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Height histograms: Use and interpretation of void and material volume
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The height histogram of a topographical image shows the statistical distribution of z-values (heights) within the image. This notes describe how the histogram can be used for quantifying the total volume of holes (void) and protrusions (material) of a surface.
The histogram is a graph of neighboring columns (or bins). Each column represents a height range. The height of each column represents the number of image pixels which have a height value in the particular range. All columns have the same width, i.e. they represent the same height span. Often the columns are so narrow that the histogram looks like a filled 2D graph, as illustrated in Figure 1.
 
The height histogram of a surface
If the surface is random, the histogram will be bell shaped with the bell top at the mean height in the image.
 
The height histogram of an image dominated by surface tilt will be flat with all columns approximately equal in height.
 
If the surface has two distinct levels as for example a height calibration standard, the histogram will have two distinct peaks each centered on the respective height levels.
 
Figure 1: Histogram of two level surface
Figure  1: Illustration of the histogram of an image with two dominant height levels. The dashed lines represent different marker positions. The numbers next to the markers correspond to each of the three situations in Figure 2.
 
Markers and volume calculation
A pair of markers (cursors) can be used to read out the positions of peaks in the height histograms in SPIP™. Each marker represents a certain height level. SPIP™ calculates the amount of material and void between the markers.
 
Material volume
This is the volume limited by the lower marker and by either the upper marker or by the surface, which ever is smallest in height at each image pixel position.
 
Void volume
This is the volume limited by the upper marker and by either the lower marker or by the surface which ever has the highest z-value at each pixel position.
 
Special cases
When the markers are placed at the boundaries of the histogram the sum of the material and void volume will be equal to the image projected area times the image z-range. When the markers are on top of each other both the material and the void volume will ideally be zero (however, due to the histogram resolution often they often end up with  a small but finite value).

Figure 2. A surface represented by a section profile, with the markers from Figure 1 and with material volume (red) and void volumes (blue) indicated. 
 
Example - Using the histogram for measuring the volume of pits
In Figure 3 is shown a 3D projection view of a two dimensional calibration grating imaged by AFM.

Figure 3. 3D projection of the AFM image of a 2D calibration grating.  
Single Pit volume
The pits are about 107 nm deep, taken as the height distance between the two peaks (not the cursors) in the histogram, see Figure 6. The lateral dimensions of each pit are about 4.1µmx4.8µm, measured from the image using a profiling tool on a singe pit. This gives a single pit volume of about 2.1µm³.  This can be verified using the polygon measure tool.
Figure 4. Section profile over a single pit along the line in Figure 5.
Color cut off
When the cursor markers are moved in the histogram (see Figure 6) the boundaries of the color bar in the active image window are moved simultaneously such that only surface between the markers are shown with the chosen color coding. What is below takes the lowest color in the color bar and what is above takes the highest color in the color bar. Therefore the surface in Figure 5 appears white.
Figure 5. The red box shows the volume measure-ment of an individual pit using the Interactive Measurement Tools in SPIP™.  It is seen that the top level surface is all white due to the cropping of the color scale at the height level of the upper (red) cursor in the histogram.

SPIP™ histogram
The height histogram of the image in Figure 1 is shown in Figure 6. In order to measure the volume of all the pits the lower cursor marker (blue triangle) has been set to the minimum height value and the upper marker (red triangle) htas been set just below the surface level. 
 
Total Void volume
The void volume calculated in the histogram is about 25 µm3.  A crude count of the number of complete pits is about 13. Multiplying this with the approximate single pit volume of 2.1 µm³ yields 27.3 µm³. This is consistent with the void volume found in the histogram, taking into account the spread in pit volumes and the uncertainty in the manual estimate of the number of complete pits.
Figure 6. Height histogram. The lower cursor is set at the minimum height value in the image and the upper cursor marker is set just below the upper peak which represents the surface level – or the bearing plane (white in Figure 5). The total void volume is 24.84 µm3.
 
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