Topological correction of brain surface meshes using spherical harmonics

Abstract

Surface reconstruction methods allow advanced analysis of structural and functional brain data beyond what can be achieved using volumetric images alone. Automated generation of cortical surface meshes from 3D brain MRI often leads to topological defects and geometrical artifacts that must be corrected to permit subsequent analysis. Here, we propose a novel method to repair topological defects using a surface reconstruction that relies on spherical harmonics. First, during reparameterization of the surface using a tiled platonic solid, the original MRI intensity values are used as a basis to select either a "fill" or "cut" operation for each topological defect. We modify the spherical map of the uncorrected brain surface mesh, such that certain triangles are favored while searching for the bounding triangle during reparameterization. Then, a low-pass filtered alternative reconstruction based on spherical harmonics is patched into the reconstructed surface in areas that previously contained defects. Self-intersections are repaired using a local smoothing algorithm that limits the number of affected points to less than 0.1% of the total, and as a last step, all modified points are adjusted based on the T1 intensity. We found that the corrected reconstructions have reduced distance error metrics compared with a "gold standard" surface created by averaging 12 scans of the same brain. Ninety-three percent of the topological defects in a set of 10 scans of control subjects were accurately corrected. The entire process takes 6-8 min of computation time. Further improvements are discussed, especially regarding the use of the T1-weighted image to make corrections.

Figure 1

Common problems include topological defects and artifacts. Topological defects such as handles (a) and holes (b) prevent the surface from being homeomorphic with a sphere. By contrast, artifacts (c, highlighted in red) may permit a correct (spherical, genus‐zero) topology but nonetheless are anatomically incorrect and should ideally be eliminated during topology correction. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 2

The full processing pipeline is illustrated using a simple shape with a handle. The original surface mesh (a) is mapped onto a sphere (b). The radii of points in the handle are modified so that they are no longer on the surface of the sphere (c). A regularly sampled grid (d, green) is then overlaid on top of this sphere to create uniform sampling in the parameter space. For each point in the regularly sampled grid (d, green), the intersecting triangle in the spherical mapping (d, blue) is found, and barycentric coordinates within this triangle are used to interpolate a spatial coordinate for this vertex lying on the original tessellated surface mesh. Points with modified radii are not favored during the search for nearest intersecting triangle. These regularly sampled points are forward‐ then reverse‐transformed using spherical harmonics. High‐frequency (e) and low‐pass filtered (f) reconstructed surfaces are derived from the spherical harmonic coefficients. In the regions containing the defect, vertices from the low‐pass filtered reconstruction are patched into the high‐bandwidth reconstruction (g). Finally, a T1 postcorrection step removes any remaining artifacts from the removal of the handle (h). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 3

Defects are labeled as either holes for filling or handles for cutting before the surface is reparameterized. In this hole, the point set is first bisected (a). The original spherical mapping with the bisected points marked is shown in (b). Inner points (previously marked as green) are then modified on the spherical mapping to have a slightly smaller radius (c). When searching for the closest triangle during the reparameterization process, the outer triangles will be strongly favored. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 4

Using a Butterworth filter to filter the coefficients results in a smoother reconstructed surface. Both a boxcar filter (a) and a Butterworth filter (b) leave some ringing artifacts, but using a Butterworth filter reduces the magnitude of the ringing artifacts. The filtered reconstructions were obtained by initially calculating coefficients for l = 1,024, then filtering such that l _l = 64. These ringing artifacts can also be reduced by using a higher order spherical harmonic approximation. The sharpness of a mesh point is the maximum angle between the normals of nearest neighbor polygons, in degrees. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 5

Some detail is lost as more coefficients are filtered. Filtering coefficients such that l _l = 32 or lower result in a surface that no longer resembles the original surface. The numbers shown are the lower value of l, and all surfaces were constructed such that l = 1,024 and then filtered with a Butterworth filter. An artifact (red arrow) is no longer visible in the surfaces reconstructed using spherical harmonics. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 6

Self‐intersections are corrected via localized smoothing. A region containing self‐intersections (a) is smoothed until the cortical surface no longer intersects itself (b). The algorithm is implemented to affect as few points as possible. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 7

Topological correction using spherical harmonics relies on a union between a high‐frequency (l = 1,024) and low‐pass filtered (l _l = 64) surface. The original surface (a) contains three holes (red arrows), two handles (yellow arrows), and two geometrical artifacts (blue arrows). The high‐frequency reconstruction (b) replaces topological defects with sharp spikes but retains more detail than the low‐pass filtered surface (c). The union of the two surfaces (d) optimizes detail retention and topological error correction. T1 postcorrection smooths out the holes and handles (e), so that the corrected surface closely matches the ideal surface (f). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 8

The forward distance error histogram is improved for surfaces corrected using the spherical harmonic method. The histograms are an average compiled for 24 hemispheres. Compared with the uncorrected surfaces, there are more points with low or zero distance error for the spherical harmonic corrected surfaces. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 9

A typical correction of topological defects by spherical harmonics is shown. Holes (a) are filled (b), and handles (c) are cut (d). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 10

Away from topological defects, surface information remains unchanged. The uncorrected surfaces (a and b) are almost identical to the spherical harmonics corrected surfaces (c and d), except in regions near topological defects. The distance in mm is the error between these surfaces and the “gold standard” averaged surface. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 11

Large defect regions in the ventricular region are filled, whereas large defects in the nonbrain area around the orbitofrontal cortex are cut. (a) In the original uncorrected surface, the ventricular defect is marked in red and the defect around the orbitofrontal cortex is marked in yellow. (b) Spherical harmonics correction fills the ventricular defect and cuts the defect around the orbitofrontal cortex. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]