Invariance to affine transformations

Abstract

Above we have seen a number of different approaches to forming functions invariant to affine image transformations: moment invariants, moment-based normalization, Fourier descriptors, differential (local) invariants and correlation invariants. We saw in chapter 1 that we would like invariant features to index the correct model in a database; normalizing an image followed by template matching appears to be robust to additive noise [36], but it achieves this at the expense of indexing into the database — each model must be matched with the normalized image. The remaining features all provide indexing, so how do their performances compare?

The experiments indicate that, of all the above invariants, the global moment invariants (rather than the point-based ones) perform best, both on binary images and on grey-level images. Nevertheless, in specific cases other invariants may provide better performance — for example, the correlation invariants classified one of the digits better than the moment invariants.

Particular emphasis was placed on the invariants' performance with coarsely sampled images; the moments perform well, and are more robust than the Fourier descriptors based on Arbter's modification [80] to the original affine arc-length parameterization

The moments were also seen to be reliable at detecting rotational symmetries, but completely unable to detect reflectional symmetry; it appears that normalization followed by template matching is necessary if one wants to distinguish between two shapes that are reflections of one another.

Finally, the permutation-invariant point-based features were seen to be considerably less stable than those that depend on a priori knowledge of the ordering, such as the affine coordinates.

Recognition schemes that make use of the above features to recognize partially occluded objects will be discussed in chapter 6; first, let us look at features invariant to projective transformations.