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Two-Dimensional Fourier Transforms

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Two-Dimensional Fourier Transforms ESS 522 2014 12. Two-Dimensional Fourier Transform So far we have focused pretty much exclusively on the application of Fourier analysis to time- series, which by definition are one-dimensional. However, Fourier techniques are equally applicable to spatial data and here they can be applied in more than one dimension. Two- dimensional Fourier transforms are used extensively in the processing of potential field data (gravity and magnetics), are a useful tool for looking at topography/bathymetry or any variable that we might plot on a map, and are also used in reflection seismology to look at record sections in which one variable is time and the other is spatial location. For time series, the Fourier transform describes the data in terms of frequency f or angular frequency ω = 2πf. For spatial data, the wave number ν = 1/λ where λ is the wavelength is equivalent to f and the circular wave number k = 2π/λ is equivalent to ω. It is common practice (but strictly incorrect) to use the term wave number when referring to the circular wave number k = 2π/λ which is equivalent to ω. I will stick to the strict terminology and use the term wave number to refer to 1/λ. Continuous 2-D FT For continuous spatial data, the one-dimensional Fourier transform pair is given by ∞ G(ν) = ∫ g(x)exp(−i2πνx)dx −∞ (12-1) ∞ g(x) == ∫ G(ν)exp(i2πνx)dν −∞ where x is the spatial coordinate and ν is the wave number. If a(x,y) is a function of two spatial variables then the two-dimensional Fourier transform is simply obtained by repeating the one dimensional Fourier transform in both dimensions ∞ A ν ,ν = A' ν , y exp −i2πν y dy ( x y ) ∫ ( x ) ( y ) −∞ ∞ ⎡ ∞ ⎤ = a x, y exp −i2πν x dx exp −i2πν y dy (12-2) ∫ ⎢ ∫ ( ) ( x ) ⎥ ( y ) −∞ ⎣−∞ ⎦ ∞ ∞ = a x, y exp ⎡−i2π ν x + ν y ⎤dxdy ∫ ∫ ( ) ⎣ ( x y )⎦ −∞ −∞ where νx and νy are the wave numbers in the x and y direction and A’ is used to refer to the variable a after it has been transformed in one of its dimensions to the wave number domain. The two-dimensional Fourier transform pair is thus ∞ ∞ A ν ,ν = a x, y exp ⎡−i2π ν x + ν y ⎤dxdy ( x y ) ∫ ∫ ( ) ⎣ ( x y )⎦ −∞ −∞ (12-3) ∞ ∞ a x, y = A ν ,ν exp ⎡i2π ν x + ν y ⎤dν dν ( ) ∫ ∫ ( x y ) ⎣ ( x y )⎦ x y −∞ −∞ 12-1 ESS 522 2014 It is perfectly possible to apply the transform to one dimension leaving the other one untouched and this is often useful for data that is a function of one spatial dimension and time. For example, you might be interesting in seeing how the frequency of a time series varied with spatial position. Discrete 2-D FT The relationship between the one and two-dimensional transforms is similar in the discrete domain. In one dimension the discrete Fourier transform of gk with k = 1, 2, … M is given by M −1 ⎛ 2πikp⎞ G = g exp − p ∑ k ⎝⎜ M ⎠⎟ k =0 (12-4) 1 M −1 ⎛ 2πikp⎞ gk = ∑ Gp exp⎜ ⎟ M p=0 ⎝ M ⎠ Two-dimensional Fourier transform of ak,l is defined for k = 1, 2, …, M and l = 1, 2, …, N, is similarly defined by M −1 N −1 ⎡ ⎛ kp lq ⎞ ⎤ Ap,q = ∑ ∑ ak,l exp ⎢−2πi⎜ + ⎟ ⎥ k =0 l =0 ⎣ ⎝ M N ⎠ ⎦ (12-5) 1 M −1 N −1 ⎡ ⎛ kp lq ⎞ ⎤ ak,l = ∑ ∑ Ap,q exp ⎢2πi⎜ + ⎟ ⎥ MN p=0 q=0 ⎣ ⎝ M N ⎠ ⎦ Aside from the complexity of keeping track of indices, the expression is relatively straightforward. Properties The properties of the 2-D transform are analogous to the one-dimensional transform. Aligned linear features in the spatial domain (i.e., valleys and ridges) are characterized by a constant ratio of the wave numbers p/q in the frequency domain (Figure 1). Convolution, correlation, time shifting and differentiation can all be defined straightforwardly in two-dimensions. For example in the continuous domain two-dimensional convolution can be written as ∞ ∞ a(x, y)**b(x, y) = ∫ ∫ a(x', y')b(x − x', y − y')dx'dy' (12-6) −∞ −∞ and in the discrete domain as c a **b a b k,l = [ ]k,l = ∑∑ p,q k − p,l −q p q 12-2 ESS 522 2014 Aliasing Aliasing is also similar to the one-dimensional case with an interesting twist. In one-dimension, aliasing leads to a repetition with a periodicity of M points so that Gp+ M = Gp (12-7) For real g(x), we also have * GM − p = G p (12-8) In two-dimensions, aliasing leads to repetition in two directions and this produces a ‘tiling’ effect (Figure 2 left) in which Ap+ M ,q+ N = Ap+ M ,q = Ap,q+ N = Ap,q (12-9) For a real function a(x,y) we also have * AM − p,N −q = A p,q (12-10) The symmetry of this form of aliasing is illustrated in Figure 2 right. Rather unintuitively we see that wave numbers in the x direction are not aliased above their one-dimensional Nyquist frequency provided the wave numbers in the y direction are sufficiently below their Nyquist and vice-versa. Filtering If you are filtering spatial data, it is generally appropriate to use a zero-phase filter. Such filters are most simply applied in the wave number domain. For two-dimensional data one would perform a 2-D Fourier transform, multiplying the spectral amplitudes by the filter amplitude response (leaving the phases unchanged) and then performing the inverse two-dimensional Fourier transform. The amplitude responses of the Butterworth filters we discussed in Lecture 10 can all be generalized to two-dimensional data by simply rotating the functions around the origin of the two-dimensional FFT. For example a two-dimensional low-pass filter can be written 2 1 (12-11) Glowpass (ν x,ν y ) = 2n ⎛ ν 2 +ν 2 ⎞ 1 x y + ⎜ ν ⎟ ⎝ c ⎠ This filter would preserve wave numbers below νc – that is it would remove long-wavelengths from the data. In Exercise 6, you apply filters to topographic data If you have two dimensional data that is a function of time and one spatial domain then the two dimensional transform will be a function of frequency f and wavenumber ν. Since the speed of a wave is given by c = λf = f /ν, lines of constant f /ν radiating from the origin correspond to waves travelling along the spatial dimension at constant speed. If you take a class in reflection seismology you will learn how to use f-k filters (k = 2 πν) to enhance seismic signals traveling at a particular speed. 12-3 .
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