Fiber Similarity Metrics

Based on point distances

Note: Not every following similarity measure is symmetric or satisfies the triangle inequality!

Average point-by-point-distance

d(F_i,F_j) = \text{mean}_{p_k\in F_i,q_k \in F_j} \| p_k - q_k \|

Closest-point-distance (Corouge)

d_c(F_i,F_j) = \min\limits_{p_k\in F_i,p_l \in F_j} \| p_k - _l \|

Mean of closest-point-distances (Corouge)

d_m(F_i,F_j) = \text{mean}_{p_k\in F_i}\min\limits_{p_l \in F_j} \| p_k - p_l \|

A derived symmetric measure is:

d_{mc}(F_i,F_j) = \frac{1}{2} \left( d_m(F_i, F_j) + d_m(F_j, F_i) \right)

This approach or just selecting the minimum or maximum is often used to make a similarity measure symmetric.

Mean of thresholded closest-point-distance (Zhang)

d_t(F_i,F_j) = \text{mean}_{p_k\in F_i, \| p_k - p_l \| \ge t}\min\limits_{p_l \in F_j} \| p_k - p_l \|

Mean weighted-closest-point-distance (Laidlaw)

Z = \sum\limits_{k=1}^m e^{|k-(m+1)/2|^2/\sigma^2}

\alpha_k = \frac{1}{Z}e^{|k-(m+1)/2|^2/\sigma^2}

d_{ij} = \text{mean}_{p_k \in F_i} \left( \alpha_i^k \cdot \min\limits_{p_l \in F_j} \| p_k - p_l \| \right)

d_w(F_i,F_j) = \max ( d_{ij}, d_{ji} )

where the tracts endings are more weighted.

Hausdorff point-distance

d_h(F_i,F_j) = \max\limits_{p_k \in F_i}\min\limits_{p_l \in F_j} \| p_k - p_l \|

Frechet point-distance

d_h(F_i,F_j) = \min\limits_{p_k \in F_i}\max\limits_{p_l \in F_j} \| p_k - p_l \|

Weighted (Chen)

d_m(F_i,F_j) = \text{mean}_{p_k\in F_i}\min\limits_{p_l \in F_j} \| p_k - p_l \|

d_{mc}(F_i,F_j) = \frac{1}{2} \left( d_m(F_i, F_j) + d_m(F_j, F_i) \right)

AvgN(F) = \text{mean} \quad \text{fractional anisotropy}

AvgR(F) = \text{mean} \quad \text{relative anisotropy}

AvgC(F) = \text{mean} \quad \text{discrete cuvature}

d_w(F_i,F_j) = \alpha\,d_{mc}(F_i, F_j) + \beta \left| AvgN(F_i) - AvgN(F_j)\right| + \gamma \left| AvgC(F_i) -AvgC(F_j) \right|

where \alpha, \beta, \gamma \in [0,1).

Maximal overlap

This is very tricky and achieved via the Gaussian-Process framework. Look in the INRIA paper: Unsupervised White Matter Fiber Clustering and Tract Probability Map Generation: Applications of a Gaussian Process framework for White Matter Fibers