# Spatial filtering

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Spatial Filtering of Surface Profile Data/1

Spatial Filtering of Surface Profile Data
Chapman Technical Note-TG-1 spat_fil.doc Rev-01-09

Spatial Filtering of Surface Profile Data/2

Explanation of Filtering
A highway can be considered a surface with filter components corresponding to hills and valleys or speed bumps. This type of road with these various topographies is shown in Figure1. One road has a large, slow rolling hill, which because it is very smooth (1A), may be acceptable for high speed travel. An alternate road has a constant series of speed bumps that are a few feet tall with 20 feet between bumps. This road would obviously be unacceptable for high speed travel (1B). The third road has a large slow rolling hill with the same speed bumps on the surface, which isalso unacceptable for high speed travel (1C). The speed bumps in Figure 1B can be considered as short spatial wavelength information and the slow rolling hill as low spatial wavelength. Surface features in general can be separated into various spatial wavelengths, similar to the example shown in Figure 1.

Figure 1

Amplitude and Spatial Wavelengths The example shown in Figure 1 illustrates theimportance of not just the amplitude or height of surface features, but also the amplitude at given intervals or spatial wavelengths. The example shown in Figure 1 shows the importance of considering the spatial wavelengths in addition to the height. The height of the road in Figure 1A is larger than the speed bumps in Figure 1B, yet the road in Figure 1A would obviously represent a better surfacefor high speed travel.

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Filtering in the Spatial Frequency Domains
There are three fundamental, analytical representations for displaying surface features in the Spatial Domain: Total Profile, Roughness and Waviness. The example of the road surface shown in Figure 1 has these three fundamental representations. The same road surface is shown inFigure 2 in the profile, waviness and roughness representations. The complex road structure shown in Figure 2A would be considered the total profile, including both the speed bumps and the rolling hill. Removing the speed bumps yields the waviness, Figure 2B. Alternatively, removing the hills yields the roughness (2C). The roughness, which shows only the speed bumps, represents the shorter spatialwavelengths. The height of the speed bumps can be calculated in the roughness series shown in Figure 2C. The height of the overall complex road structure in Figure 2A would be primarily due to the rolling hill, which is much taller than the speed bumps.

Figure 2 Total Profile The Total Profile definition is the surface information, which spans the entire bandwidth of the measuring instrument.Depending on the instrument setting, this may be from distances on the order of 1 micrometer in length to tens of millimeters. The Total Profile contains both roughness and waviness information.

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Roughness The Roughness is calculated by subtracting the Waviness from the Total Profile data. The Roughness shows the finer, or shorter spatial wavelengthfeatures, of the surface. Typically, roughness parameters are given by “Rsub” where sub is the appropriate subscript for the parameter. For example, Ra is the average roughness. Waviness Calculated from the Total Profile, the waviness represents the longer spatial wavelength features of the surface. Waviness over long distances is typically called form, figure or bow. Typically, Waviness parametersare given by “W sub” where sub is the appropriate subscript for the parameter. For example, W a is the average waviness. The example shown in Figure 3 shows how this method is applied to surface profile data. First a cutoff filter is selected to separate the roughness from the waviness. The cutoff filter is chosen so that only the small spatial features are evident in the roughness series. The...

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