Shape Defect Detection Problem

In shape defect detection, two types of measurements can be distinguished:

• Direct, or absolute, measurements, when a dimension or another quantity to be measured are specified with respect to particular shape features (corners, etc.) that are easy to identify and locate. An example is measuring the distance between centroids of two holes.
• Relative measurements, which are specified with respect to a reference shape.

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'.

  

Figure 4: Influence of shape defects on the reference pose. (a) The ideal shape and its centroid. (b) The defective shape and its centroid. (c) The desired reference pose. (d) The reference pose obtained using the least squared distance criterion.

The source of the trouble is that no reference pose (e.g., baseline) can be specified a priori because defects may deteriorate any part of the shape. It is a typical `chicken-and-egg' problem. What parts of a measured shape should be considered defective? Usually, it is assumed that those parts of the shape that coincide with the reference, or lie within the tolerance limit, are acceptable. Those parts that lie beyond the tolerance are defective. This means, however, that the defects should be recognized prior to the shape comparison whose goal is, in turn, defect recognition itself. The task is therefore to find the superposition of the measured and the reference shapes that best corresponds to the expectations of a human observer: a `correct' part should be close to the reference; a `defective' part should stand out clearly.

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:

• A reference point set, which represents the `ideal' shape. This point set may result either from precise measurement of a physical reference shape, or from computation based on a mathematical description (specification) of the reference shape. The reference point set is complete (all parts of the surface are represented), dense (any patch is densely covered by the points) and precise (the measurement/computation is precise).
• A measured point set, which may be incomplete, less dense, noisy, and distorted by defects whose deviation from the reference shape exceeds both noise level and tolerance.

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.