or p = p x1 x2 x3 x4 where xi = xi yi 1

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4 denote image coordinates for four end points of the two lines and a presents The xi i = 1 the length of vector a. To avoid the treatment of a trigonometric function in calculating the partial derivatives of function p with respect to the image coordinates, we use a simpler function: p = cos p = a b = p x1 x2 x3 x4 a b (18.10)

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Let xi yi be the true value and xi yi be the noisy observation of xi yi , then we have xi = xi + yi = yi + where the noise terms variance i2 . Hence:

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(18.11a) (18.11b)

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denote independently distributed noise terms having mean 0 and

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=0 =

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(18.12) (18.13)

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= =0

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if i = j E otherwise

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if i = j otherwise

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(18.14a) (18.14b)

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From these noisy measurements, we define the noisy parallel function, p x1 y1 x2 y2 x3 y3 x4 y4 (18.15)

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To determine the expected value and variance of p , we expand p as a Taylor series at x1 y1 x2 y2 x3 y3 x4 y4 :

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p p +

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p + yi yi xi

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p yi

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=p +

p + xi

p yi

(18.16)

Then, the variance of the parallel function becomes:

Var p = E p p

2 0 i=1

p xi

p yi

(18.17)

Hence, for a given two lines, we can determine a threshold: p = 3 E p p

(18.18)

Because the optimal p equals 1, any parallel two lines have to satisfy the following condition: 1 p p (18.19)

6.2 Normal Distance

A normal distance between any two lines is selected from among two distances d1 and d2 : dnorm = max d1 d2 where d1 = d2 = a1 xm + b1 ym + c1

2 a2 + b1 1

(18.20)

(18.21a)

a2 xm + b2 ym + c2

2 a 2 + b2 2

(18.21b)

The ai bi and ci are line coefficients for the i-line and xm ym denotes the center point coordinate of the first line, and xm ym denotes the center of the second line. Similarly to the parallel case of Section 6.1, the normal distance is also a function of eight variables: dnorm = d x1 x2 x3 x4 Through all processes similar to the noise model of Section 6.1, we obtain:

(18.22)

Var p = E p p

2 0 i=1

p xi

p yi

(18.23)

For the given two lines, we can also determine a threshold for the normal distance: d = 3 E d d

(18.24)

Because the optimal d equals 0, the normal distance for any two collinear lines has to satisfy the following condition: d d (18.25)

Energy Model for the Junction Groups

7. Energy Model for the Junction Groups

In this section, we test the robustness of the junction detection algorithm by counting the number of detected junctions as a function of the junction quality QJ of Equation (18.1). Figure 18.4 shows some images of 2D and 3D objects under well-controlled lighting conditions and a cluttered outdoor scene. We use Lee s method [19] to extract lines. Most junctions (i.e. more than 80 %) extracted from possible combinations of the line segments are concentrated in the range 0 0 1 0 of the quality measure, as shown in Figure 18.5. The three experimental sets of Figure 18.4 give similar tendencies except for a small fluctuation at the quality measure 0.9, as shown in Figure 18.5. At the quality level 0.5, the occupied portion of the junctions relative to the whole range drops to less than 1 %. When robust line features are extracted, QJ , as a threshold for the junction detection, does not severely influence the number of extracted junctions. In good conditions, the extracted lines are clearly defined along the object boundary and few cluttered lines exist in the scene, as shown in Figure 18.4(a). Therefore, the extracted junctions are accordingly defined and give junctions with a high junction quality factor, as shown in Figure 18.4(a). The parts plot in Figure 18.5 graphically shows the high-quality junctions as the peak concentrated in the neighbor range of quality measure 0.9. For QJ = 0 7, the detection ratio 1.24 (i.e. number of junctions/number of primitive lines) of Figure 18.4(a) is decreased to 0.41 for the outdoor scene of Figure 18.4(c), indicating the increased effect of the threshold (see also Table 18.1). The effect of threshold level for the number of junctions resulting from distorted and broken lines is more sensitive to the outdoor scenes. That is, junction detection

Figure 18.4 Junction extraction: the number of junctions depends on the condition of the images. Each column consists of an original image, line segments and junctions and their intersecting points for quality measure 0.5, respectively. The small circles in the figure represent the intersection points of two-line junctions. (a) Parts: a 2D scene under controlled lighting conditions; (b) blocks: an indoor image with good lighting; and (c) cars: a cluttered image under an uncontrolled outdoor road scene.