these singular points the certainty of the estimate is close to zero. A smoother estimate can be obtained using normalized averaging [KW93]. Using this technique an adaptive smoothing can be performed based on the accompanying certainty values. A new estimate of the certainty after this operation is also provided, see the 4
dashed line. The third example, see Figure 3 right, contains a test signal with one constant mean frequency with exponential increasing bandwidth from B =.5 to B =4octaves. The broader the bandwidth is the weaer the notion of instantaneous frequency for the signal get. This is why the estimates get more and more noisy when traveling to the right in the middle right figure, and why the certainty is decreasing in the bottom right figure. Also plotted in these figures are the estimates after using normalized averaging. A D example is shown in Figure 4. This images is constructed having regions from two different testpattern which are overlapping in radial sections. One component is a narrow-band noise process and the other component has increasing frequency towards the the centre middle. In the overlapping regions the alghorithm estimates the mean frequency component. If the frequency components differs a lot this is captured in the variance estimate. References [AR77] J. B. Allen and L. R. Rabiner. A unified approach to short-time fourier analysis and synthesis. In Proc. IEEE, volume 65:, pages 558 564, November 977. [Boa9a] B. Boashash. Estimating and interpreting the instantaneous frequency of a signal - part : Fundamentals. Proceedings of the IEEE, 84:5 538, 99. [Boa9b] B. Boashash. Estimating and interpreting the instantaneous frequency of a signal - part : Algorithms and applications. Proceedings of the IEEE, 84, 99. [CF37] J. Carson and T. Fry. Variable frequency electric circuit theory with application to the theory of frequency modulation. Bell Syste Tech. J., 6:53 54, 937. [Gab46] D. Gabor. Theory of communication. J. Inst. Elec. Eng., 936:49 457, 946. [Gra78] G. H. Granlund. In search of a general picture processing operator. Computer Graphics and Image Processing, 8:55 78, 978. [Knu8] H. Knutsson. Filtering and Reconstruction in Image Processing. PhD thesis, Linöping University, Sweden, 98. Diss. No. 88. [KW93] H. Knutsson and C-F. Westin. Normalized and Differential Convolution: Methods for Interpolation and Filtering of Incomplete and Uncertain Data. In Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 55 53, New Yor City, USA, June 993. IEEE. [vdp46] B. van der Pol. The fundamental principles of frequency modulation. Proceedings of the IEEE, 93:53 58, 946. Figure 4: Top: Testimage. Middle Estimated local frequency. High intensity corresponds to high frequency. Bottom: Certainty measure of the estimated frequency. 5