Conventions

Mathematical conventions

Reference libraries sometimes implement different quantities under the same name. Exacta makes these choices arguments; matching a library requires matching all relevant settings.

Choice	Exacta behavior
HD95 directions	combine="max": maximum of directional percentiles; "pooled": percentile over both directions together
ASSD directions	combine="pooled": mean over both sets of distances; "mean": mean of the two directional means
Voxel surfaces	surface="voxels": foreground boundary voxels with equal weights; face-adjacent background identifies the boundary
Surfels	surface="surfels": marching-cubes surface elements weighted by area (contour lengths in 2D); percentiles use these weights
Meshes	surface="mesh": triangle centers to nearest points on the other mesh, weighted by triangle area (contour segments in 2D); evaluated on CPU
NSD threshold	Distances at or below tolerance are included
Image exterior	Counts as background
Both masks empty	both_empty, default NaN
One mask empty	Dice/IoU: zero; HD/ASSD: one_empty, default infinity; NSD: one_empty, default zero
Undefined ratio	zero_division, default NaN, for precision/sensitivity/specificity and volume ratios

Surface equations

For two nonempty surfaces, let $d_i^P$ denote the distance from prediction surface element $i$ to the target surface, and $d_j^T$ the reverse distance. Distances use physical coordinates from spacing. Weights $w_i^P,w_j^T$ are one for voxel surfaces and area (length in 2D) for surfels or meshes, using the distance geometry described above.

For combine="max", HD95 takes the larger directional 95th percentile:

$$ \mathrm{HD95}_{\mathrm{max}} =\max\left(Q_{0.95}(d^P;w^P),Q_{0.95}(d^T;w^T)\right). $$

$Q_{0.95}$ uses linear interpolation for voxel distances and an area-weighted percentile for surfels and meshes. combine="pooled" takes one percentile after concatenating the two distance/weight sequences. These two operations need not agree.

The default pooled ASSD weights all surface elements together:

$$ \mathrm{ASSD}_{\mathrm{pooled}} =\frac{\sum_i w_i^P d_i^P+\sum_j w_j^T d_j^T} {\sum_i w_i^P+\sum_j w_j^T}. $$

For combine="mean", take each direction's weighted mean separately, then average the two means with equal weight. NSD at tolerance $\tau$ is the weighted fraction within tolerance, where $\mathbf{1}[\cdot]$ is one when the condition holds and zero otherwise:

$$ \mathrm{NSD}_{\tau} =\frac{\sum_i w_i^P\mathbf{1}[d_i^P\leq\tau] +\sum_j w_j^T\mathbf{1}[d_j^T\leq\tau]} {\sum_i w_i^P+\sum_j w_j^T}. $$

Empty surfaces use the explicit policies in the table rather than these ratios.

Reproducing reference definitions

MedPy's HD95 pools directions; MONAI's usual voxel HD95 takes their maximum. DeepMind's surface-distance uses surfel areas. MeshMetrics uses mesh geometry. ASSD changed between MedPy 0.4 (mean of directional means) and 0.5.2 (pooled mean). Record library versions as well as Exacta's arguments when reporting comparisons.

volume_similarity(convention="taha") returns 1 - abs(Vp-Vt)/(Vp+Vt). convention="itk" returns signed 2*(Vp-Vt)/(Vp+Vt). These are different quantities.

Physical axes and numerical limits

Spacing follows the array's axes. A SimpleITK array commonly uses (z, y, x), whereas GetSpacing() returns (x, y, z); reverse it when those are the layouts in use. All distances and NSD tolerances share the spacing unit.

Voxel-surface spacing components must be in [1e-100, 1e100]; surfels and meshes require [1e-50, 1e50] because their geometric calculations include higher powers of coordinates. Unsupported spacing raises even for empty masks. Small floating-point differences near an NSD threshold can change membership; do not silently replace reference precision or enable compiler transformations that change rounding.