# Naturally Combined Shape-Color Moment Invariants under Affine Transformations

## TL;DR

This paper introduces a novel set of shape-color affine moment invariants (SCAMI) that simultaneously handle shape and color deformations under affine transformations, improving robustness in image analysis.

## Contribution

It presents the first direct derivation of invariants for dual affine transformations of shape and color, eliminating manual weight selection and extending invariants to affine groups.

## Key findings

- Proposed SCAMI24 set of invariants.
- Validated effectiveness on synthetic and real datasets.
- Demonstrated robustness against affine transformations.

## Abstract

We proposed a kind of naturally combined shape-color affine moment invariants (SCAMI), which consider both shape and color affine transformations simultaneously in one single system. In the real scene, color and shape deformations always exist in images simultaneously. Simple shape invariants or color invariants can not be qualified for this situation. The conventional method is just to make a simple linear combination of the two factors. Meanwhile, the manual selection of weights is a complex issue. Our construction method is based on the multiple integration framework. The integral kernel is assigned as the continued product of the shape and color invariant cores. It is the first time to directly derive an invariant to dual affine transformations of shape and color. The manual selection of weights is no longer necessary, and both the shape and color transformations are extended to affine transformation group. With the various of invariant cores, a set of lower-order invariants are constructed and the completeness and independence are discussed detailedly. A set of SCAMIs, which called SCAMI24, are recommended, and the effectiveness and robustness have been evaluated on both synthetic and real datasets.