Once the reordering and sign adjustment of the eigenvectors have taken place, finding the closest points in the spectral domain between embeddings
and
generates a smooth correspondence map (Fig. 2 ). However, these embedded representations contain slight differences, mostly due to perturbations of the shape isometries such as small changes in distances where the surface undergoes local expansion or compression between meshes. As illustrated on Fig. 4 , nonrigid differences in the spectral embeddings become even more severe in highly convoluted surfaces such as brain cortices. Spectral representations need to be nonrigidly aligned.
Closest points in these nonrigidly aligned embedded representations would reveal corresponding points in both shapes (i.e., in the M-dimensional space (the spectral domain), if the point vi ∈
with coordinates
, is the closest point to vj ∈
with coordinates
, then vi corresponds to vj). It is at this point whereEq. (1) is extended by combining the spectral coordinates,
and
, with the feature vectors,
for nodes in model X, and
for nodes in model Y, to enable spatial regularization in the correspondence map. The extended vectors ofEq. (1) becomes:
where cx and cy are M ×M diagonal matrices that contain weights influencing each spectral coordinate, and β is a K × K diagonal matrix containing the weights for each feature (to emphasize or reflect confidence). Each feature is initially scaled, as inEq. (3) , to fit the values of the Fiedler vector, x (2) (i.e., min(f (k)) = min(x (2)) and max(f (k)) = max(x (2))). The weights c of the spectral coordinates takes into account the smoothness of an eigenvector (measured by its eigenvalue λ(u)) and the confidence in the reordering (measured by the permutation cost Q(u)). Specifically, the weight, c(u), of the uth spectral coordinate is:
where σ is a normalization factor set to
The alignment of these embeddings can be viewed as a nonrigid registration,X = φ(Y ). Fig. 4 shows the alignment challenge where the first three spectral components (x (2),x (3),x (4)) are used as 3D (x, y, z) coordinates for visualization purposes. The Robust Point Matching [18 ] with a Thin Plate Spline-based transformation is often used for 2D or 3D registration. However, with this approach, the final registration depends on the number and choice of the control points. We apply the recent Coherent Point Drift method [41 (link)] which is scalable to N dimensions, fast, and demonstrates excellent performance in this application.
To increase speed in FOCUSR, we take advantage of the property of the Coherent Point Drift method that a continuous transformation derived from a subset of the points can be applied to all nodes of the dense embeddings. In our case, we subsampleX and Y by taking randomly a few points (in our experiments we chose 1% of the total number of vertices, roughly 1000 points).
Closest points in these nonrigidly aligned embedded representations would reveal corresponding points in both shapes (i.e., in the M-dimensional space (the spectral domain), if the point vi ∈
, is the closest point to vj ∈
, then vi corresponds to vj). It is at this point where
for nodes in model X, and
for nodes in model Y, to enable spatial regularization in the correspondence map. The extended vectors of
where cx and cy are M ×M diagonal matrices that contain weights influencing each spectral coordinate, and β is a K × K diagonal matrix containing the weights for each feature (to emphasize or reflect confidence). Each feature is initially scaled, as in
where σ is a normalization factor set to
The alignment of these embeddings can be viewed as a nonrigid registration,
To increase speed in FOCUSR, we take advantage of the property of the Coherent Point Drift method that a continuous transformation derived from a subset of the points can be applied to all nodes of the dense embeddings. In our case, we subsample
