Gear pitting inspection mask

A gear pitting inspection mask specifies the tooth-flank region over which damage is counted. It separates eligibility for inspection from the classification of an eligible location as pitted. The denominator of a gear pitting area fraction depends on this mask, so a change in coverage can change the reported fraction even when the surface has not changed.

For a digital image, let $m_i\in \{0,1\}$ indicate inclusion of pixel $i$ and let $p_i\in \{0,1\}$ indicate its classification as pitted. With represented surface area $a_i$ per pixel,The arrays $m_i$ and $p_i$ are different indicator functions. Uniform pixel areas cancel, but an oblique view or a curved-surface map can require nonuniform $a_i$. A shadow, occlusion or unrecorded region should not automatically count as intact material.

An original example begins with 200 pitted pixels among 10000 eligible equal-area pixels, giving $2\%$. Cropping to a 5000-pixel region that retains 150 pitted pixels gives $3\%$. The counted damaged area fell while the fraction rose; this comparison alone cannot establish damage growth.

A gear inspection common-region comparison first uses image registration to put inspections into corresponding surface coordinates. On a common grid, the intersection mask is $m_i^{\mathrm{common}}=m_i^{(0)}{m}_i^{(1)}$. Comparing the two pitting classifications within that same region holds the denominator convention fixed. Coverage outside the intersection should still be reported: excluding a newly obscured damaged edge can hide relevant change.

Useful inspection records retain the raw image, mask, geometric mapping, image thresholding or other image segmentation rule, and the reason for excluded regions. A primary NASA gear-endurance study combined inspection history with vibration and oil-debris observations, illustrating why damage labels must remain connected to actual inspected surfaces.^[ref] The mask formalism above is an explicit measurement convention, not a claim that every historical inspection used this pixel algorithm.

A gear inspection common-region comparison evaluates repeated surface observations on the same registered portion of a tooth flank. It prevents a changed gear pitting inspection mask from being mistaken for a change in damage. The region must be common in surface coordinates, not merely occupy the same pixel positions in two camera images.

After image registration, let $m_i^{(0)}$ and $m_i^{(1)}$ be inspection-mask indicator functions on one reference grid. Their common mask isIf $p_i^{(t)}$ classifies pitting at time $t$ and $a_i$ is represented surface area, defineThe identical denominator makes the difference a comparison within the same eligible area. It does not eliminate errors in registration or image segmentation.

An original example has 10000 eligible equal-area pixels at the first inspection and 8000 at the second, with 7000 in common. In the common region, pitted counts rise from 140 to 210: fractions rise from $2\%$ to $3\%$, a change of one percentage point or $50\%$ relative to the original fraction. Suppose total pitted counts were 200 and 240 across the separate full masks. Their raw difference, 40 pixels, is not the same quantity as the 70-pixel increase within the common region.

A gear inspection coverage audit should also record the excluded areas and why they disappeared or became newly visible. An intersection-only result can miss damage on an edge that the second camera view no longer captures. Keeping the original images makes that loss of coverage visible to a later reviewer.

Common inputs can correlate the two estimated fractions. In general,This is the usual correlated-input uncertainty propagation, not an assumption that neighboring image pixels are independent.^[ref] A primary NASA endurance study linked repeated physical inspections with vibration and debris observations; the common-region formalism specifies one way to make such repeated surface measurements comparable.^[ref]

A gear inspection coverage audit records which parts of a tooth surface were eligible for observation, which were excluded and why. It accompanies a gear pitting area fraction or gear inspection common-region comparison so that a numerical damage statistic can be interpreted against the area actually inspected.

A useful record retains the original image, gear pitting inspection mask, surface-coordinate map from image registration, represented pixel areas and exclusion reasons. Shadow, occlusion and an out-of-frame edge are different from a visible region classified as intact. An audit also distinguishes changes in viewing coverage from changes in the image segmentation rule.

An original calculation shows how missing area limits interpretation. Let inspected fraction $c=A_{\mathrm{seen}}/A_{\mathrm{target}}$ and observed pitting fraction ${\varphi }_{\mathrm{seen}}$ both refer to the same target surface. If unseen area has unknown pitting fraction $0\le {\varphi }_{\mathrm{unseen}}\le 1$, thenso the gear inspection missing-area bound isAt $c=0.8$ and ${\varphi }_{\mathrm{seen}}=0.02$, the whole-target fraction lies between $0.016$ and $0.216$ without further information. This is a deterministic range over missing data, not a statistical confidence interval. A small observed fraction does not alone establish a small whole-target fraction.

For repeated inspections, record areas visible only initially, only later and at both times. A common-region result can be valid for its intersection while becoming less representative of the full target if a damaged edge is systematically lost. Increasing pixel count inside the retained region does not recover information about the missing region.

The primary NASA gear-endurance study connected inspection history to vibration and oil-debris observations, illustrating the importance of tying a damage interpretation to physical inspection rather than a signal label alone.^[ref] The audit and bound above specify an explicit measurement convention; they do not assert that the historical study used this digital-mask procedure.

A gear inspection missing-area bound limits the whole-target pitting fraction using the area actually observed and explicit constraints on the unseen area. It formalizes what a gear inspection coverage audit can establish without assuming that excluded regions resemble visible ones.

Let $c=A_{\mathrm{seen}}/A_{\mathrm{target}}$, let $p$ be the observed gear pitting area fraction, and let $u$ be the unseen fraction affected by pitting. For a consistent surface-area convention,With only $0\le u\le 1$ known,The interval width is exactly $1-c$. Improving classification precision within the visible area cannot by itself remove uncertainty from a completely unseen region.

For an original example with $c=0.9$ and $p=0.02$, the possible whole-target fraction ranges from $0.018$ to $0.118$. The observed $2\%$ therefore does not prove that the whole surface is below an illustrative $5\%$ criterion. That chosen criterion is not a gear safety or acceptance standard.

The worst-case upper bound meets a target $\theta <1$ only iffor $p<1$. At $p=0.02$ and $\theta =0.05$, at least about $96.94\%$ coverage is required for this particular worst-case guarantee.

Additional justified information can narrow the range. If a separate observation establishes $0\le u\le 0.1$, then $cp\le \Phi \le cp+0.1(1-c)$. At $c=0.8$ and $p=0.02$, this gives $0.016\le \Phi \le 0.036$. The tighter bound depends on the new evidence about the unseen region, not on a mathematical preference for a smaller interval.

These deterministic bounds assume the observed classification and represented areas are correct. Image registration, image segmentation and geometric uncertainties can require additional allowances. They are not confidence intervals and do not describe pit depth or remaining fatigue life. The primary NASA endurance study combined inspection with vibration and debris monitoring; these different observations constrain different aspects of damage.^[ref]