A Different Way to Think about Plane Fitting

9 months ago

Plane fitting in 3D computer vision is a common problem, often approached using Principal Components Analysis (PCA) to identify the plane's normal.
PCA works by analyzing the spread of data points, where the eigenvector associated with the smallest eigenvalue represents the plane's normal.
Traditional PCA-based plane fitting can be sensitive to outliers, prompting the need for robust methods like RANSAC or robust PCA.
Non-linear least squares optimization offers advantages such as joint parameter estimation, weighted inputs, and robust cost functions for outlier rejection.
Optimizing plane fitting as a least squares problem involves using a point-to-plane cost function, constrained by the unit vector requirement of the normal.
The space of 3D unit vectors (S2) is not a Lie group, but optimization can still be performed by leveraging rotations (SO3) to parameterize the normal.
By optimizing over SO3 and fixing one parameter (roll around the canonical vector), the problem reduces to an unconstrained non-linear least squares optimization.
Robust least squares fitting can be easily implemented to handle outliers, providing a more reliable plane fit compared to traditional methods.
Future exploration includes direct optimization over S2, despite it not being a Lie group, for more efficient plane fitting solutions.

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