Radial K-Space Trajectory

Generate explicit radial k-space spoke coordinates from supplied angles and signed radial sample locations; no trajectory density compensation is inferred.

Description

Radial k-space spokes map signed radial samples to Cartesian coordinates at supplied angles.

Radial k-space spokes map signed radial samples to Cartesian coordinates at supplied angles. Negative radius traverses the opposite side of the origin.

The tool returns xCoordinates (x coordinates); yCoordinates (y coordinates). Input measurements and model parameters are supplied by the caller; the tool computes the stated summary rather than acquiring those measurements.

When to use Radial K-Space Trajectory

Use Radial K-Space Trajectory when you already have spokeAnglesDegrees, radialSamplesCyclesPerMeter and want xCoordinates, yCoordinates. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.

How to use Radial K-Space Trajectory and understand its results

Enter spokeAnglesDegrees, radialSamplesCyclesPerMeter using the field table below. Match the units, ordering and data boundaries described in the calculation.
Run the tool after checking the stated bounds and matching paired observations or categories. A malformed request gives a tool error rather than a usable measurement.
Read xCoordinates, yCoordinates. Use the calculation and interpretation below to distinguish amounts, ratios, signs and decision flags where applicable.
Review the supplied parameters and assumptions before using the result in a report or comparing it with another observation.

Input field	Meaning	Value type and bounds	Requirement
spokeAnglesDegrees	Spoke angles degrees	list of number	Required
radialSamplesCyclesPerMeter	Radial samples cycles per meter	list of number	Required

Use 1–128 angles in degrees and 2–128 strictly increasing radial samples in cycles/m. Return one coordinate row per spoke.

kx = radius×cos(angle); ky = radius×sin(angle)

The symbols refer to the supplied quantities or generated values described above; unit conversions and comparison boundaries follow the stated implementation.

json

{
  "input": {
    "spokeAnglesDegrees": [
      0,
      90
    ],
    "radialSamplesCyclesPerMeter": [
      -1,
      0,
      1
    ]
  },
  "output": {
    "xCoordinates": [
      [
        -1,
        0,
        1
      ],
      [
        -6.123233995736766e-17,
        0,
        6.123233995736766e-17
      ]
    ],
    "yCoordinates": [
      [
        0,
        0,
        0
      ],
      [
        -1,
        0,
        1
      ]
    ]
  }
}

Limitations and assumptions

No gradient waveform, density compensation, spoke ordering or reconstruction is generated.

Numerical inputs must be finite. The tool checks its declared input shape and calculation constraints; those checks do not verify the quality, calibration or representativeness of the supplied observations.

Alternative or Complementary methods

K-Space Sampling Density: Screen uniformly distributed full radial spokes using ceil(πN/2) angular Nyquist coverage; golden-angle subsets and compressed-sensing reconstruction quality are not evaluated. Spiral K-Space Trajectory: Plan an Archimedean k-space spiral and its analytic gradient derivative for a supplied duration, radius and turn count; no gradient/slew constrained optimization is performed.

Methodological background: RAD229: MRI signal and sequence lecture notes.^[1]

References

RAD229: MRI signal and sequence lecture notes — Brian Hargreaves and colleagues, Stanford University

Similar or alternative tools

K-Space Sampling Density
Screen uniformly distributed full radial spokes using ceil(πN/2) angular Nyquist coverage; golden-angle subsets and compressed-sensing reconstruction quality are not evaluated.
Spiral K-Space Trajectory
Plan an Archimedean k-space spiral and its analytic gradient derivative for a supplied duration, radius and turn count; no gradient/slew constrained optimization is performed.