A MATRIZ DA ADMINISTRAÇÃO DO TEMPO

The resultant arc features from circle fitting provide the measurements for geometric shape detection. Clustered arc features define a hypothesis for the surface of a geometrically shaped object, represented by a set of points. For the detection of a known fiducial marker, the number of geometric hypotheses can be constrained with the marker parameters and domain knowledge of the environment. In this thesis, for the case of data that is acquired with a rotating laser, two points or features are close enough to be

considered for a hypothesized feature if their center points are within 10 degrees of LIDAR rotation from each other. This limits the number of points and features that are tested for fit to a 3D geometric feature.

A RANSAC approach searches for model fit of feature clusters to the fiducial marker model of interest. The sets of neighbor arc features described above are then tested for fit to a cylinder or a cone. The initial step of determines inliers by comparing the radius of the arc feature and the orthogonal distance from the center of the arc to the hypothesis axis line. Figure 7 shows the distance, d, from a 3D point being tested, X , _o

to the hypothesis line with endpoints X X . This distance is calculated as ₁, ₂



 



The symbol  denotes the vector cross product and the symbol denotes the Euclidean

norm of a vector.

Once the set of inliers is determined, the axis line is redefined with the consensus set of arc center points and performing an orthogonal distance regression. To compute this, first find the centroid of the point data and then form a matrix A , where each row are the point coordinatesx y z shifted by the mean centroid value. For the line of best fit , , to a set of 3D points, solve for the eigenvector of matrix A that corresponds to its largest eigenvalue using an SVD decomposition of A where

A=UWVt

such that U is an orthogonal matrix, the diagonal values in W are the singular values, and Vt is the transpose of the matrix containing the eigenvectors of matrix A which are also the principal components of the point data. The 3D line of best fit includes the centroid of the data and the direction vector is the eigenvector described above. For further discussion of the method of 3D line fitting with singular value decomposition see (Golub, 1996, pp. 70-73).

With an axis fit from the arc centers in the consensus set, the fiducial marker geometry is estimated using the arc feature radii. This geometric model is then tested for goodness of fit by calculating the resulting error of all the points that constitute the axis line arc features. Acceptance of the geometric model is gauged using this error estimate and thresholds set to minimize spurious geometric models that fit well to noise.

To improve the RANSAC method performance for fiducial marker detection, knowledge of the fiducial marker size and shape and the laser scan pattern of the VLS limit the number of random hypotheses. Table 7 describes the individual thresholds and Table 8 shows the entire algorithm.

Threshold Definitions

max

initial set of points for an axis hypothesis

= minimum number of arcs required for cone or cylinder hypothesis = maximum number of iterations

= error tolerance for point fit to geometr

p arcs shape N N K e  max ic shape hypothesis error tolerance for points fit to a circular arc

= max cone height threshold to eliminate tall, thin cones that fit cylinder data points

max radius to circle cone e h r   min

contain search for fiducial marker min radius to contain search for fiducial marker rotation window size for clustering circle arcs = maximum inlier distance for arc center point to an a

rot line r t t   _ xis line cosine of difference angle between orientation directions max difference in aspect ratio, the height/base_radius

horizontal gradient threshold for co

orientation aspect horizontal gradient t t t  

 nstant range flag gradient threshold for segmentation flag

segment

t 

Table 7. Threshold definitions.

Algorithm: (RANSAC approach for 3D fiducial marker detection)

1 begin initialize:

2. for all arc feature center points:

3 determine set of neighbor arcs using t_rot

4 for all sets of neighbor arc clusters of size > N_arcs: 5 while k < K_max do:

6 choose 2 arc centers

7 line  calculate a hypothesis axis line

8 for all remaining points in the neighbor arc set: 9 form a consensus set of circle arcs for an axis line 10 if center point distance to axis line < t_line

11 then include arc in the consensus set 12 if consensus set size > N_arcs:

13 axis  refit axis line from consensus set arc center points 14 for all arcs in consensus set:

15 rms_error  0

14 for all points in the arc:

15 rms_error  rms_error + point fit error to fiducial model 16 if rms_error < best fit consensus set

17 then this set is the new best fit set

18 remove consensus set arcs from the set of features for this iteration 19 next iteration, k = k + 1

20 return best fit consensus arc feature set

21 if fiducial hypothesis base_radius < r_maxand base_radius > r_min: 22 v₁  fiducial hypothesis orientation

23 v₂  fiducial known orientation 24 if v v ₁ ₂ t_orientation:

25 h height of fiducial hypothesis 26 r radius of fiducial hypothesis 27 if cylinder fiducial hypothesis: 28 then valid cylinder fiducial 29 else if cone fiducial hypothesis: 30 if h f_aspect t_aspect

r   and hhcone:

31 then fiducial hypothesis is valid cone fiducial marker 32 end

A region growing method was ruled out due to the reliance on implicit connectivity in the data. The data generated from the VLS LIDAR has a high horizontal density and a sparse vertical density. With only 64 laser scan lines, the vertical separation between scan lines increases the potential for error if vertical connectivity associations are assumed incorrectly.