alpha_expansion — SEGMENTATION graph op

• Data kinds: image2d → table

• Call: import fullseye as fs; fs.ledger.alpha_expansion(image, means: 'Optional[Sequence[float]]' = None, *, lam: 'float' = 1.0, pairwise: 'str' = 'potts', truncation: 'float' = 2.0, connectivity: 'int' = 4, max_cycles: 'int' = 10, init=None) -> 'Dict[str, object]' (to call the implementation directly, import seggraph; seggraph.alpha_expansion(image, means: 'Optional[Sequence[float]]' = None, *, lam: 'float' = 1.0, pairwise: 'str' = 'potts', truncation: 'float' = 2.0, connectivity: 'int' = 4, max_cycles: 'int' = 10, init=None) -> 'Dict[str, object]'; from the registry, opssegmentation.get("alpha_expansion"))

Usage

Multi-label α-expansion (Boykov–Veksler–Zabih 2001): E(f) = Σ (I_p − μ_(f_p))² + λ Σ w_pq V(f_p, f_q), where V is a metric (alpha_expansion).

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

`means = 各ラベルの平均 μ_1..μ_L(L ≥ 2。省くと画素の値の 1/6・1/2・5/6 分位の 3 ラベル)。V は "potts"([α ≠ β]、c = 1)か "truncated_linear"`

(min(|α − β|, T)、T ≥ 1 なら計量、c = min(L − 1, T))。1 周 = α = 1..L の expansion を順に。各移動は 2 値の graph cut で

厳密に解き、エネルギーが厳密に下がる時だけ 受け入れる。1 周で 1 度も受け入れなければ止まる(論文と同じ)。

返す: `labels(1..L)、energy、energies(受け入れの列、単調減少)、move_minima`(各移動の 2 値問題の最小、

今の値以下 = 門)、`c、bound_factor` = 2c(Theorem 6.1: E(f̂) ≤ 2c E(f*))。

Detailed usage guide

• halcon_segmentation family guide

Background guides (the physics and conventions behind this op)

• graph_hierarchy_and_thresholds — グラフ・階層・閾値の定理でつくる分割 — 何が厳密で、どこで割れ方が倒れるか

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.

• The canonical algorithm (author, year) and its uses are named in the family usage guide above.

Runnable examples (verified samples that actually call this op)

• poc_graph_hierarchy_segmentation — py -3.11 examples/poc_graph_hierarchy_segmentation.py

Ops the type connects to (they accept table as input)

class_ndim_norm

Same category (graph)

graph_cut_binary · statistical_region_merging · max_tree · area_opening_attr · quasi_flat_zones · alpha_tree · hierarchical_watershed · ultrametric_contour_map

*Provenance: seggraph.py — SEGMENTATION 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.