Line Fitting in 3D

Orthogonal Regression is the process of finding the line that best fits a set of points by minimizing their squared orthogonal distances to it.

This is more powerful than traditional least squares in that it is invariant to global rotation.

Because of this, however, it has no analytic solution in 3D and above. There are a variety of iterative solutions available, but they often depend on an operation called the Singular Value Decomposition or SVD. The SVD is a complex matrix decomposition that is usually only found in heavy-weight matrix libraries. Avoiding the inclusion of these libraries is the motivation for the technique outlined in this post.

The Centroid

The first key insight is that the orthogonal regression line will always pass through the average (or centroid) of the points.

The centroid can be computed by adding the points together, and then dividing by the number of points.

centroid = Vector3.zero;
foreach point in points {
  centroid += point;
}
centroid /= points.count;

Now that we have a point that the line passes through, we just need to calculate its direction.

Getting Closer

Since this algorithm is iterative, we need an iteration function that gets us closer to the true line direction at each step.

We can choose almost any** direction to start, and we will iteratively work toward the real direction.

The key to calculating the next direction estimate is to multiply each point by the dot product of it and the current estimate. This moves all the points to the same hemisphere as the current guess, allowing you to simply sum them and normalize them for the new direction estimate.

// Solve for the centroid (See Above)
nextDirection = Vector3.zero;
foreach point in points {
  centeredPoint = point - centroid;
  nextDirection += dot(centeredPoint, direction) * centeredPoint;
}
direction = nextDirection.normalize();

In formal mathematics, this is known as Power Iteration.

A Virtuous Cycle

We only need to run the stepping routine repeatedly to converge on the line of best fit.

// Solve for the centroid (See Above)
for(i = 0; i < iterations; i++){
 // Step the best fit line direction (See Above)
}

No SVD’s required!

Sequential Down Projection for Secondary Axes

Typically one might extend the line fitting code to fit planes by projecting the points onto the plane defined by the primary axis, and then fitting the secondary axis on that flattened set of points.

// Solve for the centroid (See Above)
// Fit Primary Axis with normal points(See Above)
foreach(point in points){
  flattenedPoints.push(projectOnPlane(point, primaryAxis));
} 
// Fit Secondary Axis with flattened points (See Above)
normal = cross(primaryAxis, secondaryAxis).normalized;

Abusing Singularities for Fun and Profit

** The observant among you might have noticed that there exist starting angles where the progress towards the true fitting line is 0. This is actually a “singularity” that occurs when the starting guess is perfectly orthogonal to the true answer. One should check for these cases, though it’s rare in practice.

However, it turns out we can abuse this behaviour to fit all of the axes simultaneously! Instead of projecting the points onto the plane defined by the primary axis, we project the current estimation of the secondary axis itself! This lets us solve for both the primary and secondary axes simultaneously!

// Solve for the centroid (See Above)
for (int iter = 0; iter < iters; iter++) {
  newPrimaryAxis = Vector3.zero, newSecondaryAxis = Vector3.zero;
  foreach(point in points) {
    point = worldSpacePoint - origin;
    newPrimaryAxis += dot(primaryAxis, point) * point;
    newSecondaryAxis += dot(secondaryAxis, point) * point;
  }
  primaryAxis = newPrimaryAxis.normalized;
  secondaryAxis = projectOnPlane(newSecondaryAxis, 
                                 primaryAxis).normalized;
}
normal = cross(primaryAxis, secondaryAxis).normalized;

From here, it is trivial to see how one might add additional secondary axes of fit. In formal math terminology, this is equivalent to alternating Power Iteration and Gram-Schmidt Orthonormalization.

There might come a point when numerical instability will prevent the lower-order axes from fitting nicely in this paradigm. If that happens, I recommend just projecting your points down to the plane (defined by the previous axis) after solving each axis.

I hope these techniques will come in handy whenever someone wants to fit lines or planes without incurring large dependencies within their projects.

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