tb_inertia_tensor — 2D `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])

tb_inertia_tensor: input → output

*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)

Usage

Inertia tensor of the point cloud (3,3) (from the central second moment, assuming equal mass and a total mass of 1).

> The detailed description below is the original text — the summary and the headings are translated.

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)。

共分散 C を使うと I = tr(C)·E₃ − C(E₃ は単位行列)と等価。対称・半正定値。

重心中心化のため並進不変。

Returns

-------

np.ndarray, shape (3, 3)

対称な慣性テンソル。

補足:

• 単位は長さ²(質量 1 の等質量点とみなすので密度は入らない)。点群を回転で回すと `R I Rᵀ に写り、固有値(principal_moments)が回転不変量、固有ベクトルが主軸(moment_axes`)。

• 入力は (N,3)、N >= 1(1 点なら零行列)。形状不正・非有限は `ValueError`。

• 共分散 C とは `I = tr(C)·E₃ - C の関係で、C と I の固有ベクトルは同じ、固有値は tr(C) - c_i`。

• 実体(体積)のモーメントではなく サンプル点 のモーメントなので、同じ形でも点密度の偏りで値が変わる。密度を均すなら前段で `voxel_grid_downsample`。決定論的。

2-D 進化レジストリへ橋渡しした 3d の op `inertia_tensor。実装は同じで、呼び出し規約だけ op(v, a, b) に合わせてある。この op に調整点は無く、a も b` も使われない。

References (sample data, literature)

• 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.

Try it in Studio

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

Runnable examples (verified samples that actually call this op)

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

Ops the type connects to (they accept `matrix` as input)

identity · tb_mat_pinv · tb_mat_cond · tb_stat_covariance · tb_stat_correlation · matrix_to_img

Same category (`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).*

tb_inertia_tensor — 2D typed op