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

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”

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

RESOLUTION AND SPATIAL FREQUENCY A “COMPLICATED” STRUCTURE Chiu et al. (1993) Biophys J. 64: 1610-1625 LOW-RESOLUTION FEATURES HIGH-RESOLUTION FEATURES RESOLUTION, “d”, AND SPATIAL FREQUENCY, “s = 1/d” ARE “THE SAME THING”

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

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

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

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

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

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

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)

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”

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

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

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