In shape defect detection, two types of measurements can be distinguished:
Most of direct measurements are straightforward, as they usually have precise mathematical definitions in terms of images. Such computational definitions are relatively easy to translate into a computer algorithm. The above mentioned circular magnet ring measurement system (section 6) belong to this category.
Relative measurements are much more challenging, as the definitions are, in fact, implicit. Here, one compares a measured shape to a reference (ideal) shape. For shift- and rotation-invariant comparison, optimal reference position and orientation (pose) of the measured shape is to be found, which requires invariant matching of the two shapes.
Figure 4 illustrates the pitfall of the matching problem. Assume we match a reference and a defective shapes by superimposing their centroids and minimizing the sum of the squared local distances between the two shapes. The centroid of the measured defective shape is shifted compared to the position needed for proper comparison. Consequently, the measured shape is rotated with respect to the desired position. Differences are measured in all dimensions, although real defects are only in the upper part of the legs: the defects are `smeared'.
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A related problem is addressed in robust regression and outlier detection [4], when the parameters of a linear model are estimated based on a set of measured points. However, in this case it is assumed that a majority of these points can be brought into strict correspondence with the reference (model) points, while the outliers are random. In manufactured shapes, both the acceptable and the defective parts of the measured contour usually have systematic deviations from the reference shape. In the acceptable parts, the deviations are within the tolerance limits, while in the defective parts the deviations exceed these limits. In addition, no simple analytical expression for the reference shape can usually be given.
Formally, the general problem of shape matching for defect detection can be stated as follows. Two point sets are given:
Under these condition, one should find the superposition (matching) of the two point sets, in which the non-defective parts coincide, while the defective parts stand out.