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Joachim Frank

Print publication date: 2006

Print ISBN-13: 9780195182187

Published to Oxford Scholarship Online: April 2010

DOI: 10.1093/acprof:oso/9780195182187.001.0001

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(p.319) Appendix 1 Some Important Definitions and Theorems

Source:
Three-Dimensional Electron Microscopy of Macromolecular Assemblies
Publisher:
Oxford University Press

Figure A1.1 Some examples of the two-dimensional sine waves which form the basis of the Fourier representation of an image (white represents +1, black –1). (a) ϕ = 0°; l = 1, m = 2; (b) ϕ = 90°; l=1, m = 2; (c) ϕ = 90°; l= –1, m = 2; (d) ϕ = 90°; l=1, m = –2; (e) ϕ = 180°; l = 2, m = 2; (f) ϕ = 45°; l=5, m = 3.

1. The Discrete Two-Dimensional Fourier Transform

An image (such as the projection of a molecule) is defined as a two-dimensional (2D) array representing samples of the optical density distribution on a regular grid:

(A1. 1)
(p.320) The coordinates are multiples of the sampling distance Δx = Δy; x i = iΔx; yk = kΔx. The Fourier transform is a mathematical representation of the image using sine waves or complex exponentials as basis functions. One can see the Fourier transform as an alternative representation of the information contained in the image. What singles out the Fourier transform, among many other expansions using sets of orthonormalized basis functions, is that it allows the influence of instrument aberrations and the presence of periodicities in the object to be readily analyzed. It also provides a key to an understanding of the relationship between the projections and the object they originate from, and to the principle underlying three-dimensional (3D) reconstruction.

The basis of the Fourier representation of an image is a set of “elementary images”—the basis functions. In the case of the sine transform, to be used as an introduction here, these elementary images have sinusoidal density distribution:

(A1.2)
Each of these images (examples in figure A1.1), represented by a discrete set of pixels at positions (x i, yk) of a square lattice, is characterized by its discrete spatial frequency components (u l, v m)—describing how many full waves fit into the frame in the horizontal (l) and vertical (m) directions:
(A1.3)
Any discrete image {p ik; i = 1, …, I; k = 1, …, K} can be represented by a sum of such elementary images, if one uses appropriate amplitudes a lm and phases ϕlm.. The amplitudes determine the weights of each elementary image, and the phases determine how much it has to be shifted relative to the normal position (first zero assumed at x i = y k = 0; note that in figure A1.1 this occurs in the upper left corner since a left-handed coordinate system is used):
(A1.4)
[Here, we have rewritten the elementary image (A1.2) by substituting u l by l/(IΔx) and x by iΔx, etc.) To understand equation (A1.4), consider one of its elementary images, indexed (l=1, m = 0): as i increases from 1 to I, the argument of the sine function goes from 0 to 2π exactly once: this elementary image is a horizontally running sine wave that has the largest wavelength (or smallest spatial frequency) fitting the image frame in this direction.

The 2D sine transform is a 2D scheme, indexed (l, m), that gives the amplitude a lm and the phase shifts ϕ lm of all elementary sine waves with spatial frequencies (u l, v m), as defined in equation (A1.3), that are required to represent the image.

The Fourier representation generally used differs from the definition (A1.4) in that it makes use of “circular” complex exponential waves:

(A1.5)
(p.321) (Note that “i” is the symbol for the imaginary unit, to be distinguished from the index i.) The reason is that this representation is more general and leads to a more tractable mathematical formulation. Hence, the Fourier representation of an image is
(A1.6)
or, symbolically,
(A1.7)
Conversely, given an image, the Fourier coefficients F lm are obtained by a very similar reciprocal expression:
(A1.8)
or, symbolically,
(A1.9)

2. Properties of the Fourier Transform

Linearity. This property follows from the definition: if an image is a linear combination of two images,

then its Fourier transform is
(A1.9a)
where F 1(k), F 2(k) are the respective Fourier transforms of the images f 1(r) and f 2(r)

Scaling of argument. Given two images of the same object with different magnifications, f 2(r) = f 1(s r), where s is a scaling factor, then their Fourier transforms are related by F 2(k) = F 1(1/s k).

Symmetry property. Images are normally real-valued. It is easy to see from equation (A1.8) that for real-valued functions, the normally complex-valued Fourier transform has the property:

(A1.10)
which is known as Friedel symmetry in X-ray crystallography. The asterisk stands for taking the complex conjugate:
(A1.11)

(p.322)

Figure A1.2 Representation of a complex number Fas a vector in the complex plane. A (amplitude, or modulus) is the length of the vector, while ϕ (the phase) is the angle (counterclockwise, or mathematically positive) that the vector is rotated by with respect to the real axis.

Thus, since for real-valued images, half the Fourier transform determines the values of the other half, only one half is normally required in all computations and storage.

Amplitude and phase. The complex coefficient F lm can be represented by a vector in the complex plane (figure A1.2). Its length,

(A1.12)
is the amplitude of the Fourier transform, while its phase is
(A1.13)

Shift Theorem. When the image is shifted by a vector Δr = (Δx, Δy), its Fourier transform changes according to equation (A1.8) by multiplication with an exponential factor:

(A1. 14)

3. Convolution Product and Convolution Theorem

Suppose we were able to form an image of a single point of the object, located at (x i, yk). Its image will appear as an extended disk, on account of resolution limitations, at the corresponding point (x i, yk) in the image plane. In bright-field electron microscopy (EM), applied to weak phase objects (which is a good approximation in cryo-electron microscopy of biological macromolecules), image formation can be described by linear superimposition: the image of two object points can be obtained by adding the images that would be obtained by imaging (p.323) the two points independently and separately. An even more restrictive property holds in EM, the property of isoplanasy: the image of a point looks the same irrespective of its location in the image field. Under those conditions, the image is related to the object by

(A1.15)
where i, k, l, m relate to positions on a regular sampling grid and h(x, y) is the point-spread function—it is the image of a single object point placed at (0, 0). The relationship (A1.15) follows immediately from the properties of linearity and isoplanasy. The right-hand side of equation (A1.15) represents the convolution product of o(x, y) with h(x, y). We will make use of the symbolic notation o(x, y)o h(x, y). The delta-function is a particular point spread fuction consisting of a single point, h(x l – xi, y myk) = δ(x l – xi, y myk), which has the value one at xl = x i, y m = yk and zero elsewhere. Use of this function in (A1.15) gives the result p(xi, yk) = o(xi, yk); i.e., convolution with the delta-function reproduces the original function.

The convolution theorem is of fundamental importance for the computation of convolution expressions. The theorem states that the Fourier transform of a convolution product of two functions is equal to the product of their Fourier transforms.

Hence, the computation of the convolution product of two functions o(x, y) and h(x, y) can be computed in the following way:

1. (i) Compute H(u, v) = F{h(x, y)}

2. (ii) Compute O(u, v) = F{o(x, y)}

3. (iii) Form the scalar product of these two transforms

(A1.16)

4. (iv) compute the inverse Fourier transform of the result:

(A1.17)

This is one of the many instances where the seemingly complicated step of Fourier transformation proves much more efficient than the direct evaluation of a linear superimposition summation. The time saving is due to the very economic organization of the fast Fourier transform (FFT) algorithm in numerical implementations (see Cooley and Tukey, 1965). Another example will be found in the evaluation of the auto- and cross-correlation functions below.

4. Resolution in the Fourier Domain

There exists a practical limit to the number of sine waves to be included in the Fourier representation (A1.6) of an image. Obviously, sine waves (or complex (p.324) exponentials) with wavelengths smaller than the size of the smallest features contained in the image carry no information.

This information limit can be expressed in terms of the spatial frequency radius R=√u2+v 2=0 beyond a certain radius R = R 0 (i.e., outside of a roughly circular domain in the Fourier plane) no meaningful Fourier components are encountered. This boundary is called the bondlimit or resolution limit. [Note that usage of the term resolution is inconsistent in the literature; it is used to denote either the smallest distance resolved (in real space units, dimension length) or the resolution limit defined above (in spatial frequency units, dimension 1/length.]

The resolution limitation invariably can be traced back to a physical limitation of the imaging process: either there is an aperture in the optical system that limits the spatial frequency radius of the object’s Fourier components (e.g., the objective lens aperture in the electron microscope), or the recording of the image itself may give rise to some blurring (as, for instance, the lateral spread of electrons in the photographic emulsion).

By virtue of the convolution theorem, any such spread or blurring in the image is expressed by a multiplication of the object’s Fourier transform with a function that has a finite radius in the spatial frequency domain.

5. Low-Pass and High-Pass Filtration

For a noncrystalline object, which we are exclusively dealing with, the signal and noise components of the Fourier transform are superimposed and normally inseparable. However, the two components frequently have different behaviors as a function of spatial frequency radius R: while the signal component falls off at the resolution limit (see section 4), the noise component has significant contributions beyond that limit. Hence, multiplication of the Fourier transform of such an image with a cutoff function:

(A1.18)
will eliminate part of the noise, thus enhancing the signal-to-noise ratio. Application of such a function, or variants of it having a smooth transition instead of a sharp cutoff, is termed low-pass filtration, as it passes only Fourier components with low spatial frequencies.

Functions with smooth radial transition are usually preferred, since application of equation (A1.18) would cause an artificial enhancement of image features whose size corresponds to the spatial frequencies at the cutoff (e.g., see Frank et al., 1985). The most “gentle” function used for Fourier filtration is one with a Gaussian profile (e.g., Frank et al., 1981b):

(A1.19)
(p.325) Its falloff behavior is controlled by the parameter R 0: at a spatial frequency radius of $R = u 2 + v 2 = R 0$, the filter function reduces the Fourier amplitude to 1/e of its original value.

6. Correlation Functions

Translational cross-correlation function. The cross-correlation function is defined in the following way:

(A1.20)

It can be shown that this function can also be evaluated by using a Fourier theorem closely related to the convolution theorem. Thus, again the tedious calculation of the sum can be bypassed: the Fourier transform of the cross-correlation function is equal to the conjugate product of the Fourier transforms of these images. This suggests a fast way of computing equation (A1.20):

1. (i) Compute

2. (ii) Compute

3. (iii) Take the complex conjugate of P 2(u, v)

(A1.21)

4. (iv) form the conjugate product of the two Fourier transforms:

(A1.22)

5. (v) finally, take the inverse Fourier transform of the result, which gives

(A1.23)

Autocorrelation function. As a special case, the autocorrelation function is obtained by letting p 2(x, y) = p 1(x, y). In this case, the computational sequence above is reduced to the following:

(i) Compute

(ii) Compute its absolute square, A(u, v) = |P 1(u, v)|2

(iii) Take the inverse Fourier transform of the result

(p.326)