typed op• Data kinds: points → matrix
• Call: fullseye.apply(img, "tb_inertia_tensor", a=0.5, b=0.5) (the 2-D model is one image plus two scalar knobs a,b∈[0,1])
*The figure is the actual output on a synthetic 128×128 input. Left: input, right: output. Point clouds are drawn as a top-down scatter (brightness = z), 1-D series as a line plot, volumes as the maximum-intensity projection along z, videos as the middle frame, complex images as magnitude; return values that are not pictures are shown as the values themselves.*
*Knob a does not change the output (measured: identical at 0.1 / 0.5 / 0.9).*
*Knob b does not change the output (measured: identical at 0.1 / 0.5 / 0.9).*
Stages (the ops that come before → this op, left to right):
▸ tb_inertia_tensor: stages (docs site)
On other images (synthetic scene / photo / coins. Top row: inputs, bottom row: their outputs. Knobs at default):
▸ tb_inertia_tensor: other inputs (docs site)
Inertia tensor of the point cloud (3,3) (from the central second moment, assuming equal mass and a total mass of 1).
I_xx = mean(y²+z²), I_yy = mean(x²+z²), I_zz = mean(x²+y²), I_xy = -mean(xy), I_xz = -mean(xz), I_yz = -mean(yz). With the covariance C this is equivalent to I = tr(C)·E₃ − C (E₃ is the identity matrix). Symmetric, positive semi-definite. Translation-invariant because of centroid centring.
np.ndarray, shape (3, 3)
Symmetric inertia tensor.
Supplement:
• Units are length² (points are treated as equal masses of mass 1, so density does not enter). Rotating the point cloud maps it to R I Rᵀ; the eigenvalues (principal_moments) are rotation invariants and the eigenvectors are the principal axes (moment_axes).
• The input is (N,3) with N >= 1 (1 point gives the zero matrix). Invalid shape or non-finite values raise ValueError.
• Relation to the covariance C: I = tr(C)·E₃ - C; C and I have the same eigenvectors, and the eigenvalues are tr(C) - c_i.
• These are moments of the sample points, not of the solid (volume), so the same shape gives different values with uneven point density. To even out density, apply voxel_grid_downsample beforehand. Deterministic.
The 3d op inertia_tensor bridged into the 2-D evolution registry. The implementation is the same; only the calling convention is adapted to op(v, a, b). This op has no tuning points; neither a nor b is used.
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
The program below has been verified to run (same input as the figure). In Studio's help this block becomes buttons that load and run it on the spot.
img_to_points 0.50 0.50 tb_inertia_tensor 0.50 0.50
▸ Load this pipeline · Load & run
The examples below call the underlying ledger op inertia_tensor. This bridge op is the same implementation adapted to the fn(v, a, b) convention, so the behaviour carries over unchanged (only the call form differs).
• moment_invariants — py -3.11 examples_3d/moment_invariants.py
matrix as input)identity · tb_mat_pinv · tb_mat_cond · tb_stat_covariance · tb_stat_correlation · tb_intrinsics_to_fullseye · tb_intrinsics_to_carla · matrix_to_img
typed)tb_points_to_voxel · tb_estimate_point_normals · tb_iss_keypoints · tb_project_points · tb_render_point_depth · tb_statistical_outlier_removal · tb_radius_outlier_removal · tb_voxel_grid_downsample
*Provenance: ops.py — 2D operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.