1. FOURIER THEORY: KEY CONCEPTS IN 2D & 3D
ROBERT M. GLAESER
Electron Crystallography Workshop
UC Davis
Sept. 7-13, 2008

2. FOURIER TRANSFORMS: ANY OBJECT CAN BE THOUGHT OF AS A SUM OF SINUSOIDAL FUNCTIONS
The sum of sine-functions shown in panel (a) produces the complex, repeating motif shown in (b)
Not obvious in the figure, the repeat-period of the “nth” sine function is 1/n of the period of the first sine function
Successive sine functions are “harmonics” of the fundamental
When the sum becomes an integral – i.e. a “continuous sum” – we refer to the integral as a “Fourier Transform”

3. The two sine functions differ only in their amplitude
EACH SINE FUNCTION HAS ITS OWN AMPLITUDE (“WEIGHT”), PHASE (“STARTING POINT”), AND FREQUENCY (“PERIODICITY”)
The two sine functions differ only in their amplitude
Their “weight” in a Fourier Series would be different
The two sine functions have the same amplitude but they differ in their phase
They “start” at different points
The two sine functions have the same amplitude and the same phase (starting point), but they differ in their frequency
The repeat-period (resolution) is not the same

4. RESOLUTION AND SPATIAL FREQUENCY
A “COMPLICATED”
STRUCTURE
Chiu et al. (1993)
Biophys J. 64:
LOW-RESOLUTION
FEATURES
HIGH-RESOLUTION
FEATURES
RESOLUTION, “d”, AND SPATIAL FREQUENCY, “s = 1/d”
ARE “THE SAME THING”

5. This integral “picks out” the value of the sth weight
THE FOURIER-TRANSFORM INTEGRAL GENERATES COMPLEX WEIGHTS (STRUCTURE FACTORS) FOR EACH SINE-FUNCTION IN THE SUM
The Fourier-Transform integral is defined as:
This integral “picks out” the value of the sth weight
in the sum of sine functions

6. DIFFRACTION PATTERNS OF IMAGES [OPTICAL OR COMPUTED] ARE FOURIER TRANSFORMS
Each diffraction spot represents a different (spatial) frequency
The amplitude of each spot is unique to the structure of the object
When a Fourier transform like this one is calculated in a computer, one also gets the phase at each spatial frequency
Note: high radius
= high frequency
= high resolution

7. THE PROJECTION THEOREM EXPLAINS WHY 3-D STRUCTURE ANALYSIS REQUIRES VIEWS FROM MULTIPLE DIRECTIONS
When a 3-D object is projected to produce a 2-D image, its Fourier transform is a 2-D slice (NOT a projection) of the 3-D transform of the oject
These 2-D slices always pass through the origin
and thus are called”central sections”
When data from many different “central sections” are combined
one builds up the full, 3-D Fourier transform
Real Space Fourier (reciprocal) Space

8. Fourier (reciprocal) Space
INFORMATION ABOUT THE OBJECT IS THE SAME IN THE REAL WORLD AND IN THE FOURIER TRANSFORM OF THE OBJECT
The “undo button” turns out to be just doing another Fourier transform
Real Space
Fourier (reciprocal) Space

9. Similarly, the FT of {f(x)*g(x)} is F(s).G(s)
THE FOURIER TRANSFORM OF A PRODUCT IS THE CONVOLUTION OF TWO RESPECTIVE FOURIER TRANSFORMS
The FT of {f(x).g(x)) is F(s)*G(s), where
Similarly, the FT of {f(x)*g(x)} is F(s).G(s)
Note that F and G are shifted relative to one another by an amount “s”, and then you find the area under the curve when the two are multiplied
Doing this over and over again for new values of “s” results in a new, continuous function, whose values are determined by the values of the two functions that are being convoluted together

10. Convolution generally “broadens” the original object
But convolution of one function with a
single-point function (“delta function”)
replicates the original function
Convolution of a “rectangle function” with itself
results in a “broader” function with a new shape * =
Convolution of one unit cell with a 2-D lattice
produces a 2-D crystal

11. Since V2-Dcrystal = Vunit cell *(2-D lattice-of-points)
HOW IS THE FT OF A 2-D CRYSTAL RELATED TO THE FOURIER TRANSFORM OF ONE UNIT CELL (e.g. ONE MOLECULE)?
Since V2-Dcrystal = Vunit cell *(2-D lattice-of-points)
the convolution theorem tells us that
FT{V2-Dcrystal} = FT{Vunit cell}.FT{2-D lattice-of-points}
Pulling “a rabbit from a hat”, let me tell you that the FT{lattice-of-points} is itself a lattice of parallel lines
Think of this as a kind of “comb”, or even better, a “brush”
Thus,
The Fourier transform of a 2-D crystal is a regular array of one-dimensional samples of the Fourier transform of the unit cell
Where each of the samples is a continuous, 1-D function, extracted from the original, 3-D Fourier transform of the unit cell (e.g. one molecule)

12. WHY IS THE DIFFRACTION PATTERN OF A 2-D CRYSTAL ALWAYS A 2-D ARRAY OF SPOTS?
Diffraction patterns
represent values of
the Fourier transform
lying on the surface
of the “Ewald sphere”
The array of “disks” represents a
regular array of positions in the
sx, sy plane, and the continuous
gray-scale variation represents
the sampling along the lattice lines
We normally approximate this spherical
surface by a PLANE that passes through
the origin of reciprocal space, i.e. by a
“Central Section”

13. HAD ENOUGH? STILL TO COME:
The secret of the Fast Fourier Transform (FFT)
The cross-correlation function is a special example of convolution

14. THE “FAST” FOURIER TRANSFORM (FFT) USES MATHMATICAL IDENTITIES TO REDUCE THE AMOUNT OF COMPUTATION THAT IS REQUIRED
The FFT is based on the Fourier “Shift” theorem [which has many other applications, so please remember this theorem!]
The trick is to
Divide the image in half and do two separate Fourier transforms
Then add the two results together, after applying an additional “phase ramp”, exp(-i2ps.a), where “a” is the relative shift between the two halves
This trick is based on the fact that FT{f(x-a)} = FT{f(x)} exp(-i2ps.a)
But don’t stop there!
Divide each half of the data by two, and apply this trick again
But first divide each “quarter” in half, etc., until subdivision is no longer possible
The computational work increases as n log2n rather than as n2 when you use this trick – THIS RESULTS IN AN ENORMOUS REDUCTION OF WORK WHEN n IS A LARGE NUMBER OF PIXELS

15. A QUICK COMMENT ABOUT THE HEAVILY USED CROSS-CORRELATION FUNCTION
If F(s) = FT{V(x)} and H(S) = FT{h(x)},
then FT-1{F(s).H*(s)} = V(x)*h(-x)
i.e. V(x) convoluted with
an inverted form of h(x)
This instance of convolution produces
the well-known and useful
Cross-correlation function
Point-spread functions in image theory
Matched filtering (one form of template matching)
Unbending; real-space averaging