# Algebraic and projective invariants

## Abstract

We have looked at invariants of binary forms and of ternary forms; the former play a crucial role in allowing one to use moments to form features invariant to affine transformations, while the latter are important in obtaining invariance to projective transformations.

We saw that the quadratic, cubic and quartic binary forms have nine independent invariants, but also a large number of irreducible invariants. Many of the irreducible invariants are high order polynomials in the coefficients with a large number of terms, which means that the moment invariants based on them are more susceptible to image distortions, as will be demonstrated in the following chapter. Nevertheless, we will see that a number of moment invariants are robust and useful features, in particular when recognizing partially occluded objects in chapter 6.

A ternary form defines a polynomial plane curve, from which we saw that polynomial plane curves of a given order remain polynomial curves of the same order under planar projection. To obtain invariant functions of the polynomial's coefficients one needs more coefficients than degrees of freedom (eight); hence five lines provide two independent invariants, as do two conics; a cubic provides one and a quartic provides six, one of which was listed explicitly. Chapter 5 discusses how these invariants can be used to recognize projectively transformed images.

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