 # Least-Squares Crossing Lines in Matlab

## Transcribed Text

Problem: Least-Squares Crossing Lines In this problem, you will be given a set of data that are on the XY plane and a matching set of data in space. The spatial data also happen to lie on a plane, which is tilted and offset from the XY plane. The two data sets represent a common configuration: the output of image-processing where the spatial data have been sensed as the planar data using an X-ray imaging device. For simplicity, we will model the X-rays as coming from a point source. The data given to you do not perfectly cross. You must: (i) Estimate the focus as the point that minimizes line error, and (ii) Summarize your results The instructor used a uniform distribution to deviate known planar points from their nominal positions. You are encouraged to also do this, to gain experience in generating random data for testing your numerical code. (If you do this, you need to clearly state your methods.) For example, you can add small random numbers to the X and Y values of PlanarPoints to study the behavior of your estimator. The base data, the resulting lines, and the data provided to you with the nearly crossing lines are given in Figure 1. (A) (B) (C) Figure 1: Data to fit to a set of lines. (A) The base data, stars on the X-Y plane and circles in space. (B) The data fit lines that exactly cross. (b) Randomly deviated data produce lines that cross imperfectly or not at all. The data for the assignment is in a file of Matlab code that creates the data for you. If you execute the Matlab command a4data then you will find the variables PlanarPoints and SpatialPoints that are the data. In your report, clearly and concisely summarize your findings. You may analyze the errors as the distances from the least-squares focal point to the lines, or the skew-line distance, or any other reasonable error measure.

## Solution Preview

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PlanarPoints = ...
[[0.96; 1.05; 0] [1.00; 8.02; 0] [8.07; 1.99; 0] ...
[7.95; 7.95; 0] [5.00; 2.93; 0]];

SpatialPoints = ...
[[3.75; 3.75; 13.75] [3.60; 6.05; 12.98] [7.14; 2.86; 5.71] ...
[7.73; 7.73; 1.82] [5.00; 4.05; 10.48]];
figure()
for i=1:5
j=3*(i-1);
%    line([PlanarPoints(j+1);SpatialPoints(

X(i)=PlanarPoints(j+1);
Y(i)=PlanarPoints(j+2);
Z(i)=PlanarPoints(j+3);
Xsp(i)=SpatialPoints(j+1);
Ysp(i)=SpatialPoints(j+2);
Zsp(i)=SpatialPoints(j+3);
%line([X(i);Xsp(i)],[Y(i);Ysp(i)],[Z(i);Zsp(i)]);

t=0:1:20;
x=X(i)+(Xsp(i)-X(i))*t;
y=Y(i)+(Ysp(i)-Y(i))*t;
z=Z(i)+(Zsp(i)-Z(i))*t;
axis([0 10 0 10 0 25])
%plot3([X(i),Xsp(i)], [Y(i), Ysp(i)], [Z(i), Zsp(i)])
plot3(x,y,z)
hold on
xlabel('x')
ylabel('y')
zlabel('z')
end...
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