subsequently focus on the candidates with strong curvature, as specified by the
second derivative tensor [5]. Similarly, strong anisotropic diffusion, which also
bases on the second derivative tensor, and subsequent thresholding for the prevailing features has been used for initial detection of candidates, which were
afterwards filtered according to the volume of the objects [6]. Subsequently, the
candidates were clustered according to efficiently computed, rotation-invariant
properties in an entirely data-driven workflow.
9.3.2 Template Matching
In many cases, however, structures of the complexes of interest are at least
approximately known. Template-based identification and detection is then preferred
because it is more specific and efficient. Cross-correlation based pattern recognition
algorithms use a template structure (‘template matching’) [7] depicting the complex
of interest for particle localization. In essence, the tomographic volume is screened
for features that closely resemble the 3D template. For optimal performance, the
provided template structure has to be adapted to the acquisition conditions. In
particular, the average defocus modulating the phase contrast and the information
loss due to the missing wedge has to be considered. To this end, the template
structure is convoluted with a point-spread function (PSF), i.e., a function that
describes how a point is deformed in the tomographic imaging.
In general, template matching provides a measure of the local similarity of the
template and the tomogram. The most common similarity metric in image processing is cross-correlation. For this purpose, the template structure is shifted
through the tomogram in all three spatial directions, voxel by voxel. This can be
accomplished efficiently with the help of Fast Fourier Transforms (FFT) [7, 8]. In
the cross correlation volume, the cross-correlation coefficient (CCC) of the template
and the local tomogram is assigned to the respective central voxel. To account for
the missing wedge and contrast variations within the tomogram, the CCC is constrained to commonly sampled segments in Fourier space and computed on volumes locally normalized within a tight mask [8, 9]. Because the macromolecular
complexes of interest can be oriented arbitrarily in the tomogram, the voxel-wise
cross-correlation is determined using a predefined set of 3D orientations for the
template structure. To avoid excessive computational demands for template
matching the angular sampling must not be too fine. The required sampling depends
on the voxel size used for the search and on the diameter of the template. In many
cases, the angular difference between two adjacent orientations is above 10°. From
the computed cross correlation volumes only the highest CCC is typically retained
for each voxel because the required storage space would be enormous otherwise.
A list of peaks can be compiled from the resulting 3D correlation volume, indicating positions and corresponding orientations of candidate particles [7].
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second derivative tensor [5]. Similarly, strong anisotropic diffusion, which also
bases on the second derivative tensor, and subsequent thresholding for the prevailing features has been used for initial detection of candidates, which were
afterwards filtered according to the volume of the objects [6]. Subsequently, the
candidates were clustered according to efficiently computed, rotation-invariant
properties in an entirely data-driven workflow.
9.3.2 Template Matching
In many cases, however, structures of the complexes of interest are at least
approximately known. Template-based identification and detection is then preferred
because it is more specific and efficient. Cross-correlation based pattern recognition
algorithms use a template structure (‘template matching’) [7] depicting the complex
of interest for particle localization. In essence, the tomographic volume is screened
for features that closely resemble the 3D template. For optimal performance, the
provided template structure has to be adapted to the acquisition conditions. In
particular, the average defocus modulating the phase contrast and the information
loss due to the missing wedge has to be considered. To this end, the template
structure is convoluted with a point-spread function (PSF), i.e., a function that
describes how a point is deformed in the tomographic imaging.
In general, template matching provides a measure of the local similarity of the
template and the tomogram. The most common similarity metric in image processing is cross-correlation. For this purpose, the template structure is shifted
through the tomogram in all three spatial directions, voxel by voxel. This can be
accomplished efficiently with the help of Fast Fourier Transforms (FFT) [7, 8]. In
the cross correlation volume, the cross-correlation coefficient (CCC) of the template
and the local tomogram is assigned to the respective central voxel. To account for
the missing wedge and contrast variations within the tomogram, the CCC is constrained to commonly sampled segments in Fourier space and computed on volumes locally normalized within a tight mask [8, 9]. Because the macromolecular
complexes of interest can be oriented arbitrarily in the tomogram, the voxel-wise
cross-correlation is determined using a predefined set of 3D orientations for the
template structure. To avoid excessive computational demands for template
matching the angular sampling must not be too fine. The required sampling depends
on the voxel size used for the search and on the diameter of the template. In many
cases, the angular difference between two adjacent orientations is above 10°. From
the computed cross correlation volumes only the highest CCC is typically retained
for each voxel because the required storage space would be enormous otherwise.
A list of peaks can be compiled from the resulting 3D correlation volume, indicating positions and corresponding orientations of candidate particles [7].
9 Structural Biology in Situ Using Cryo-Electron …
239