What is the cost of adding a constraint in linear least squares?

TL;DR

This paper derives an exact formula quantifying how imposing a constraint in linear least squares affects the fitting error, and introduces a method to analyze the impact of constraints in practical applications like color calibration.

Contribution

It provides a precise expression linking constraints to error increase and a technique to separate data components affected by the constraints, with application to color calibration.

Findings

Exact error increase formula for constrained least squares

Method to decompose data into constrained and unconstrained components

Application demonstration in camera color calibration

Abstract

Although the theory of constrained least squares (CLS) estimation is well known, it is usually applied with the view that the constraints to be imposed are unavoidable. However, there are cases in which constraints are optional. For example, in camera color calibration, one of several possible color processing systems is obtained if a constraint on the row sums of a desired color correction matrix is imposed; in this example, it is not clear a priori whether imposing the constraint leads to better system performance. In this paper, we derive an exact expression connecting the constraint to the increase in fitting error obtained from imposing it. As another contribution, we show how to determine projection matrices that separate the measured data into two components: the first component drives up the fitting error due to imposing a constraint, and the second component is unaffected by the constraint. We demonstrate the use of these results in the color calibration problem.