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CHAPTER 6 CONTOURS
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for curve fitting that are useful when some of the edge points have been incorrectly linked into the contour These incorrectly assigned points are called outliers
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Total Regression
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Classical linear regression minimizes the difference between a data point and the model in only one dimension, the dimension of the dependent variable For example, a functional model of the form
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relating the dependent variable y to the independent variable x, with the p model parameters al through a p , assumes that there are no errors in the independent variable x In machine vision, errors in the x and y coordinates of location are equally likely and the curve model may be a vertical line, for instance, which cannot be represented in functional form In machine vision, lines and other curve models are fitted to edges using total regression, which minimizes the sum of the squares of the perpendicular distances of the data points from the regression model The advantage of this technique is that it compensates for errors in both the x and y directions Total regression has actually already been presented in 2 where it was used to derive the equations for determining the orientation of a blob, although the term total regression was not used at the time To avoid problems when the line is vertical, represent the equation for a line by using polar coordinates:
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x cos () + y sin () - p = O
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Minimize the sum of the squared perpendicular distances of points (Xi, Yi) from the line: (641) The solution to the total regression problem is
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= x cos () + Ysin ()
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(642)
68 CURVE APPROXIMATION
with
- LXi n i=l
(643) (644)
- LYi' n i=l
The orientation of the total regression line is tan2e with
e, given by
(645)
2 LX~Y~
i=l n n
(646) (64 7)
LX - LY
i=l i=l
I xi = Xi - X
Yi = Yi - y
(648) (649)
Total regression uses a least-squares error norm that is optimal if the errors are from a normal distribution, but is not suitable if there are outliers present in the data In the case of fitting a curve model to edge data, outliers would occur if the edge linking procedure incorporated one or more edges from other contours into the edge list for a contour Outliers can occur even if the edge linking procedure performs flawlessly For example, consider a list of edges from two adjacent sides of a rectangle The corner must be identified in order to segment the edges into the two sides before fitting a line to the edges If the corner point is not identified correctly, some edges may be assigned to the wrong side, and these edges are outliers In general, errors in classification introduce errors into the regression problem that are not normally distributed In such a case, the errors may be modeled by a mixture distribution that combines a Gaussian distribution for modeling the normal errors with a broad-tailed distribution for modeling the outliers due to imperfect classification
CHAPTER 6 CONTOURS
Estimating Corners
The best method for estimating corners is to use one of the methods for fitting a line to edge points and then compute the intersection of the lines This method compensates for the error introduced by edge detection operators that round off the corners and is more accurate than using a corner detector which only uses local information Given the implicit equations for two lines,
alx a2 x
+ b1y + Cl
(650) (651)
+ b2y + C2
the location of the intersection is (652)
(653)
If a 1 b2 - a2bl is close to zero, then the lines are nearly parallel and cannot be intersected A good method for detecting corners is to try to fit pairs of lines over successive sublists of 2n + m edge points along the contour The parameter n is the number of edge points required for an accurate line fit, and the parameter m is the number of edge points to skip between the sides of the corner The gap skips over the edge points in the rounded part of the corner A corner is detected by testing the magnitude of a 1b2 - a2bl against a threshold
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