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**3D (Areal) Filtering**

**F-Operator****Long Wavelength Pass Filter****Short Wavelength Pass Filter****Band Pass Filter****Notch Filter**

The measurement dataset consists of an array of values which represent heights at various points along the surface. Depending on the measurement technology such as stylus diameter, optical lens performance, camera resolution and electronics, a minimal spatial wavelength structure may be measured. Typically the smallest spatial wavelength that can be measured consists of 5 measured resolution elements along a given direction. The largest spatial wavelength that can be measured is limited to the full extent of the measured field. For example if a measurement is made over a 1mm lateral region then the largest spatial wavelength that can be ideally measured is 1mm long.

Many applications require only certain spatial wavelengths be included in the calculation of the various texture parameters. For example, a sealing surface may be better understood by considering a limited number of shorter spatial wavelength components since longer spatial wavelength structures may be easily conformed to by a compliant sealing material. Furthermore, to support correlation between different measurement systems it is imperative that the bandwidth of the spatial wavelength structures being measured are the same.

Once a surface is measured a mathematical operation is applied to remove any base form such as Tilt, Cylinder, Sphere, etc. if necessary. This mathematical operation is referred to as an F-operator in that the “form” is removed from the “raw” measurement prior to any additional filtering operations.

After the F-operator, a Gaussian filter is applied to the data which limits the short spatial wavelength (S-Filter) and long spatial wavelength (L-Filter) structures prior to analysis.

The S-Filter and L-Filter are characterized by cutoff lengths:

For example, a 0.8 mm cutoff-length L-Filter, implies that for structures with a spatial wavelength of 0.8 mm the amplitude presented for analysis is 50% of the amplitude of the unfiltered measured data. For spatial wavelengths greater than 0.8 mm the surface structure amplitudes are further attenuated such that spatial wavelengths components greater than 1.1 mm are attenuated by more than 90% prior to analysis. For spatial wavelengths less than 0.8 mm, surface structure amplitudes are minimally attenuated with wavelength components less than 0.5 mm being attenuated by 10% or less.

As another example, an 8 um cutoff-length S-Filter implies that for structures with a spatial wavelength of 8 um, the amplitude presented for analysis is 50% of the amplitude of the unfiltered measured data. For spatial wavelengths less than 8 um, the surface structures amplitudes are further attenuated such that spatial wavelength components less than 2 um are totally eliminated prior to analysis. For spatial wavelengths greater than 8 um, the surface features are minimally attenuated with wavelength components greater than 11 um being attenuate by about 10% or less.

The following figure depicts the “bandwidth” and attenuation rate (rolloff) for an S-Filter of 8um and L-Filter of 0.8mm. The ASME B46.1-2002 reference includes graphs depicting the attenuation rates (rolloff) for various filter cutoff values.

F-Operator

The F-Operator is used to remove the base form comprising the surface. As displayed in the figure below, a spherical surface is rendered “flat” after an F-Operator for spherical shape is applied. A number of F-Operators are available such as Tilt, Curvature (i.e. parabolic fit), Spherical (i.e. true spherical fit), Cylinder and various combinations. The F-Operator may be set to find the best fitting form based on a least squares fit or user-defined values for the radius of curvatures may be selected.

Long Wavelength Pass Filter

The S-Filter is used to attenuate the short
spatial wavelength structures that may be present in the measurement from
sources such as electronic noise or other measurement artifacts. Additionally,
some surfaces may physically contain short spatial wavelength structures
which are not relevant to the attended application and may affect subsequent
analyses. The figure below demonstrates an

S-filter used to eliminate the shorter spatial wavelength components comprising
the texture.

When only an S-filter is used, the type
of filtering is termed **Long Wavelength Pass.**

Short Wavelength Pass Filter

The L-Filter is used to attenuate the longer spatial wavelength structures that may be present in the measured surface, which are not relevant to the attended application and may affect subsequent analyses. The figure below demonstrates an L-Filter used to remove the longer spaced spatial structure revealing the finer spaced texture and peaked features..

When only an L-filter is used, the type
of filtering is termed **Short Wavelength Pass.**

Band Pass Filter

The combination of an S-filter and an L-Filter may be used to establish a bandwidth of spatial wavelengths which are used for analysis.

When the spatial wavelength of the structure
to be analyzed is contained between the S-Filter cutoff and the L-filter
cutoff, the filter is termed a **Band Pass Filter.**

The figure below demonstrates the use of a Band Pass Filter which is comprised of an S-Filter to remove short spatial wavelength structures and an L-Filter used to remove the longer spatial wavelength structures. The final surface consists of a band of spatial wavelengths relevant to the application.

Notch Filter

The combination of an S-filter and an L-Filter may be used to eliminate a bandwidth of spatial wavelengths from analysis, maintaining the spatial structure with wavelengths less than the S-Filter Cutoff and spatial wavelengths greater than the L-filter Cutoff.

When the spatial wavelengths of the structure
to be analyzed consists of spatial structure finer spaced that the S_Filter
Cutoff and spatial structure with coarser spaced features than the L-Filter
cutoff, the filter is referred to as a **Notch Filter.**

The figure below demonstrates that with a Notch Filter, features such as the isolated wider peaked regions (circled in the images) and moderate spatial structure lengths between the dominant mountain/valley features are eliminated.