11.2.3 Gaussian Filter
The Gaussian function is arguably the most used spatial frequency weighting
function, due to its scale-space property of not introducing spurious details in the
processed image (see paragraph 3.2). Furthermore, Gaussian spatial frequency
weighting is usually included in any image processing toolkit, and can be implemented in both Fourier space and real space. In the 2D case this can be obtained by
convoluting the original image Iðx; yÞ with the 2D Gaussian function
G r ðx; yÞ ¼
1
2r 2 p
exp À
x
2
þ y
2
2r
ð11:6Þ
In Fourier space, this convolution can be obtained with (11.5) by performing a
multiplication between the Fourier transform of the original image, FfIðx; yÞg,
with the Fourier transform of the Gaussian function,
FfG r ðx; yÞg ¼
1
2r 2 p
exp À
r
2
2
u
2
þ v
2
À
Á
;
ð11:7Þ
which is also a Gaussian. The challenge of using Gaussian filters for signal optimization is to select the optimal value of the parameter r able to minimize the
signal loss, while maximising noise loss. Larger r increases contrast in the processed image.
An example of a cryo-ET volume processed with a Gaussian diffusion filter is
shown in Fig. 11.2. Influenza A virus was imaged as described in Sect. 11.2.2. The
original FBP tomogram is shown in Fig. 11.2a. The original tomogram blurred with
r ¼ 1, r ¼ 2, and r ¼ 4 are shown in Fig. 11.2b–d (respectively). Larger r (i.e.
r ¼ 4) produces stronger contrast images (see Fig. 11.2d). In Fig. 11.2d it is
possible to identify the structures not visible in the unprocessed image (Fig. 11.2a),
for example the ribonucleoprotein particles (RNPs) that package the genome segments in the virus interior (white arrows in Fig. 11.2a–d). High frequency data loss
in the filtering process can be assessed using Fourier Shell Correlation (FSC), a
frequency-by-frequency agreement measure between two images that has value 1
when the two images are in full agreement [20, 21]. The FSC between Gaussian
filter processed and unprocessed data (Fig. 11.2e) shows that the increase of r
results in a decrease of FSC between the processed and unprocessed image [21].
11.3 Signal Optimization in Real Space
Real space signal optimization is used to enhance local spatial features by managing
gray value fluctuations within a neighbourhood of each image point.
288
M. Maiorca and P. B. Rosenthal
The Gaussian function is arguably the most used spatial frequency weighting
function, due to its scale-space property of not introducing spurious details in the
processed image (see paragraph 3.2). Furthermore, Gaussian spatial frequency
weighting is usually included in any image processing toolkit, and can be implemented in both Fourier space and real space. In the 2D case this can be obtained by
convoluting the original image Iðx; yÞ with the 2D Gaussian function
G r ðx; yÞ ¼
1
2r 2 p
exp À
x
2
þ y
2
2r
ð11:6Þ
In Fourier space, this convolution can be obtained with (11.5) by performing a
multiplication between the Fourier transform of the original image, FfIðx; yÞg,
with the Fourier transform of the Gaussian function,
FfG r ðx; yÞg ¼
1
2r 2 p
exp À
r
2
2
u
2
þ v
2
À
Á
;
ð11:7Þ
which is also a Gaussian. The challenge of using Gaussian filters for signal optimization is to select the optimal value of the parameter r able to minimize the
signal loss, while maximising noise loss. Larger r increases contrast in the processed image.
An example of a cryo-ET volume processed with a Gaussian diffusion filter is
shown in Fig. 11.2. Influenza A virus was imaged as described in Sect. 11.2.2. The
original FBP tomogram is shown in Fig. 11.2a. The original tomogram blurred with
r ¼ 1, r ¼ 2, and r ¼ 4 are shown in Fig. 11.2b–d (respectively). Larger r (i.e.
r ¼ 4) produces stronger contrast images (see Fig. 11.2d). In Fig. 11.2d it is
possible to identify the structures not visible in the unprocessed image (Fig. 11.2a),
for example the ribonucleoprotein particles (RNPs) that package the genome segments in the virus interior (white arrows in Fig. 11.2a–d). High frequency data loss
in the filtering process can be assessed using Fourier Shell Correlation (FSC), a
frequency-by-frequency agreement measure between two images that has value 1
when the two images are in full agreement [20, 21]. The FSC between Gaussian
filter processed and unprocessed data (Fig. 11.2e) shows that the increase of r
results in a decrease of FSC between the processed and unprocessed image [21].
11.3 Signal Optimization in Real Space
Real space signal optimization is used to enhance local spatial features by managing
gray value fluctuations within a neighbourhood of each image point.
288
M. Maiorca and P. B. Rosenthal