Fourier Transform Henning Stahlberg Introduction The tools provided by the Fourier transform are helpful for the analysis of 1D signals (time and frequency (or Fourier) domains), as well as 2D/3D signals (real space and spatial frequency (or Fourier) domains). A Fourier series is an expansion of a periodic function f(x) of period L, in terms of an infinite sum of sines and cosines (Fig. 1):! f (x) = a 0 / 2 + "(a n cos(n x) + b n sin(n x)), where! = 2" / L and a n =! / " b n =! / " " /! n=1 f (x)cos(n! x)dx (n=0,1,2 ) " /! " /! f (x)sin(n! x)dx (n=1,2 ) " /! Fig. 1 Fourier series of periodic functions having a period L (from Mathematica). Simply by linear combinations of cos and sine functions, any periodic function can be approximated with arbitrary accuracy. The Fourier transform is a generalization of the complex Fourier series in the limit as L!!. Replace the discrete a n with the continuous F(h)dh while letting n/l!h. Then change the sum to an integral, use the identity cos n2!x/l + i sin n2!x/l = exp i n2!x/l, and the above equation is rewritten as in the definition below: Definition 1D Fourier transform (often t (time) is used instead of x and " (frequency) instead of h: F(h) =! f (x)exp(!i(2"hx)dx Inverse 1D Fourier transform: f (x) = " F(h)exp(i(2!hx)dh 2D Fourier transform: 1
F(h, k) =!! f (x, y)exp(!2"i(hx + ky))dxdy Inverse 2D Fourier transform: f (x, y) = " " 3D Fourier transform: F(h, k,l) = F(h, k)exp(2!i(hx + ky))dhdk f (x, y, z)exp(!2"i(hx + ky + lz))dxdydz Inverse 3D Fourier transform: f (x, y) = """ F(h, k,l)exp(2!i(hx + ky + kz))dhdkdl Theorems Examples are taken from Optics, i.e. 2D Fourier transforms are considered. The theorems are valid for nd Fourier transforms. Given two 2D Fourier pairs, f (x, y)! F(h,k) g(x, y)! G(h,k) the following holds: 1) Linearity theorem (Fig. 2) FT(a! f (x,y) + b! g(x, y)) = a! F(h,k) + b! G(h,k) Fig. 2. Examples of real space functions and their linear superposition (top row) and the corresponding Fourier transforms (represented are only the amplitudes; bottom row) 2) Similarity theorem 2
f (s! x,q! y) = 1 s! q F( h s, k q ) Fig. 3. Similarity theorem. Large extension in real space (top row) leads to narrow structures in Fourier space (bottom row). Note that this example relates to the resolution of an optical system (top row left: a lens with a large numerical aperture has a high resolution, i.e. a narrow impulse response function, bottom left; top row right: a lens with a small numerical aperture has a low resolution, i.e. a large impulse response function, bottom right) 3) Translation in the real space (shift theorem) f (x! x 0, y! y 0 ) = F(h,k)exp(!2"i(hx 0 + ky 0 )) 4) Parseval s theorem (conservation of energy) " " f (x, y) 2 dxdy = F(h,k) 2 dhdk!"!" 5) Convolution theorem % & % & " "!"!" [ f! g](x, y) = f (", )g(x ",y )d"d % % [ f! g](x, y) " F(h,k) G(h,k) This is a most useful theorem. It relates to the Whittacker-Shannon sampling theorem (see Fig. 8), it allows lattices to be described in an elegant manner (, it provides the foundation for the optical image formation, i.e. the concept of the contrast transfer function (CTF), just to give a few examples. Fig. 4 demonstrates how a lattice is generated from a point lattice and a motif. The Fourier transform of the motif is sampled on the reciprocal point lattice. 6) Rotation in the real space <> rotation in Fourier space 3
Fig. 5 When the motif rotates, the Fourier transform simply rotates as well. As a link to the optical diffraction theory we write h,k as h =! / (" f );k = / (" f )see Optical Diffraction & Image Formation, p6, Eqs. 9-11 Transform pairs To know such pairs is most useful for attempts to apply the Fourier transform concept in practical cases. 1) Delta function: (x) = 0, x! 0 (x)! 0, x = 0 and " (x) dx = 1!(x " x 0 ) exp("2ihx 0 ) 2) Comb function The comb function represents a point lattice; its Fourier transform is a point lattice again! +% &!(x " m x) ' &!(h " m / x) n="% +% n="% 3) Rect function The 1D rect function is a slit, the 2D rect function an open rectangle. rect(x) = 1, if x 1/2 rect(x) = 0, otherwise rect(x)! sin("h) / "h = sin c(x) 4) Triangle function The 2D triangle function is a pyramid. (x) =1- x, x 1 (x) = 0 otherwise!(x) "!(y) (sin(h) / h) 2 " (2sin(k) / k) 2 4
5) Gauss function exp(!"(x 2 + y 2 )) exp(!"(h 2 + k 2 )) 6) Circle function The circ function describes the aperture of a lens. circ(r) = 1, r1; (r=(x 2 +y 2 )) circ(r) = 0 otherwise circ(r)! J 1 (2") / Building crystals We use the comb function +% &!(x " m x) that is a infinite set of functions, which are n="% equally spaced. The convolution operation reproduces the motif exactly at each peak to yield a 1D crystal * = Fig. 6. Convolution theorem. The motif is convoluted with the point lattice to yield a 1D crystal. We build a 2D crystal simply using a 2D comb function and a 2D motif. Now we use the convolution theorem to understand the Fourier transform of this crystal. Fig. 7. Convolution theorem. The motif is convoluted with the point lattice to yield a 2D crystal (top row). The Fourier transform of the motif is a complex function (molecular transform, bottom left). Multiplication with the reciprocal point lattice yields sampling of the molecular transform (bottom right, shown is only the real part). 5
Space invariant linear system An ideal lens is a space invariant linear system. Its properties are given by the impulse response h(x,y) in real space, or its contrast transfer function (CTF) H(h,k) in Fourier space: h(x, y)! H(h,k) The image of an object g(x,y) generated by a microscope with CTF H(h,k) is then % & % & [g! h](x, y) = g(", )h(x ", y )d"d % % and its Fourier transform is written accordingly: G(h,k)! H(h,k) In other words, amplitude and phase of the Fourier transform G(h,k) (a complex function) giving all the information on the object g(x,y), are change by multiplication with the CTF H(h,k). Hence, the image is not anymore the full representation of the object. Sampling theorem To digitize an image, the scanner measures the object g(x,y) at discrete points x m,y n, described by a 2D comb function. This corresponds to acquire a single number for the density distribution on each point of the sampling raster. The process may be also represented by multiplication of the 2D density distribution g(x,y) with the comb(x,y). In the Fourier space this process is the convolution of the object transform G(h,k) with the transform of the comb (see transform pair 2)). If %x=%y is small, the separation 1/%x in the Fourier space will be large. If the extension of the Fourier transform G(h,k) is less than 1/%x, the function g(x,y) is properly sampled. 6
Fig. 8. The Wittacker Shannon sampling theorem predicts that the sampling is sufficient, when 1/(2%x) > h max. 1/(2%x) is also termed Nyquist frequency. Fourier transform in real life MP3 compression (from Wikipedia) There are some further tricks to optimize compression one wants to keep as much of the important frequencies, but not the noise. Compression thus gives the max frequency to take along and then takes those frequencies fmax that are above a certain threshold. JPEG is another example. The Fourier transform of the image to be compressed is analyzed. Depending on the quality, fmax is reduced and only strong Fourier terms are transmitted. JPEG shows edge ringing much as is seen with MP3, and if quality is low, the image appears decomposed into blocks. Fourier filtering and correlation averaging The goal of the Fourier exercise is to use the tools for eliminating the noise from electron micrographs recorded at low dose. The flexibility image processing offers is tremendous: many different approaches can be used to improve the visibility of a structure. One striking example is that of a 2D crystal embedded in ice and recorded at a dose of 5 electrons/å 2. 7
The first processing step is to compute the diffraction pattern, i.e. the FT 2, often referred to as power spectrum (PS). If the crystals is well ordered, diffraction spots will emerge above the background. The next step is to index the diffraction pattern. This means to determine the lattice vectors. Most image processing system allow this to be done using the mouse. The program calculates the FT and multiplies it with an array of small windows that are positioned on the lattice determined. The thus processed FT is then back transformed to obtain the filtered image. Fig. 11: Fourier peak filtering of a 2D crystal submerged in noise. Diffraction spots are better visible in the insert. The filter transmits their information while blocking the noise. Cross correlation averaging and lattice unbending. Often the crystal quality is not optimal. Some unit cells can be well located on the lattice, but others may be damaged. Often the lattice is bent, so the unit cells are gradually rotated along the lattice line. Clearly one should avoid averaging unit cells that are rotated with respect to one another. Also, damaged unit cells should be eliminated. Both imperfections are measured by the cross correlation function, XCF. This function measures the overlap between two functions (similar to the convolution, but the complex conjugate of the second function is taken; in case of real functions, the two operations are identical). Since the overlap as maximal when the two functions are centered, maxima (the correlation peaks) indicate the locations of the unit cells. Deviations of these positions from an ideal crystal lattice indicate distortions, which can be corrected. The steps required are illustrated in Fig. 2. 8
Fig.12: Cross correlation averaging and crystal unbending of a GlpF 2D crystal. Steps will be discussed in lectures and exercises. 9