dc_rpca_lowrank — 2D `decomposition` op

• Data kinds: image → image

• Call: fullseye.apply(img, "dc_rpca_lowrank", a=0.5, b=0.5) (the 2-D model is one image plus two scalar knobs a,b∈[0,1])

dc_rpca_lowrank: 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.*

Sweeping knob a (0.1 / 0.5 / 0.9, the other knob at its default):

▸ dc_rpca_lowrank: knob a sweep (docs site)

*Knob b does not change the output (measured: identical at 0.1 / 0.5 / 0.9).*

On other images (synthetic scene / photo / coins. Top row: inputs, bottom row: their outputs. Knobs at default):

▸ dc_rpca_lowrank: other inputs (docs site)

*The 4th column is a colour (H,W,3) input. This op handles colour without crossing channels (it does not convolve the colour channel as a third spatial axis).*

Usage

Robust-PCA low-rank (background) part.

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

画像行列 `M を M = L + S(低ランク L + スパース S`)に分解する

Principal Component Pursuit を inexact ALM(Lin/Chen/Ma 2010)で解き、`L`

を返す。スパース項の重みは `λ = (0.5 + 1.5*a) / sqrt(max(m, n))(m, n`

は分解時の行列寸法)で、`a が大きいほど S が疎になり、その分 L` に

残る成分(ランク)が増える。`a=0 では λ` が小さく、ほとんどの変動が

`S に吸われて L はのっぺりする。b` は未使用。反復は最大 60 回、

収束判定は `||M - L - S||_F <= 1e-7 * ||M||_F`。

長辺が 64 を超える画像は 64 に縮小して分解し、`L` だけを線形補間で元の

大きさに戻す(低ランク部は縮小に耐えるため)。入力は [0,1] の 2 次元 float64

に揃え(3 次元はチャネル平均)、返り値も同形・[0,1] に clip。全零画像は入力を

そのまま返す。BLAS のスレッド数は分解中だけ `fsthreads` で絞る(小行列の

SVD ではスレッドが多いほど遅いため)。例外時は fail-soft で入力のクリップ版が

返り、台帳に記録される。

「行や列にわたって繰り返す構造」(縞、グラデーション、周期パターン)を背景と

みなす分解なので、単一の 2 次元画像でも織物・シート・ディスプレイ画素などの

周期背景から孤立欠陥を分離するのに向く。自然画像のような非周期背景では

`L` は単なる低ランク近似で、意味のある背景にならない。対になる欠陥側は

`dc_rpca_sparse(L + (S - 0.5) == 入力`)。エッジ保存の平滑で構造を

取りたいなら `dc_structure_texture`。

Detailed usage guide

• gallery2d_texture_freq family guide

Background guides (the physics and conventions behind this op)

• blas_threads_and_memory — 行列分解が遅い理由の知識 — BLAS スレッド・キャッシュ・メモリ配置

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.

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.

dc_rpca_lowrank 0.50 0.50

▸ Load this pipeline · Load & run

Runnable examples (verified samples that actually call this op)

• blas_thread_budget — py -3.11 examples/blas_thread_budget.py

• gallery2d_texture_freq — py -3.11 examples/gallery2d_texture_freq.py

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

identity · gaussian · mean_box · bilateral · unsharp · median · min_filter · max_filter

Same category (`decomposition`)

dc_structure_texture · dc_texture_residual · dc_rpca_sparse · dc_retinex · dc_local_contrast_norm · dc_homomorphic

*Provenance: ops.py — 2D operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*

dc_rpca_lowrank — 2D decomposition op