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represent the edge list
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Definition 63 A boundary is the closed contour that surrounds a region
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In this chapter, the term edges will usually refer to edge points The edge orientation is not used by most curve fitting algorithms In the few cases where the algorithm does use the edge orientation, it will be clear from the context that the term edges refers to edge fragments
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CHAPTER 6 CONTOURS
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Geometry of Curves
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Planar curves can be represented in three different ways: the explicit form y = f(x), the implicit form f(x,y) = 0, or the parametric form (x(u),y(u)) for some parameter u The explicit form is rarely used in machine vision since a curve in the x-y plane can twist around in such a way that there can be more than one point on the curve for a given x The parametric form of a curve uses two functions, x (u) and y (u ), of a parameter u to specify the point along the curve from the starting point of the curve at PI = (X(UI),y(UI)) to the end point P2 = (X(U2),y(U2)) The length of a curve is given by the arc length:
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The unit tangent vector is
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dX) 2 (d Y ) 2 ( du + du duo
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(61)
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p/(U)
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t (u) =
Ip' (u ) I '
(62)
where p( u) = (x( u), y( u)) The curvature of the curve is the derivative of the tangent: n(u) = p"(u) Consider three points along the curve: p(u + 6), p(u), and p(u - 6) Imagine a circle passing through these three points, which uniquely determine the circle In the limit as 6 --+ 0, this circle is the osculating circle The osculating circle touches the curve at p( u), and the center of the circle lies along the line containing the normal to the curve The curvature is the inverse of the radius of the osculating circle
Digital Curves
In this section, we present a set of algorithms for computing the elements of curve geometry, such as contour length, tangent orientation, and curvature, from the list of edge points Slope and curvature are difficult to compute precisely in the digital domain, since the angle between neighboring pixels is quantized to 45 increments The basic idea is to estimate the tangent orientation using edge points that are not adjacent in the edge list This allows a larger set of possible
62 DIGITAL CURVES
tangent orientations Let Pi = (Xi, Yi) be the coordinates of edge i in the edge list The k-slope is the (angle) direction vector between points that are k edges apart The left k-slope is the direction from Pi-k to Pi' and the right k-slope is the direction from Pi to Pi+k The k-curvature is the difference between the left and right k-slopes Suppose that there are n edge points (Xl, Yl), , (Xn' Yn) in the edge list The length of a digital curve can be approximated by adding the lengths of the individual segments between pixels:
V(Xi - Xi-l)2
+ (Yi
- Yi-l)2
(63)
A good approximation is obtained by traversing the edge list and adding 2 along sides and 3 along diagonals, and dividing the final sum by 2 The distance between end points of a contour is
(64)
Chain Codes
Chain codes are a notation for recording the list of edge points along a contour The chain code specifies the direction of a contour at each edge in the edge list Directions are quantized into one of eight directions, as shown in Figure 61 Starting at the first edge in the list and going clockwise around the contour, the direction to the next edge is specified using one of the eight chain codes The direction is the chain code for the 8-neighbor of the edge The chain code represents an edge list by the coordinates of the first edge and the list of chain codes leading to subsequent edges A curve and its chain code are shown in Figure 62 The chain code has some attractive properties Rotation of an object by 45 can be easily implemented If an object is rotated by n x 45 , then the code for the rotated object is obtained by adding n mod 8 to the original code The derivative of the chain code, also called difference code, obtained by using first difference, is a rotation-invariant boundary description Some other characteristics of a region, such as area and corners, may be directly computed using the chain code The limitation of this representation is
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