fft(2).pdf

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Correlation-Based Search,
Filtering, and
Fast Fourier Transform
© 2002, Pablo Chacón and Willy Wriggers.
Dept. of Molecular Biology, The Scripps Research Institute
ρ
atomic
(
r
)
ρ
em
(
r
)
probe
target
6D rigid-body search
match
A Typical Exhaustive Search Scenario
ρ
em
(
r
)
target density on lattice
ρ
atomic
(
rR
)
rotated probe molecule density projected to the lattice:
lattice
projection
low-pass
filter
g
⊗
ρ
atomic
(
rR
)
≡
ρ
calc
(
r
)
tri-linear interpolation
g
= Gaussian kernel
C
(
T
)
=
ρ
em
(
r
)
×
ρ
calc
(
r
+
T
) d r
3
Fitting criterion: e,g, linear cross-correlation,
evaluate for every rotation
R
and translation
T
∫
Enhancing the Fitting Contrast with
Density Filtering
For low-resolution smooth density maps (e.g. from EM) we wish to enhance
the
fitting contrast
(difference between correct and spurious fits) by adding
“contour” information to the criterion.
A suitable filter would assign negative values to the interior, positive values
to the molecular contour. Both volume and contour matches would provide
positive contributions to the correlation criterion:
Filter
Docking
How can we characterize the molecular
“contour” in a low-resolution map?
Laplacian filter
∂
2
∂
2
∂
2
∇ =
2
+
2
+
2
∂
x
∂
y
∂
z
2
FFT Acceleration of the Translational Search
C
(
T
)
=
∫
(
e
⊗
ρ
)
(
r
)
×
(
e
⊗
ρ
)
(
r
+
T
) d r =
3
em
calc
f
(
e
⊗
ρ
em
)
×
−
1
f
f
(
e
⊗
ρ
calc
)
*
Fourier Convolution Theorem:
Direct approach (top): N
2
multiplications
FFT approach (bottom): N log N
multiplications
(N = number of voxels,
f
= Fourier Transform,
e
= optional filter, e.g. Laplacian)

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