typed op• Data kinds: qimage → qimage
• Call: fullseye.apply(img, "tb_qft2", a=0.5, b=0.5) (2-D model = एक image + दो scalar knobs a,b∈[0,1])
*Figure synthetic 128×128 input पर सच में चलाकर मिला output है। बाईं ओर input, दाईं ओर output। Point clouds ऊपर से देखे गए scatter के रूप में (brightness = z), 1-D series line के रूप में, volumes z दिशा में maximum-intensity projection के रूप में, videos बीच के frame के रूप में, complex images amplitude के रूप में, और जो return values चित्र नहीं बन सकते वे स्वयं values के रूप में दिखाए जाते हैं।*
*Knob a output नहीं बदलता (मापा गया: 0.1 / 0.5 / 0.9 पर एक जैसा)।*
*Knob b output नहीं बदलता (मापा गया: 0.1 / 0.5 / 0.9 पर एक जैसा)।*
Stages (पहले आने वाले ops → यह op, बाएँ से क्रम में):
▸ tb_qft2: stages का figure (docs site)
दूसरी images पर भी (synthetic scene / photo / coins। ऊपर की row में inputs, नीचे की row में उनके outputs। Knobs default पर):
▸ tb_qft2: दूसरी images पर output (docs site)
Quaternion (hypercomplex) 2-D Fourier transform। → (H, W, 4) centred spectrum।
> नीचे का विस्तृत विवरण मूल text में है — सारांश और headings का अनुवाद हो चुका है।
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.
• Sample data catalog (DL URL / license) — 2-D के लिए skimage.data (BSD/public) + synthetic, 3-D के लिए real data sources (Stanford/PDS आदि) के DL URL।
• Operators का स्रोत और संदर्भ — इस op family के मूल research/methods के स्रोत।
नीचे का program सच में चलता है, यह जाँचा गया है (figure वाला ही input)। Studio की help में यह block buttons बन जाता है, जिनसे इसे वहीं load करके चलाया जा सकता है।
img_to_rgb 0.50 0.50 tb_rgb_to_quaternion 0.50 0.50 tb_qft2 0.50 0.50
▸ यह pipeline load करें · Load करके चलाएँ
नीचे के उदाहरण मूल ledger op qft2 को call करते हैं। यह bridge op वही implementation है, बस fn(v, a, b) convention के अनुसार ढाला गया है, इसलिए व्यवहार वैसा ही लागू होता है (केवल call का रूप अलग है)।
• quaternion_monogenic — py -3.11 examples/quaternion_monogenic.py
qimage को input के रूप में लेते हैं)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. यह per-op note tools/opdocs.py md से अपने-आप generate होता है (हाथ से edit न करें)।*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.