typed op• 資料種類:qimage → qimage
• 呼叫:fullseye.apply(img, "tb_qft2", a=0.5, b=0.5)(2-D 的模型是一張圖 + 兩個純量旋鈕 a,b∈[0,1])
*無圖: 該運算子以 qimage 為輸入。從影像開始的 Studio 程式無法到達該型別，請參見下方的可執行範例。*
四元數（超複數）二維傅立葉變換。→ (H, W, 4) 置中頻譜。
> 以下的詳細說明為原文 —— 摘要與標題已翻譯。
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. :func: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.
How it is computed, and why that is not a shortcut
--------------------------------------------------
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 :func:_mu_basis`).
Honest accounting against the channelwise baseline
--------------------------------------------------
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.
• 範例資料目錄(下載 URL / 授權) —— 2-D 用 skimage.data(BSD/公有領域)加合成圖,3-D 給出真實資料源(Stanford/PDS 等)的下載 URL。
• 運算子來歷與參考文獻 —— 該運算子族所依據的研究/方法出處。
• (尚無)
qimage 作為輸入)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_angle_3points · tb_project_points · tb_render_point_depth · tb_statistical_outlier_removal · tb_radius_outlier_removal
*Provenance: ops.py — 2D 運算子登記表。本條目由 tools/opdocs.py md 自動產生(請勿手動編輯)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.