All 3D points recorded in the C3D file have the capability of recording a residual measurement value – this is a number that provides information about the relative accuracy of the 3D measurement of the associated point

Although the concepts behind the calculation of the 3D point residual are based on optical photogrammetry, the general principals are applicable to most 3D measurement systems and can be applied to many 3D measurement techniques.

The illustration below demonstrates the situation when two observers see a single point in 3D space.  Observer C1 measures the point to be in the direction C1 to D1, and observer C2 determines the point to be in the direction C2 to D2.  Thus, we know that the point lies somewhere on the line C1-D1, and that it must lie on the line C2-D2.  This is possible only if the point lies at the intersection of the two rays; thus, the 3D reconstruction process must calculate the locations of intersections of rays from different observers. 

However, due to small errors in the measuring system, the measured rays from the two observers to any single point will not, in general intersect.  This invariably results in the measurement software making a decision about the most probable location for the point under observation when the rays fail to intersect.  For the two rays shown, the point location is set at the mid-point of the line forming the shortest distance between them. 

Figure 23 - Point residual determination with two cameras.

The distances from the assumed point location to each ray are related to the uncertainty of the point’s calculated location, and are termed the residuals for the measurement.  Generally, inaccurate measurements or calibration will produce large residuals although in the case of two-observer measurements, small residuals do not necessarily mean that the measurements were of high accuracy.  If the errors happen to be in the plane containing the two rays (containing C1-D1 and C2-D2), then small residuals will result no matter how large the actual errors are. 

For this reason, three observer measurements are usually more reliable.  A three-observer measurement involves a third ray (C3-D3) which will normally pass in the vicinity of the intersection of the other two rays and as a result, the problem of determining the point’s most probable position becomes somewhat more complicated.

Figure 24 – Point residual determination with three cameras.

A least-squares technique should be used to calculate the location of a point in space such that the sum of the squares of the shortest distances from that point to each ray is a minimum.  This calculated point then represents the best estimate of the observed point’s center.  The individual residual components are the shortest distances (perpendiculars) from the calculated point to each ray.  Application software that calculates 3D point coordinates should also store the average value of the residuals for each 3D point in each frame.  This value is a useful indicator of the reliability of the marker location determination.

In a three-observer measurement the probability of obtaining an inaccurate point location with low residuals is quite small.  Two of the observers must have errors of exactly the right magnitude in both horizontal and vertical components of their ray directions if a three-ray intersect with very small residuals and a large error is to be produced.  Hence, the average residual value is a much better indicator of 3D point location accuracy if more than two observers contribute to the measurement.  In general, the residuals obtained for two observer measurements will be smaller than those obtained from measurements made by three or more observers – this does not imply that two observer measurements are more accurate.

By convention, 3D point residuals can also act as flags for modified or invalid data points.  A residual value of –1.0 is used to indicate that a point is invalid while a value of 0.0 indicates that the 3D point coordinate is the result of modeling calculations, interpolation or that the data has been filtered. Valid residual values are always positive numbers.