riesz_transform — QUAT `riesz` op

• Data kinds: image2d → qimage

• Call: import quatimage; quatimage.riesz_transform(image) -> 'np.ndarray' (or opsquat.get("riesz_transform"))

Usage

The 2-D Riesz transform of an image, as a pure quaternion field. → (H, W, 4).

The isotropic generalisation of the Hilbert transform: a *pair* of filters

with frequency responses `-i*u/|w| and -i*v/|w|`, returned as the

quaternion `(0, R1 f, R2 f, 0)`. It is the 2-D object that has no complex

analogue — the 1-D analytic signal needs a direction to say which way "90

degrees later" is, and in 2-D there is no single direction, so the answer

needs two components and therefore an algebra with room for them.

Closed form, which is how this is tested rather than eyeballed

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For a grating `cos(2*pi*(u0*x + v0*y))` sampled on the DFT grid,

`R1 = (u0/|w0|) * sin(2*pi*(u0*x + v0*y))`,

`R2 = (v0/|w0|) * sin(...)`

exactly. Measured over a table of eight grid-exact orientations from 0 to

159.4 degrees on a 64x64 frame, the largest absolute deviation from that

closed form is 6.1e-15, and the orientation recovered through

:func:monogenic_orientation matches the grating's to 3.6e-15 rad at

every one of them. There is no tolerance to choose.

Note the scalar component is 0, so this is the Riesz *transform* and not the

monogenic signal — feeding it to :func:monogenic_phase gives `pi/2`

everywhere, correctly but uselessly. Use :func:monogenic_signal, which

keeps the band-pass image in the scalar slot.

Raises `ValueError: *image* is not a finite real (H, W)` array with

`H, W >= 2, or exceeds :data:MAX_PIXELS`.

Detailed usage guide

• quaternion_monogenic family guide

References (sample data, literature)

• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).

• Operator provenance and references — the sources of the research/methods this op family came from.

• The canonical algorithm (author, year) and its uses are named in the family usage guide above.

Runnable examples (verified samples that actually call this op)

• quaternion_monogenic — py -3.11 examples/quaternion_monogenic.py

Ops the type connects to (they accept `qimage` as input)

quaternion_to_rgb · quat_norm · quat_conjugate_image · quat_normalize_image · quat_image_multiply · monogenic_amplitude · monogenic_phase · monogenic_orientation

Same category (`riesz`)

monogenic_signal · monogenic_amplitude · monogenic_phase · monogenic_orientation

*Provenance: quatimage.py — QUAT operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*

riesz_transform — QUAT riesz op