For optical 3-D measurement systems, camera noise is the dominant uncertainty factor when optically cooperative surfaces are measured in a stable and controlled environment. In industrial applications repeated measurements are seldom executed for this kind of measurement system. This leads to statistically suboptimal results in subsequent evaluation steps as the important information about the quality of individual measurement points is lost. In this work it will be shown that this information can be recovered for phase measuring optical systems with a model-based noise prediction. The capability of this approach will be demonstrated exemplarily for a fringe projection system and it will be shown that this method is indeed able to generate an individual estimate for the spatial stochastic deviations resulting from image sensor noise for each measurement point. This provides a valuable tool for a statistical characterization and comparison of different evaluation strategies, which is demonstrated exemplarily for two different triangulation procedures.

For optical 3-D measurement systems, like fringe projection systems, the achievable accuracy mainly depends on the environmental conditions and the properties of the measurement object. In a production-related environment, systematic deviations caused for example by temperature fluctuations or vibrations are the dominant factors. In contrast, a well-controlled environment leads to a strong influence of the surface properties of the object under test. While the deviations are correlated with the micro-topography in the case of non-cooperative surfaces, stochastic deviations caused by camera noise come to the fore in the case of cooperative surfaces. Depending on the local lighting conditions, these can differ by up to an order of magnitude for different measurement points on the object.

In industrial applications of optical 3-D measurement systems, the measurements are usually not repeated for estimating repeatability. The deviations of the measurement process may be characterized by measurements of spherical and planar artefacts in the context of acceptance tests and re-verifications according to VDI 2634 (VDI, 2012). It has to be pointed out however that this is not to be confused with a measurement uncertainty as it is only a method to ensure that the system is working in conformity with the specifications. The experimental determination of a task-specific measurement uncertainty, for example according to VDA5 (VDA, 2011), is only worth the effort in the case of a small measurement object portfolio and large lot sizes. In addition this method characterizes the whole measurement process only in a general way. As an alternative, the use of “virtual fringe projection systems” (Haskamp et al., 2012) is being discussed as a tool for task-specific error analysis. Although this approach can be seen as state of the art for tactile coordinate measuring machines (Trenk et al., 2004), in the case of fringe projection systems it is not yet established in the industry.

Fringe pattern with locally varying visibility

Physical model of signal generation in a CCD-camera pixel according to EMVA 1288 (EMVA, 2010).

Mathematical model for the signal generation according to EMVA 1288 (EMVA, 2010).

Propagation of phase deviations into the object space in a fringe
projection system with two cameras. The implementation is based on a passive
triangulation along a search vector in the

The missing assessment of the quality of each individual measurement point leads to statistically suboptimal results in subsequent evaluation steps as for example this information is not available as a weighting factor for the matching of geometric primitives. Because of the use case outlined above, i.e. the measurement of cooperative surfaces in a well-controlled environment, the geometric deviations of the measured points are dominated by camera noise, and they can be estimated directly from the measurement data if the noise characteristic of the measurement process is modelled. This approach will be described in the following sections and is a first important step towards a task-specific uncertainty estimation for this kind of measurement system.

Optical measurement systems that use intensity patterns to spatially encode
an object are referred to as “structured illumination” techniques. Among
others, a widely applied coding approach is based on multiple phase-shifted
sinusoidal fringe patterns. In this case the actual coding takes place in the
time domain, as the phase-shifted patterns are usually recorded sequentially.
The recorded intensity

Cropped camera views of the measured sphere with the fringe pattern projected onto it.

Visualization of the distribution of parameter

Visualization of the distribution of parameter

Achieved coverage of the

With the recorded intensities

For symmetric

Empirically determined reference data for the stochastic
deviations of the measured

Estimated value according to the noise model

It can be seen that the fundamental process is a conversion, first from the
number of photons

For the four-step algorithm a simple relation between the recorded
intensities

Plot of a single data column of

Relative deviation of the estimated value

Relative stochastic deviation (precision) of the estimated value

Propagation of phase deviations into the object space in a fringe projection system with two cameras. The implementation is based on a passive triangulation along a viewing ray of camera 1.

The propagation of the stochastic phase deviations into the object space
depends on the actual implementation of the triangulation strategy. In this
first example a passive triangulation along a search vector has been
implemented, where “passive” in this context means that the projection is
only used for the optical coding of the surface, whereas the actual
triangulation is done with two cameras. In this example an iterative method
is used, where each object point is found on a regular

The spatial coding of the surface via fringe projection is realized by a
combination of phase shifting and heterodyne evaluation. The whole image
sequence consists of 12 images: three slightly different fringe widths

Empirically determined reference data for the stochastic
deviations of the measured

Because the image coordinates are calculated with subpixel resolution by
means of a resection into the image plane, the corresponding phase values
have to be interpolated. In the case of a bilinear interpolation that has
been applied here, the interpolated phase value

For the propagation of the resulting estimated phase noise in both camera
views

For the experimental validation a photogrammetric fringe projection system
has been used, composed of two cameras with a resolution of
1024

The resulting parameter space of combinations of

In addition to the object point

Estimated value according to the noise model

Plot of a single data row of

It has been deduced from the data that thermal influences lead to varying
inner and outer orientation of the cameras over the course of the measurement
series, resulting in an observed movement of the point cloud. For the
centroid of the point cloud, a mean displacement in the

From the whole series of 800 measurements, the empirical standard deviation
of the

For a better assessment of the prediction quality over the whole measured
geometry, Fig. 12 shows the relative deviation of the estimated value

The relative deviation of the estimation is

The precision of the estimation method can be characterized by the
distribution of

Relative deviation of the estimated value

Relative stochastic deviation (precision) of the estimated value

Similarly to the first example discussed in Sect. 4, where object points are
found on a regular grid, the triangulation along a viewing ray is also a
passive technique where the 3-D information is calculated from two camera
views and the projector is only used as an optical surface coding device.
Viewing rays in this context are the principal rays for the given imaging
geometry from each pixel through the common projection centre of a pinhole
camera model, including deviations from the ideal imaging geometry by means
of distortion terms. In contrast to the triangulation along a vector in the

The spatial coding of the surface via fringe projection is realized with the
same method as described in Sect. 4.1, i.e. by a combination of phase
shifting and heterodyne evaluation with an image sequence of 12 images: three
slightly different fringe widths with four

In the case of a viewing-ray-based triangulation method, there is no need for
subpixel interpolation in the first camera view because the starting points
of the viewing rays are usually defined on pixel centre coordinates. The
phase value and quality metric for the first camera can then simply be
calculated from these pixel values:

As described above, the passive triangulation along a viewing ray uses the
same iterative approach to find an object point, but in contrast to example 1
in this case it is bound to a specific viewing ray in the first camera. The
point search is done by moving the object point along this viewing ray and
minimizing the phase difference

The experimental set-up and methodology for the second example are the same
as described in Sect. 4.4; in fact, the raw images of the identical
measurement have been used for both implementations. In this case the viewing
rays of the first camera define the measurement grid, which is non-regular in
object space due to optical distortions and the geometry of the measurement
object. The resulting distribution of the distance between adjacent object
points has a median value of about 25

The experimental findings for this second example are in accordance with the
findings of the first example described in Sect. 4.5, and the results are
qualitatively equivalent. The empirical standard deviation of the

In this work a model-based approach for the estimation of stochastic coordinate deviations, caused by camera noise in phase-measuring optical 3-D measurement systems, has been proposed. In the case of a fringe projection system measuring an optically cooperative surface in a well-controlled environment this influence is the dominating factor.

Starting from a general phase-noise estimation the propagation into the
object space has been derived for two different triangulation methods. For
both triangulation methods the noise estimation has been experimentally
validated and the estimation quality, i.e. the precision and accuracy of the
estimated values, has been assessed. The precision of this estimation is
better than 10 % for most of the measured points, which can be seen as
feasible for practical applications, especially given the fact that such
information is generally not available at all for most industrial
applications. These estimated spatial noise values can be used for further
data processing as a point quality metric. Although in this work the
deviations have been simplified to scalar

The data presented in this paper have been generated by means of repeated measurements with a fringe projection system and consist of several hundred images of a ball-bearing sphere. The image series itself does not carry any scientifically relevant meaning besides demonstrating the prediction method described in this work. The experiments can easily be reproduced with other fringe projection systems. Therefore, the image series is not available online, but the authors will provide sample data upon request.

The authors declare that they have no conflict of interest.

This work is partly based on prior research that has been funded by the Deutsche Forschungsgemeinschaft (DFG) under grant Pe1402/21. The authors gratefully acknowledge the funding. Edited by: U. Neuschaefer-Rube Reviewed by: two anonymous referees