how the intersection of two 2D lines can be expressed as their cross product, assuming the lines are expressed as homogeneous coordinates. 1\. If you are given more than two lines and want to find a point \(\tilde{\boldsymbol{x}}\) that minimizes the sum of squared distances to each line, $$ D=\sum_{i}\left(\bar{x} \cdot \bar{l}_{i}\right)^{2} $$ how can you compute this quantity? (Hint: Write the dot product as \(\tilde{\boldsymbol{x}}^{T} \overline{\boldsymbol{l}}_{i}\) and turn the squared quantity into a quadratic form, \(\overline{\boldsymbol{x}}^{T} \boldsymbol{A} \tilde{\boldsymbol{x}}\).) 2\. To fit a line to a bunch of points, you can compute the centroid (mean) of the points as well as the covariance matrix of the points around this mean. Show that the line passing through the centroid along the major axis of the covariance ellipsoid (largest eigenvector) minimizes the sum of squared distances to the points. 3\. These two approaches are fundamentally different, even though projective duality tells us that points and lines are interchangeable. Why are these two algorithms so apparently different? Are they actually minimizing different objectives?

Step by step solution

01

Intersection of Two 2D Lines

In the homogeneous coordinates, intersection point \( \tilde{\boldsymbol{p}} \) of two lines \( \tilde{\boldsymbol{l_1}}\) and \(\tilde{\boldsymbol{l_2}} \) can be calculated as their cross product. \( \tilde{\boldsymbol{p}} = \tilde{\boldsymbol{l_1}} \times \tilde{\boldsymbol{l_2}} \).

02

Minimizing the Sum of Squared Distances

Given multiple lines, we can find the point \( \tilde{\boldsymbol{x}} \) that minimizes the sum of squares of distances to each line, expressed as \( D=\sum_{i}\left(\breve{\boldsymbol{x}} \cdot \breve{\boldsymbol{l}}_{i}\right)^{2} \). We can rewrite the dot product as \( \tilde{\boldsymbol{x}}^{T} \breve{\boldsymbol{l}}_{i} \) and turn the square quantity into a quadratic form, \( \breve{\boldsymbol{x}}^{T} \boldsymbol{A} \tilde{\boldsymbol{x}} \) where A is the coefficients matrix.

03

Fitting a Line to Points

For a set of points, the centroid (mean) and covariance matrix can be calculated. The line that passes through the centroid along the major axis of the covariance ellipsoid (largest eigenvector) minimizes the sum of squared distances to the points.

04

Comparison of Different Approaches

Although points and lines are interchangeable due to projective duality, the two algorithms look different because one minimizes the distance between a point and lines while the other minimizes the distance from points to a line. They are minimizing different objectives.

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