fourier op• 数据种类:qimage → qimage
• 调用: import fullseye as fs; fs.ledger.qft2(qimage, side, mu=None) -> 'np.ndarray'(要直接调用实现,import quatimage; quatimage.qft2(qimage, side, mu=None) -> 'np.ndarray';从台账取用则 opsquat.get("qft2"))
四元数(超复数)2-D 傅里叶变换。→ 中心化的 (H, W, 4) 频谱。
> 以下的详细说明为原文 —— 摘要与标题已翻译。
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.
• 示例数据目录(下载 URL / 许可证) —— 2-D 用 skimage.data(BSD/公有领域)加合成图,3-D 给出真实数据源(Stanford/PDS 等)的下载 URL。
• 算子来历与参考文献 —— 该算子族所依据的研究/方法出处。
• 算法的正典(作者、年份)与用途见上面的族使用指南。
• quaternion_monogenic — py -3.11 examples/quaternion_monogenic.py
qimage 作为输入)quaternion_to_rgb · quat_norm · quat_conjugate_image · quat_normalize_image · quat_image_multiply · monogenic_amplitude · monogenic_phase · monogenic_orientation
fourier)*Provenance: quatimage.py — QUAT 算子登记表。本条目由 tools/opdocs.py md 自动生成(请勿手工编辑)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.