HausdorffDistanceMetric percentile returns NaN for a missed prediction, dropping failures from the dataset average

Describe the bug

HausdorffDistanceMetric(percentile=...) returns nan when one of the two masks is empty, where percentile=None returns inf for the same input.

nan is not a quieter way of saying inf here. It is this metric's "not applicable" sentinel — it is what both-masks-empty returns — and do_metric_reduction excludes it from the average. A prediction that missed the structure entirely is therefore removed from a dataset score rather than counted as the worst case, and the reported HD95 improves as the model finds fewer structures.

To Reproduce

import torch
from monai.metrics import HausdorffDistanceMetric

gt = torch.zeros(1, 1, 64, 64)
gt[0, 0, 20:28, 20:28] = 1.0
pred = torch.zeros(1, 1, 64, 64)          # the model found nothing

for percentile in (None, 50, 95, 99, 100):
    m = HausdorffDistanceMetric(include_background=True, percentile=percentile)
    m(y_pred=pred, y=gt)
    print(percentile, m.get_buffer().flatten().tolist())

# None -> [inf]
# 50   -> [nan]
# 95   -> [nan]
# 99   -> [nan]
# 100  -> [nan]

The effect over a dataset. 100 images, every non-empty prediction identically 4px off, so the only variable is how many images the model misses:

A model that finds nothing in 99 images out of 100 reports the same HD95 as one that finds the structure every time. A runnable script that prints the whole table is at https://github.com/asifuddin01/A-PR.

Expected behavior

inf, matching percentile=None, so the case is scored rather than discarded. nan stays reserved for both masks empty, where there is genuinely nothing to measure.

Cause

get_surface_distance reports an infinite distance for every boundary voxel when a mask is empty, so an all-infinite tensor reaches _compute_percentile_hausdorff_distance. torch.quantile interpolates linearly between the two order statistics straddling the requested rank:

inf + (inf - inf) * 0.95  ->  nan

The quantile of a constant sequence is that constant, so the nan is an artefact of the interpolation rather than a property of the distances. percentile=None escapes it because .max() does not interpolate.

Environment

MONAI version: 1.6.0
Numpy version: 2.5.2
Pytorch version: 2.14.0
MONAI flags: HAS_EXT = False, USE_COMPILED = False, USE_META_DICT = False
MONAI rev id: eccefc57550b111ed781d82249dfe77872a0e918

Also reproduced on current dev at c0d1ec1.

Additional context

Only the percentile path is affected. SurfaceDistanceMetric returns inf for the same input and SurfaceDiceMetric returns 0.0; both are already honest about a miss.

Separately, #9033 fixes percentile=0 being treated as unset by the if not percentile: guard. It does not cover this: it adds a .min() branch above the torch.quantile call and leaves the interpolation as it is. The two are independent.

I have a fix and regression tests ready and will open a PR against this issue.