Data Interpolation

We start by reconstructing a continuous tensor field in the volume through trilinear interpolation. In this scheme the value of a tensor at any point inside the voxel is a linear combination of the $8$ values at its corners and is completely determined by them. Since the coefficients of this linear combination are independent of the tensor indexes, the linear combination of the tensors can be done component-wise.

$${{\mathbf T}(x,y,z) = {\mathbf T}_{ijk}~(1-x)(1-y)(1-z)+}$$

$$+ {\mathbf T}_{i+1,jk}~x(1-y)(1-z) + {\mathbf T}_{i,j+1,k}~(1-x)y(1-z)$$

$$+ {\mathbf T}_{ij,k+1}~(1-x)(1-y)z + {\mathbf T}_{i+1,j,k+1}~x(1-y)z$$

$$+ {\mathbf T}_{i,j+1,k+1}~(1-x)yz + {\mathbf T}_{i+1,j+1,k}~xy(1-z)$$

$$+ {\mathbf T}_{i+1,j+1,k+1}~xyz$$

We can use trilinear component-wise interpolation because symmetric tensors form a linear subspace in the tensor space: any linear combination of symmetric tensors remains a symmetric tensor, i.e., symmetric tensors are closed under linear combination (the manifold of symmetric tensors is not left). Component-wise interpolation is sufficient for our purposes; more sophisticated interpolation methods, however, would better preserve the eigenvalues along an interpolation path [14]. On the other hand, component-wise interpolation of eigenvectors and eigenvalues themselves would not lead to correct results, since a linear interpolation between two unit vectors is not a unit vector anymore - the interpolated eigenvector value would leave the manifold of unit vectors. In addition, there can be a correspondence problem in the order of the eigenvalues. Various types of tensor interpolation are discussed, for example, in [11].

 

 

Figure 3: Comparison of non-filtered (top) and MLS filtered fibers (bottom). Note the more regular and smoother behavior of the filtered fibers