Nyquist Ghost Correction

Apply supplied linear odd/even phase calibration exp(−i(a+bx)) to an even EPI line; calibration estimation, timing correction and ghost removal efficacy are not inferred.

Description

A linear phase model can describe an odd-even EPI line mismatch.

A linear phase model can describe an odd-even EPI line mismatch. Multiplying a complex even-line signal by the negative phase rotates it while preserving magnitude.

The tool returns correctedReal (corrected real); correctedImaginary (corrected imaginary). Input measurements and model parameters are supplied by the caller; the tool computes the stated summary rather than acquiring those measurements.

When to use Nyquist Ghost Correction

Use Nyquist Ghost Correction when you already have sampleCoordinates, evenLineReal, evenLineImaginary, phaseInterceptRadians, phaseSlopeRadiansPerCoordinate and want correctedReal, correctedImaginary. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.

How to use Nyquist Ghost Correction and understand its results

Enter sampleCoordinates, evenLineReal, evenLineImaginary, phaseInterceptRadians, phaseSlopeRadiansPerCoordinate 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 correctedReal, correctedImaginary. 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
sampleCoordinates	Sample coordinates	list of number	Required
evenLineReal	Even line real	list of number	Required
evenLineImaginary	Even line imaginary	list of number	Required
phaseInterceptRadians	Phase intercept radians	number	Required
phaseSlopeRadiansPerCoordinate	Phase slope radians per coordinate	number	Required

Use aligned nonempty real, imaginary and coordinate lists with supplied phase coefficients in radians. Apply the complex rotation point by point.

correctedComplex = evenLineComplex×exp(−i(intercept+slope×coordinate))

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

json

{
  "input": {
    "sampleCoordinates": [
      -1,
      0,
      1
    ],
    "evenLineReal": [
      1,
      1,
      1
    ],
    "evenLineImaginary": [
      0,
      0,
      0
    ],
    "phaseInterceptRadians": 1.5707963267948966,
    "phaseSlopeRadiansPerCoordinate": 0
  },
  "output": {
    "correctedReal": [
      6.123233995736766e-17,
      6.123233995736766e-17,
      6.123233995736766e-17
    ],
    "correctedImaginary": [
      -1,
      -1,
      -1
    ]
  }
}

Limitations and assumptions

Coefficients are supplied rather than fitted. The tool does not reverse readout direction, estimate navigators or reconstruct an image.

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

Dixon Water Fat Separation: Separate ideal complex in-phase/opposed-phase signals into water=(IP+OP)/2 and fat=(IP−OP)/2; assumes corrected common phase and a single fat peak, not magnitude-only Dixon or B0 solving. Echo Spacing: Budget echo-center spacing from explicitly nonoverlapping readout, turnaround and settling intervals; waveform overlaps require a different timing model.

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

Dixon Water Fat Separation
Separate ideal complex in-phase/opposed-phase signals into water=(IP+OP)/2 and fat=(IP−OP)/2; assumes corrected common phase and a single fat peak, not magnitude-only Dixon or B0 solving.
Echo Spacing
Budget echo-center spacing from explicitly nonoverlapping readout, turnaround and settling intervals; waveform overlaps require a different timing model.