Smoothness FWHM

Infer coordinate-axis Gaussian-equivalent residual FWHM and determinant-based resel density from residual variance and physical gradient covariance; not the applied smoothing-kernel width.

Description

Residual-gradient covariance relates image smoothness to fluctuations of residual derivatives.

Residual-gradient covariance relates image smoothness to fluctuations of residual derivatives. Axis widths and volumetric resel density summarize that fitted covariance.

The tool returns axisFwhmMm (axis fwhm mm); reselDensityPerMm3 (resel density per mm3). Input measurements and model parameters are supplied by the caller; the tool computes the stated summary rather than acquiring those measurements.

When to use Smoothness FWHM

Use Smoothness FWHM when you already have residualVariance, gradientCovariancePerMm2 and want axisFwhmMm, reselDensityPerMm3. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.

How to use Smoothness FWHM and understand its results

Enter residualVariance, gradientCovariancePerMm2 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 axisFwhmMm, reselDensityPerMm3. 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
residualVariance	Residual variance	number	Required
gradientCovariancePerMm2	Gradient covariance per mm2	list of list of number	Required

Require positive residual variance and an exactly symmetric positive-definite 3×3 gradient covariance in inverse-mm² units. Validate leading minors and determinant before evaluation.

FWHMᵢ = sqrt(4 ln2 × variance/covᵢᵢ); density = sqrt(det(cov)/variance³)/(4 ln2)^1.5

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

json

{
  "input": {
    "residualVariance": 1,
    "gradientCovariancePerMm2": [
      [
        1,
        0,
        0
      ],
      [
        0,
        4,
        0
      ],
      [
        0,
        0,
        9
      ]
    ]
  },
  "output": {
    "axisFwhmMm": [
      1.6651092223153954,
      0.8325546111576977,
      0.5550364074384652
    ],
    "reselDensityPerMm3": 1.2996400069926133
  }
}

Limitations and assumptions

Covariance is supplied rather than estimated from images. Axis widths alone do not capture rotation or nonstationary smoothness.

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

Resel Count: Estimate the volume resel term R3 from a fractional search mask and axis-aligned anisotropic FWHM; boundary and lower-dimensional resel terms are not included. Spatial Smoothing FWHM: Build a normalized truncated 1D Gaussian kernel from physical FWHM and voxel spacing; distinct from estimating residual smoothness from derivative covariance.

Methodological background: An Introduction to Random Field Theory.^[1]

References

An Introduction to Random Field Theory — Brett, Penny and Kiebel; University College London

Similar or alternative tools

Resel Count
Estimate the volume resel term R3 from a fractional search mask and axis-aligned anisotropic FWHM; boundary and lower-dimensional resel terms are not included.
Spatial Smoothing FWHM
Build a normalized truncated 1D Gaussian kernel from physical FWHM and voxel spacing; distinct from estimating residual smoothness from derivative covariance.