Fourier shell correlation (FSC) criterion [5], defined below, for the
windowed volumes. The second is a purely Fourier approach which
compares conical sectors of Fourier space of the half-maps for
consistency [6]. It may be thought of as an averaged (and therefore
smoother) version of 3D SSNR that was introduced earlier [7]. We
can therefore assign a direction to each resolution curve (the direction being down the axis of the aforementioned cone); we have
used the term “directional resolution” for this reason. The problems in a reconstructed map cannot be accurately captured using a
single global resolution criterion and should be quantitatively evaluated using both local and directional resolution measures. Highquality reconstructions will show consistency using the given
measures.
The global resolution of the reconstructed map across all spatial
frequencies is typically determined using the Fourier shell correlation (FSC) criterion [5]. To obtain an FSC, it is necessary to split
the dataset into two equal independent halves and reconstruct
individual “half-maps”:
FSC k
ð Þ
P
k
!0
%k
F k
! 0
G
∗
k
! 0
Norm F k
!
Norm G k
!
,
Norm F k
ð Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
X
k
!0
%k
F k
! 0
F
∗
k
! 0
v
u
u
u
t
Á Á Á,
and Norm G (k) defined similarly. Here, F and G refer to the Fourier
transforms of each of two half-map reconstructions, and k is the
spatial frequency vector. The FSC is the cross-correlation of the
half-maps divided by the self-correlation evaluated over some fixed
radius of Fourier space. It is the three-dimensional equivalent of the
two-dimensional Fourier ring correlation [8, 9]. The nominal
global resolution of the map can be obtained based on some
criterion for determining a spatial frequency cutoff
[10, 11]. There have been other resolution metrics previously
proposed in the literature, but due to the widespread acceptance
of the FSC, those will not be addressed here. A detailed background on the FSC and other measures can be found in multiple
reviews [12, 13].
The local resolution is obtained by calculating the FSC in small
fields (patches) and scanning across the map. Several approaches
make use of this “windowed” local resolution evaluation approach
[4]. The central voxel within each windowed patch would then be
assigned the resolution value computed within this patch. The
patches would be scanned across the map, enabling the assignment
of local resolution to the entirety of the reconstructed density.
162
Sriram Aiyer et al.
windowed volumes. The second is a purely Fourier approach which
compares conical sectors of Fourier space of the half-maps for
consistency [6]. It may be thought of as an averaged (and therefore
smoother) version of 3D SSNR that was introduced earlier [7]. We
can therefore assign a direction to each resolution curve (the direction being down the axis of the aforementioned cone); we have
used the term “directional resolution” for this reason. The problems in a reconstructed map cannot be accurately captured using a
single global resolution criterion and should be quantitatively evaluated using both local and directional resolution measures. Highquality reconstructions will show consistency using the given
measures.
The global resolution of the reconstructed map across all spatial
frequencies is typically determined using the Fourier shell correlation (FSC) criterion [5]. To obtain an FSC, it is necessary to split
the dataset into two equal independent halves and reconstruct
individual “half-maps”:
FSC k
ð Þ
P
k
!0
%k
F k
! 0
G
∗
k
! 0
Norm F k
!
Norm G k
!
,
Norm F k
ð Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
X
k
!0
%k
F k
! 0
F
∗
k
! 0
v
u
u
u
t
Á Á Á,
and Norm G (k) defined similarly. Here, F and G refer to the Fourier
transforms of each of two half-map reconstructions, and k is the
spatial frequency vector. The FSC is the cross-correlation of the
half-maps divided by the self-correlation evaluated over some fixed
radius of Fourier space. It is the three-dimensional equivalent of the
two-dimensional Fourier ring correlation [8, 9]. The nominal
global resolution of the map can be obtained based on some
criterion for determining a spatial frequency cutoff
[10, 11]. There have been other resolution metrics previously
proposed in the literature, but due to the widespread acceptance
of the FSC, those will not be addressed here. A detailed background on the FSC and other measures can be found in multiple
reviews [12, 13].
The local resolution is obtained by calculating the FSC in small
fields (patches) and scanning across the map. Several approaches
make use of this “windowed” local resolution evaluation approach
[4]. The central voxel within each windowed patch would then be
assigned the resolution value computed within this patch. The
patches would be scanned across the map, enabling the assignment
of local resolution to the entirety of the reconstructed density.
162
Sriram Aiyer et al.