Asperity Density, Summit Radius, and RMS Height Calculator

Estimate areal RMS height, summit density, and local principal summit radii from a rectangular surface-height grid.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Surface and analysis inputs

Surface file (CSV or TXT)Choose a local rectangular numeric matrix. Files stay in this browser.Surface height matrix z_ijAt least 5 × 5 finite values; commas, tabs, spaces, or semicolons are accepted. Missing values are rejected rather than interpolated.

X spacing ΔxColumn-to-column sample spacing; must be greater than zero.Y spacing ΔyµmRow-to-row spacing; uses the selected lateral unit.Height unitUnit of every matrix height and height threshold.Leveling methodSubtracts a least-squares reference before statistics.Smoothing radius r_ssamplesSquare moving-average half-width, 0–5; zero means no smoothing.Summit neighborhoodA summit must be strictly higher than each selected neighbor.Minimum prominence p_minµmMinimum rise above the highest neighboring sample; non-negative.Minimum summit height h_minµmSigned elevation relative to the leveled mean plane; positive is upward.Radius-fit half-window r_fsamplesFits a (2r_f + 1) square neighborhood. Candidates too near an edge are excluded.

Surface statistics

Areal RMS height Sq—µm

Accepted summits N_s—

Summit density η—µm⁻²

RMS summit height h_rms—µm

Median R₁—µm

Median R₂—µm

Median R_eq—µm

Grid —Area —µm²Rejected fits —Edge exclusions —

Processed height map; red rings mark accepted fitted summits. Color is relative within the current grid.

How to use

Upload a CSV/TXT surface or paste a rectangular matrix with at least 5 rows and 5 columns. Blank, missing, and non-numeric cells are not interpolated and must be corrected first.
Enter Δx and Δy and select their common lateral unit. Select the unit used by every height value in the matrix.
Select mean removal, plane leveling, or quadratic detrending. Use stronger detrending only when it represents form rather than roughness.
Set an optional moving-average radius, the 4- or 8-neighbor summit rule, minimum prominence, signed minimum summit height, and local fit half-window.
Select Calculate to update Sq, summit count and density, summit-height RMS, radii, diagnostics, and the detected-summit map.
Use Reset example to restore the supplied 9 × 9 analytical surface and its units and controls.
Interpret density in inverse lateral-unit squared, heights in the chosen height unit, and fitted radii in the chosen lateral unit. Review rejected fits and edge exclusions before reuse.
Repeat with defensible threshold, smoothing, and fit-window choices. Materially changing results indicate sensitivity that must be reported.
Do not use this model for nonuniform spacing, unordered point clouds, unresolved tip convolution, surfaces dominated by form, or safety-critical contact predictions without independent metrology and model validation.

Worked example

Use the supplied 9 × 9 paraboloid with Δx = Δy = 1 µm, heights in µm, mean removal, no smoothing, 8 neighbors, 0.01 µm prominence, and a 2-sample fit half-window. Expected results are Sq = 0.6540 µm, one accepted summit, η = 0.015625 µm⁻², R₁ = 5.000 µm, R₂ = 10.000 µm, and R_eq = 7.071 µm. The recovered radii identify the imposed principal curvatures; this analytical example is a code check, not a claim about a manufactured surface.

Equations used

Let z_ij be the preprocessed height at row i and column j, N the number of grid points, N_s the number of accepted fitted summits, Δx and Δy the physical sample spacing, and A = (n_x − 1)Δx(n_y − 1)Δy the sampled rectangular area.

Sq = √[(1/N) Σ_i,j z_ij²]

η = Nₛ / A

h_rms = √[(1/Nₛ) Σ_k=1^Nₛ h_k²]

Each accepted neighborhood is fitted by least squares to z = ax² + bxy + cy² + dx + ey + f. The local Hessian is H = [[2a,b],[b,2c]]. Both eigenvalues κ₁ and κ₂ must be negative for a concave-down summit. Principal radii use R_i = −1/κ_i after converting height and lateral coordinates to one unit. This calculator reports R₁ ≤ R₂ and R_eq = √(R₁R₂).

Positive height is upward from the leveled reference plane. Mean removal subtracts one constant; plane leveling subtracts α + βx + γy; quadratic detrending subtracts α + βx + γy + δx² + εxy + ζy². Optional smoothing is a clipped-edge square moving average. A strict local maximum must exceed every selected neighbor and both thresholds.

Theory and method

Discrete summit statistics are not intrinsic constants of a surface: sampling interval, instrument bandwidth, leveling, filtering, neighborhood definition, thresholds, and fit window all alter the reported population. The implementation therefore exposes these choices and reports edge and curvature-fit exclusions. It does not compensate for probe-tip convolution, measurement uncertainty, anisotropic acquisition artifacts, or scale-dependent roughness.

Summit-height RMS uses only accepted summit elevations relative to the reference plane; it is not Sq over all samples and is not the standard deviation about the summit population mean. For the supplied example it is 1.000 µm, while Sq is 0.6540 µm. Hessian radii are a small-slope curvature approximation; they are not full geometric principal radii for a significantly tilted fitted surface. The sampled area convention spans the center-to-center extent of the grid. Sq uses the final leveled and optionally smoothed grid. Summit heights are measured from the leveled reference plane. Radius statistics include only finite negative-definite Hessian fits; saddle points and concave-up fits are rejected. The geometric mean is the only equivalent-radius convention used here.

Applicability limits: use rectangular uniformly spaced areal grids with finite values and adequate lateral/vertical resolution. Verify instrument calibration, remove physically justified form, inspect outliers, and repeat the analysis across plausible settings. ISO 25178-2 supplies areal surface-texture terms but does not make this discrete summit detector interchangeable with every commercial implementation.

References

ISO 25178-2, Geometrical product specifications (GPS)—Surface texture: Areal—Part 2: Terms, definitions and surface texture parameters.
J. A. Greenwood and J. B. P. Williamson, “Contact of nominally flat surfaces,” Proceedings of the Royal Society A 295 (1966), 300–319. doi:10.1098/rspa.1966.0242.

Continue in TriboSolver

Use defensible Sq and summit-radius statistics as screening inputs for a rough-surface contact study in TriboSolver. Preserve the preprocessing settings and sensitivity range so the contact model can be independently checked.

Input definitions

Surface height matrix (zᵢⱼ): Rectangular CSV, tab-delimited, or whitespace-delimited areal height grid; every row must have the same number of finite values. Unit: selected height unit.
X sample spacing (Δx): Physical center-to-center spacing between adjacent columns. Unit: selected lateral unit.
Y sample spacing (Δy): Physical center-to-center spacing between adjacent rows. Unit: selected lateral unit.
Leveling method (—): Mean removal, least-squares plane removal, or least-squares quadratic detrending before statistics. Unit: dimensionless.
Smoothing radius (rₛ): Optional square moving-average half-width in samples; zero disables smoothing. Unit: samples.
Summit neighborhood (—): Four- or eight-connected strict local-maximum test. Unit: neighbors.
Minimum prominence (pₘᵢₙ): Minimum summit height above the highest selected neighbor after preprocessing. Unit: selected height unit.
Minimum summit height (hₘᵢₙ): Minimum accepted elevation relative to the leveled mean plane; positive is above the plane. Unit: selected height unit.
Radius-fit half-window (r_f): Half-width of the local quadratic-fit window around each detected summit. Unit: samples.