typed op• データ種: qimage → qimage
• 呼び出し: fullseye.apply(img, "tb_qft2", a=0.5, b=0.5) (2-D は 1 画像 + 2 スカラつまみ a,b∈[0,1] のモデル)
*図は合成の入力 128×128 で実際に走らせた出力。左が入力、右が出力。点群は上から見た散布(明るさ = z)、1-D 列は折れ線、体積は z 方向の最大値投影、動画は中央フレーム、複素画像は振幅、絵にならない返り値は値そのもの。*
*つまみ a は出力を変えない(実測: 0.1 / 0.5 / 0.9 で同一)。*
*つまみ b は出力を変えない(実測: 0.1 / 0.5 / 0.9 で同一)。*
段階(前置きの op → この op。左から順):
▸ tb_qft2: 段階の図 (docs サイト)
別の画像でも(合成シーン / 写真 / 硬貨。上段が入力、下段がその出力。つまみは既定):
▸ tb_qft2: 別の画像での出力 (docs サイト)
四元数（超複素数、hypercomplex）2次元フーリエ変換。→ (H, W, 4) の中心化されたスペクトル。
> 以下の詳細説明は原文のままです —— 要約と見出しは訳出済み。
The colour analogue of complexops.cx_fft: a colour image is transformed as one hypercomplex signal rather than three unrelated real ones. The kernel is exp(-mu * 2*pi*(u*x/W + v*y/H)) for a unit pure quaternion *mu* (default: the grey axis of RGB, Sangwine's choice), and because quaternions do not commute the kernel can be applied on either side:
• side="left" — F[u,v] = sum_{x,y} E(x,y,u,v) * f[x,y]
• side="right" — F[u,v] = sum_{x,y} f[x,y] * E(x,y,u,v)
The argument is required. Left and right are not a sign convention. On the fuzzer's (32, 32) dichromatic render they differ by max|F_L - F_R| = 19.11 against a peak modulus of 1045 (1.8 % of full scale) and on a random colour field by 33.35 against 892.9 (3.7 %; another seed gives 34.05 against 892) — with no exception and no NaN to mark the difference. Mixing them across a round trip is much worse, because there the disagreement is not attenuated by the spectrum's dynamic range: iqft2(qft2(q, "left"), "right") returns an image whose error reaches 1.113 on data whose own range is 0.9994 — a completely different picture that still looks like a picture. (On the grey-axis-dominated dichromatic render the same mistake costs only 0.054 against a range of 1.076, which is the dangerous case: a 5 % error is exactly the size that survives a visual check.)
The spectrum is returned centred (DC at the array centre, via fftshift), matching the convention complexops.cx_fft established for the cimage sort. iqft2 un-centres before inverting, and the round trip is exact: measured max|iqft2(qft2(q, s), s) - q| = 2.22e-15 for both sides on a standard-normal (32, 32, 4) field.
Every quaternion splits as q = A + B*nu with A, B in the commutative subfield generated by mu (the *symplectic decomposition*, Ell & Sangwine 2007). The kernel commutes with A and *anti*-commutes past nu, so the whole transform reduces to two ordinary complex FFTs — for the left transform both with the standard kernel, for the right one of them with the conjugate kernel. That reduction is what makes left and right differ, and it is verified against a brute-force O(N^2) quaternion DFT written straight from the definition, on a 4x4 image, for three different mu and both sides: the largest disagreement over all six combinations is 8.2e-15. The fast path is checked against the definition, not against itself. The choice of the internal nu is likewise verified not to matter (two different nu, max difference 1.4e-14 — see _mu_basis).
Because the decomposition above is *linear* in the channels, the QFT is a fixed recombination of the three per-channel complex FFTs: rebuilding qft2(q, "left") from three numpy.fft.fft2 calls on the R, G and B planes agrees to max|err| = 1.14e-13. So this transform buys no information a channelwise FFT does not already contain, and this module does not claim it does. It also does not buy speed — it moves four real transforms' worth of data where the channelwise route moves three, and pays for the symplectic pack/unpack on top: measured on (256, 256), best of 20, 8.246 ms against 3.409 ms, i.e. 2.42x slower (run to run, 2.3x-2.4x). What it buys is that the four numbers stay one algebraic object, so a rotor can be applied to the spectrum and the colour meaning of mu survives the transform.
Raises ValueError: *qimage* is not a valid (H, W, 4) field; *side* is not 'left' / 'right'; *mu* is not a finite non-zero 3-vector.
Typed bridge of the quat op qft2 into the 2-D evolution registry: the same implementation, called under the op(v, a, b) convention. This op has no tunable parameter; a and b are unused.
• サンプルデータ カタログ(DL URL / ライセンス) — 2-D は skimage.data(BSD/public)+ 合成、3-D は実データ源(Stanford/PDS 等)の DL URL。
• 演算子の来歴・参考文献 — この op 族の元になった研究/手法の出典。
下のプログラムは実際に走ることを確かめてある(図と同じ入力)。Studio のヘルプではこのブロックがボタンになり、その場で読み込んで実行できる。
img_to_rgb 0.50 0.50 tb_rgb_to_quaternion 0.50 0.50 tb_qft2 0.50 0.50
次の例は元の台帳 op qft2 を呼ぶもの。この橋渡し op は同じ実装を fn(v, a, b) 規約に合わせただけなので、挙動はそのまま当てはまる(呼び出し形だけ違う)。
• quaternion_monogenic — py -3.11 examples/quaternion_monogenic.py
qimage を入力に取れる)identity · tb_quaternion_to_rgb · tb_quat_norm · tb_quat_conjugate_image · tb_quat_normalize_image · tb_monogenic_amplitude · tb_monogenic_phase · tb_monogenic_orientation
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. この per-op ノートは tools/opdocs.py md が自動生成(手編集しない)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.