^{1}

^{2}

^{1}

^{2}

This paper analyzes linear least squares problems with absolute quadratic constraints. We develop a generalized theory following Bookstein's conic-fitting and Fitzgibbon's direct ellipse-specific fitting. Under simple preconditions, it can be shown that a minimum always exists and can be determined by a generalized eigenvalue problem. This problem is numerically reduced to an eigenvalue problem by multiplications of Givens' rotations. Finally, four applications of this approach are presented.

The least squares methods cover a wide range of applications in signal processing and system identification [

Let

By replacing

The set

In the following, we set

In the following two sections, we introduce the theoretical basics of this optimization. The main result is the solution of a generalized eigenvalue problem. Afterwards, we reduce this system numerically to an eigenvalue problem. In Section

If

The real regular matrix

The minimization problem in (

If

Let

Let

If

With

Because of

The set of all eigenvalues of a matrix

In case of

The following lemma is a modified result of Fitzgibbon [

The signs of the generalized eigenvalues of (

With

For the following proofs, we need the lemma of Lagrange (see, e.g., [

For

Let

Let

We consider

In fact, in Lemma

The matrix

Let

For the smallest value

The matrix

We minimize

In numerical applications, a generalized eigenvalue problem is mostly reduced to an eigenvalue problem, for example, by multiplication with

Many times,

With Theorem

Let

and a generalized eigenvector

First, we would like to find an ellipse for a given set of points in

A numerically stable noniterative algorithm to solve this optimization problem is presented by Halir and Flusser [

Instead of ellipses, O'Leary and Zsombor-Murray want to find a hyperbola to a set of scattered data

In Bookstein's method, the conic constraint is restricted to

After the molding process in optical applications, the shrinkage of rotation-symmetric aspheres is implicitly defined for

The coefficients

In this paper, we present a minimization problem of least squares subject to absolute quadratic constraints. We develop a closed theory with the main result that a minimum is a solution of a generalized eigenvalue problem corresponding to the smallest positive eigenvalue. Further, we show a reduction to an eigensystem for numerical calculations. Finally, we study four applications about conic approximations. We analyze Fitzgibbon's method for direct ellipse-specific fitting, O'Leary's direct hyperbola approximation, Bookstein's conic fitting, and an optical application of shrinked aspheres. All these systems are attribute to the general optimization problem.