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”
EACH SINE FUNCTION HAS ITS OWN AMPLITUDE
(“WEIGHT”), PHASE (“STARTING POINT”),
AND FREQUENCY (“PERIODICITY”)
a) The two sine functions differ
only in their amplitude
•
Their “weight” in a Fourier
Series would be different
b) The two sine functions have
the same amplitude but they
differ in their phase
•
They “start” at different points
c) 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”
THE FOURIER-TRANSFORM INTEGRAL GENERATES
COMPLEX WEIGHTS (STRUCTURE FACTORS)
FOR EACH SINE-FUNCTION IN THE SUM
• The Fourier-Transform integral is defined as:
 V( x) exp( i 2s  x)dx
• This integral “picks out” the value of the sth weight
in the sum of sine functions
 V( x) exp( i 2s  x)dx  F (s)
 F( s ) exp( i )
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
• The projection theorem
– 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
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
F ( s )   V( x) exp( i 2s  x)dx
V( x)   F ( s ) exp( i 2s  x)ds
Real Space
Fourier (reciprocal) Space
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
F ( s )  G ( s )   F ( s ' )G ( s  s ' )d s '  Q( s )
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
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(-i2s.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(-i2s.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