As we saw in the previous articles on circle fitting, different algorithms are available for fitting circles for different purposes. One concern with data fitting is what happens in the presence of outliers – isolated points that have significantly different deviation than the surrounding points. This article will examine the effect of an outlier on various different circle fitting algorithms.
Initial Data Setup
• A special dataset was prepared, with perfect points with the exception of one large outlier
• The points are based on a circle of diameter 100, with center point X=0, Y=0
• One outlier point deviates inward (mainly in the –Y direction) by 10 units
Least Squares Fitting
• This is the default fitting algorithm in most CMM software
• A circle is fit that minimizes the sum of the squared deviations
• In this case, the single 10 unit outlier point decreases the diameter by 1.1 units and moves the center by 1.1 units
Minimum Circumscribed Circle
• Finds the smallest perfect circle that fits around all the data points
• Because the outlier deviates inward, it has no effect on the circumscribed circle
Maximum Inscribed Circle
• Finds the largest perfect circle that fits inside all the data points
• Because the outlier deviates inward, it dominates the inscribed circle
• The single 10 unit outlier has decreased the size of the circle by the full 10 units
• The center of the circle has been displaced by 5 units, or half of the outlier’s deviation
The Least Squares algorithm has the effect of attenuating the outlier. The size and center of the circle are affected, but to a relatively small degree.
The extremity-based algorithms (Minimum Circumscribed and Maximum Inscribed) were affected to a large degree or not at all. The result depended on the direction of the outlier’s deviation.
The effectiveness of each approach can depend on the nature of the outlier – whether it represents real deviation in the surface of the feature, or false deviation due to measurement error.