Intersection of three points

Detection accuracy can be further improved by using intersection points of three lines for each endpoint. This is done by utilising the peak coordinates (ρ0, θ0), without imposing any extra computational burden. Note that peak

information should not be neglected but used alongside the butterfly bound- ary information. It is easy to further increase the reliability of the result by introducing a third straight line. Because the peak corresponds to the line passing through both of the endpoints, it can be considered as a valuable cri- terion. Note that, as a part of the MLFRFT processing steps, this peak will

be captured via the peak detection process; therefore, no extra computation will be imposed to the system.

Figure 4.4: Figure 4.3 is cropped and the intersection area for pf is enlarged.

The intersection area forms a triangular shape. The true endpoint is defined as the geometrical centre of the triangle.

Figure 4.3 shows the intersection of the lines corresponding to the bound- ary points and the peak (coloured in red) in the image domain. In theory, straight-lines represented by (ρf 1, θ1), (ρf 2, θ2) and (ρ0, θ0) have one commu-

nal intersection which is the first endpoint (pf in Figure 4.2(a)). The same

scenario applies for the straight-lines represented by (ρs1, θ1), (ρs2, θ2) and

(ρ0, θ0) to detect the second endpoint (ps in Figure 4.2(a)). However, in

practice, these lines do not meet at an exact point and a triangular shape forms in the intersection area. This is due to rounding errors caused by discretisation of the Hough space.

For instance, if we crop the Figure 4.3 and enlarge the intersection area of pf, a triangular shape can be seen as shown in Figure 4.4. A fair ap-

endpoint. Geometric barycentre or centroid of a triangle is the average (arith- metic mean) position of the three vertices. Simply speaking, it is a single point where the three medians1 of a triangle intersect. If (x1, y1), (x2, y2) and

(x3, y3) are three vertices of the triangle then the centroid is given by

c =  (x1+ x2+ x3) 3 , (y1+ y2+ y3) 3  (4.4)

A formal statement for the LSBB algorithm is given as follows:

Step 1) Curve fitting. A number of adjacent sinusoids on each boundary will be fitted into a single curve using a Robust Least Squares method. This is done by choosing multiple columns along θ. Empirical studies showed twenty columns on each side of the peak can provide sufficient accuracy for curve fitting.

Step 2) Boundary point selection. Define the ∆θ and seek the rising

and falling edges along the two column vectors θ1 and θ2 and extract

(ρf 1, θ1), (ρf 2, θ2), (ρs1, θ1) and (ρs2, θ2).

Step 3) Solving simultaneous equations. Find the line intersections of straight-lines represented by (ρf 1, θ1), (ρf 2, θ2), and (ρ0, θ0) in the

image domain. A triangular shape appears in the intersection area. The target endpoint is found by locating the centroid of the triangle using Equation 4.4.

Step 4) Repeat Step 3 for the second endpoint. Solving the simul- taneous equations to find the intersections of (ρs1, θ1), (ρs2, θ2), and

(ρ0, θ0).

Step 5) Draw the segment. Connect the two endpoints to identify the straight-line segment.

1_{Median of a triangle is a straight line cutting through a vertex and the midpoint of}

(a) (b)

Figure 4.5: (a) Mixed shape image. (b) Its overlapping sinogram.

Problem

Despite its accuracy, the LSBB algorithm is highly dependent on the num- ber of lines in the image space and detection precision can be unfavourably affected when the number of segments increases. This will lead to a complex Hough space with superimposed butterfly wings. That is a recurrent problem in image processing applications because there are typically many line seg- ments in an image. For instance, Figure 4.5(a) shows a test image reported in [70, 2, 71], consisting of a mixture of three distinguishable straight-line segments and a circle, with its corresponding superimposed butterflies in Figure 4.5(b). It can be seen, that the sinusoids of L2 are overlapped with

those of L1, L3, and the circle. Accordingly, seeking boundary points for each

peak will be a problematic task and will eventually yield false endpoints. In fact, the essential condition for the LSBB algorithm is to have a transparent single butterfly when the impressions of the other peaks are eliminated. This suggests decomposing the Hough space in a way that, each butterfly appears in a separate Hough space.

Solution

Basically, each butterfly has to be treated separately. There are two ways to do this: either operations in Hough space or enhancements in image space. To apply operations in Hough space a typical solution could be to define a “window-of-interest” around each peak and its surrounding sinusoids [63, 64]; however, it would not guarantee the complete elimination of adjacent peaks and their overlapping sinusoids. Instead for each peak a window can be defined in image space around the straight-line representing that peak. The concept of neighbourhood of a straight-line can be used to construct such a window in the image plane [67]. However, one must go through the entire process of building a proper neighbourhood and compromise on the radius selection for each segment. The neighbourhood approach also requires tedious parameter tuning. In the following, a new window-of-interest in the image domain is defined and the butterfly of each segment is isolated without neighbourhood identification. Because computation of each window and its corresponding butterfly is independent the algorithm can be implemented in parallel.