The distance-transform subsystem is topology-only and requires a non-empty regular 2D tree domain. It measures every node support against its foreground A4 contour in the original image grid. Tree-construction adjacency and node altitudes do not change this metric.

Two coordinated families are public:

unsuffixed names, such as MAX_DIST, select the adaptive-A8 DIFT approximation;
names ending in _EXACT, such as MAX_DIST_EXACT, select the exact Euclidean distance transform, whose internal cost field is squared.

AttributeGroup::DistTransf in C++ and Attribute.Group.DIST_TRANSF in Python contain the 29 approximate scalars. AttributeGroup::DistTransfExact and Attribute.Group.DIST_TRANSF_EXACT contain the 29 matching exact scalars. Request both groups to obtain all 58 attributes. Results follow canonical scalar-ordinal order; see Attributes for result layouts.

Mathematical contract

For a node support X, write z(x) = d(x)^2, where d(x) is the Euclidean distance in pixels from x to the foreground A4 contour of X. In the approximate family, z denotes the corresponding adaptive-A8 DIFT cost. Let n = |X|.

The maximum and squared-distance summaries are:

MAX_DIST = sqrt(max z) = max d;
MAX_SQUARED_DIST = max z;
DIST_SQUARED_SUM = sum z;
DIST_SQUARED_MEAN = (sum z) / n;
DIST_RMS = sqrt(DIST_SQUARED_MEAN);
DIST_SQUARED_VARIANCE = mean(z^2) - mean(z)^2.

The real-distance summaries use d = sqrt(z):

DIST_SUM = sum d;
DIST_MEAN = (sum d) / n;
DIST_VARIANCE = mean(d^2) - mean(d)^2.

The maximum center is the smallest row-major support pixel among tied maximizers. Maximum-plateau area counts every maximizer, and plateau centroid coordinates are arithmetic means, so they may be fractional.

The sparse histogram stores (z, count) bins in increasing squared-distance order. DIST_MEDIAN, DIST_Q25, DIST_Q75, and DIST_Q90 are lower empirical quantiles of d: the smallest distance whose cumulative count reaches ceil(p*n). DIST_MODE chooses the smallest distance when frequencies tie. DIST_ENTROPY is Shannon entropy in bits. DIST_POSITIVE_AREA counts samples with z > 0, and DIST_LEVEL_COUNT counts represented squared-distance levels, including zero.

Distance-weighted geometry

Spatial descriptors use weight w(x) = d(x). The row and column centroids are sum(w*coordinate) / sum(w). If every distance is zero, the centroid falls back to the ordinary support centroid and all weighted central moments are zero.

DIST_WEIGHTED_CENTRAL_MOMENT_20 is the unnormalized weighted column spread, DIST_WEIGHTED_CENTRAL_MOMENT_02 is the unnormalized weighted row spread, and DIST_WEIGHTED_CENTRAL_MOMENT_11 is the unnormalized row-column mixed moment. Axis orientation is reported in degrees. Eccentricity is the major/minor second-moment eigenvalue ratio: isotropic or point-degenerate fields return 1, and line-degenerate fields saturate at 1e6.

Every name above has an exact counterpart obtained by appending _EXACT.

Units

Quantity	Unit
maximum distance	pixels
maximum squared distance, squared mean	pixel²
squared sum	pixel² accumulated over support samples
squared-distance variance	pixel⁴
real-distance sum, mean, quantiles, mode, RMS	pixels
real-distance variance	pixel²
entropy	bits
areas and level counts	counts
centroids	zero-based pixel coordinates
weighted central moments	pixel³
orientation	degrees
eccentricity	dimensionless

Related guides

Distance-transform architecture: exact and approximate providers, scheduling, queue policy, oracles, and complexity.
Attribute catalog: scalar names, groups, and concise public descriptions.
Benchmark guide: sensitivity and timing protocols for this attribute family.