typed op• 数据种类:cimage → feature
• 调用:fullseye.apply(img, "tb_cplx_cr_residual", a=0.5, b=0.5)(2-D 的模型是一张图 + 两个标量旋钮 a,b∈[0,1])
*图为在 128×128 合成输入上实际运行的输出。左为输入，右为输出。点云以俯视散点显示(亮度 = z)，一维序列为折线，体数据为沿 z 的最大值投影，视频为中间帧，复数图像为幅值；无法成像的返回值直接显示数值。*
*旋钮 a 不改变输出(实测: 0.1 / 0.5 / 0.9 相同)。*
*旋钮 b 不改变输出(实测: 0.1 / 0.5 / 0.9 相同)。*
阶段(前置算子 → 本算子，从左到右):
▸ tb_cplx_cr_residual: 处理阶段 (文档网站)
采样复数场的 Cauchy-Riemann 残差——将“这个场是否全纯？”量化为一个数值。
> 以下的详细说明为原文 —— 摘要与标题已翻译。
With f = u + i v sampled on a uniform grid, holomorphy means u_x = v_y and u_y = -v_x (Cauchy-Riemann). This returns the relative residual max(|u_x - v_y|, |u_y + v_x|) / max|grad| (central differences, numpy.gradient): 0 = the samples satisfy CR to the discretisation limit, 2 = the field is the conjugate of a holomorphic one (conj(z) gives exactly 2), values in between = partly analytic or noisy.
Grid convention (it decides the sign of the answer): f[i, j] is the field at z = x0 + j*spacing + i*spacing*1j — rows index the *increasing imaginary* axis, columns the real axis. Image arrays usually run rows *downward*; feeding one directly measures the conjugate field, whose residual is 2, not 0. Flip rows (f[::-1]) to use image data.
Discretisation, honestly: central differences are exact for polynomials of degree <= 2, so f = z**2 returns exactly 0; for higher order the residual floors at O(h^2 * |f'''|) (measured: f = z**3 on a [-1,1]^2 grid returns 1.7e-3 at h and 4.2e-4 at h/2 — a factor 4.00, the expected second order). Read a small value as "consistent with holomorphic at this resolution", never as proof.
A constant field returns 0.0 (it is holomorphic; the 0/0 of the normalisation is resolved by that limit, and stated here rather than left to numpy).
Raises ValueError: not a 2-D array, either dimension below 3 (no central difference exists), non-finite/masked input, over-cap size, non-finite or non-positive *spacing*.
HALCON: no operator (derivate_gauss supplies the real-valued derivatives one would build this from).
Typed bridge of the math op cplx_cr_residual into the 2-D evolution registry: the same implementation, called under the op(v, a, b) convention. a drives spacing (default 1); b is unused.
• 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。
下面的程序已确认可以运行(与图相同的输入)。在 Studio 帮助中，此块会变成按钮，可当场加载并运行。
img_to_cimage 0.50 0.50 tb_cplx_cr_residual 0.50 0.50
下面的示例调用的是底层账本算子 cplx_cr_residual。此桥接算子只是把同一实现适配为 fn(v, a, b) 约定，行为完全相同(只是调用形式不同)。
• math_complex — py -3.11 examples/math_complex.py
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
feature 作为输入)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 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.