The above expressions refer to continuous functions. In practical image analysis
problems, however, the input image is discretely sampled, and a discrete spectrum
is recorded. In image processing and analysis, the discrete Fourier transform
(DFT) is considered [18]. Specifically, given the natural numbers k; l and u; v, the
discrete Fourier transform F of the image I is
Fðk; lÞ
1
MN
X NÀ1
v¼0
X
MÀ1
u¼0
Iðu; vÞe
Ài2p
uK
M þ
vl
N
ð
Þ
ð11:3Þ
The inverse discrete Fourier transform becomes:
Iðu; vÞ
1
MN
X NÀ1
l¼0
X
MÀ1
k¼0
Fðk; lÞe
i2p
uK
M þ
vl
N
ð
Þ
ð11:4Þ
Fast computations of the DFT can be achieved using a method called the Fast
Fourier Transform (FFT) [4].
An important application of the FFT is the efficient solution of the convolution
problem. Using 3-D generalization, the convolution of two image volumes
I 1 ðx; y; zÞ and I 2 ðx; y; zÞ is known to be a computationally onerous problem in real
space [23]. In the frequency domain it can be solved by a computationally inexpensive multiplication [34], by application of the well-known convolution theorem
which states that convolution in real space is equivalent to a multiplication in the
Fourier domain
I 1 ðx; y; zÞ Ã I 2 ðx; y; zÞ ¼ F
À1
FfI 1 ðx; y; zÞgFfI 2 ðx; y; zÞg
f
g
ð11:5Þ
where à is the convolution operator.
11.2.2 Bandpass Filters
In the Fourier spectrum of an image, coarse features are associated with low spatial
frequency signal, while fine features are associated with high spatial frequency.
Noise may predominate over signal at high spatial frequency, impairing the interpretation of both coarse and fine features. Denoising can be obtained by filtering
specific frequency bands associated with noise. Specifically, filtering high spatial
frequencies may reduce the contribution of high resolution noise. Filtering is
achieved by weighting functions that vary from crude frequency truncation to
strictly monotonic functions, with the choice designed to avoid the creation of
spurious detail in the image.
An example of crude noise reduction by removing high frequency information is
shown for a cryo-tomogram of frozen-hydrated influenza virus in Fig. 11.1. Images
in a tilt series were acquired from À60
to þ 60
, 3
increment, and reconstructed
286
M. Maiorca and P. B. Rosenthal
problems, however, the input image is discretely sampled, and a discrete spectrum
is recorded. In image processing and analysis, the discrete Fourier transform
(DFT) is considered [18]. Specifically, given the natural numbers k; l and u; v, the
discrete Fourier transform F of the image I is
Fðk; lÞ
1
MN
X NÀ1
v¼0
X
MÀ1
u¼0
Iðu; vÞe
Ài2p
uK
M þ
vl
N
ð
Þ
ð11:3Þ
The inverse discrete Fourier transform becomes:
Iðu; vÞ
1
MN
X NÀ1
l¼0
X
MÀ1
k¼0
Fðk; lÞe
i2p
uK
M þ
vl
N
ð
Þ
ð11:4Þ
Fast computations of the DFT can be achieved using a method called the Fast
Fourier Transform (FFT) [4].
An important application of the FFT is the efficient solution of the convolution
problem. Using 3-D generalization, the convolution of two image volumes
I 1 ðx; y; zÞ and I 2 ðx; y; zÞ is known to be a computationally onerous problem in real
space [23]. In the frequency domain it can be solved by a computationally inexpensive multiplication [34], by application of the well-known convolution theorem
which states that convolution in real space is equivalent to a multiplication in the
Fourier domain
I 1 ðx; y; zÞ Ã I 2 ðx; y; zÞ ¼ F
À1
FfI 1 ðx; y; zÞgFfI 2 ðx; y; zÞg
f
g
ð11:5Þ
where à is the convolution operator.
11.2.2 Bandpass Filters
In the Fourier spectrum of an image, coarse features are associated with low spatial
frequency signal, while fine features are associated with high spatial frequency.
Noise may predominate over signal at high spatial frequency, impairing the interpretation of both coarse and fine features. Denoising can be obtained by filtering
specific frequency bands associated with noise. Specifically, filtering high spatial
frequencies may reduce the contribution of high resolution noise. Filtering is
achieved by weighting functions that vary from crude frequency truncation to
strictly monotonic functions, with the choice designed to avoid the creation of
spurious detail in the image.
An example of crude noise reduction by removing high frequency information is
shown for a cryo-tomogram of frozen-hydrated influenza virus in Fig. 11.1. Images
in a tilt series were acquired from À60
to þ 60
, 3
increment, and reconstructed
286
M. Maiorca and P. B. Rosenthal