Homework 3 The sphere localized

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(a) (b)
Figure 1: (a) the sphere localized in Q1. (b) the cylinder localized in Q2.
PA Q1: In this question, you must use RANSAC to localize a sphere given
point cloud data. The point cloud contains a ball that is only partially visible
to the sensor. The position of the center of the ball is unknown. The radius
is unknown, but between 5cm (0.05m) and 11cm (0.11m). Write a function
in Q1.m that calculates these two quantities. You can test your code using
the cropped point cloud (see commented hw3.m). But, you should submit
code that works on the entire point cloud without cropping. Important:
please do NOT use the built-in Matlab function for sphere fitting.
Hint: one way to generate sphere hypotheses is as follows:
1. sample a point from the cloud;
2. sample a radius of the candidate sphere between 5 and 11cm;
3. project a vector from the sampled point in the direction of the associated surface normal for a distance equal to the sampled radius. This
point would be at the center of the candidate sphere.
PA Q2: Same as Q1 except for a cylinder. This question is harder because
you need to calculate the center, the orientation, and the radius (between
0.05m and 0.1m). The orientation should be returned in the form of a unit
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vector pointing along the axis of the cylinder (either direction is fine). You
only need to solve this problem for the segmented cloud, as implemented in
the code.
Hint: one way to generate cylinder hypotheses is as follows:
1. Sample a radius for the candidate cylinder between 5 and 10 cm.
2. Sample two points from the cloud
3. Set the cylinder axis direction equal to the direction of the cross product between the surface normals associated with the two sampled
points.
4. Pick one of the sampled points from the cloud and use it to estimate
a candidate center, just as you did in Q1.
5. Project the points in the cloud onto the plane orthogonal to the axis
you just calculated. You can do this projection by multiplying the
points in the cloud by this matrix: I − aˆaˆ
T
, where ˆa is equal to the
axis of the cylinder. Also project the candidate center into this plane
in the same way.
6. Evaluate number of inliers in the plane for a circle with the given
projected center and the sampled radius.
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