Computer Aided Engineering Design

300 COMPUTER AIDED ENGINEERING DESIGN

after threshold operation for a pixel i is given as Ii = 1 if Ii≤Ith else Ii = 0. Figure 10.4 (a) shows the
intensity values before and after the threshold operation and the corresponding images are shown in
Figure 10.4(b).

Figure 10.4 Threshold operations on an image

(a) Intensity graph before and after threshold (b) Image before and after threshold

The subsequent step is of edge detection. The operation takes an intensity image I procured
previously as its input, and returns a binary image BW of the same size as I, with 1’s where the
function finds edges in I and 0’s elsewhere. Edge pixels are those where the intensity gradients are
above a fixed threshold. A gradient threshold function is shown in Figure 10.5(a) and the edge
detection operation is illustrated with an example in Figure 10.5(b). Once the edge points are identified,
they are tracked to form a closed loop forming the contour on a plane. A traditional neighborhood

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0.5

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0 0.5 1

(a) The edge detection function (b) Edge detection

Figure 10.5 Illustration of edge detection

detection algorithm would work well except for regions where some branching may occur. A way to
overcome this is to follow the edge point having the lower angle of curving (Figure 10.6 a). This
ensures that the wrong edge points are not selected to form the edge loop. Once the edge pixels
are detected, in many applications, a smooth closed B-spline curve is interpolated through them
(Figure 10.6b).

After reconstructing all contours, a tiling or skinning surface is created. In some applications, the
reconstructed contours can be used as the CAD representation as in case of layered manufacturing.
However, most other CAD/CAM operations require a B-rep model. For an object with Mcontours,
Ci(ui), 1 ≤i≤ Mwith respective parameterization ui, we wish to fit a surface S(u,v) through the
collection of contours. The surface will be closed and periodic in the udirection (direction along the
given contours) and open in the vdirection (direction through the contours). We may reparameterize
the contours using a common global parameter uto fit the tiling surface. An elementary method to
determine parameter correspondence is to assume a one-to-one relationship between given contours.
We may assume all the contours to be parameterized between 0 and 1 (or any other bounds) and that