Algebraic Invariance of Image Characteristics

Abstract

In the preceding chapter, we considered image characteristics that were transformed linearly under camera rotation. Such linear transformations defined representations of SO(3). By taking linear combinations, we rearranged such image characteristics into groups such that each had independent transformation properties. In mathematical terms, this process is the reduction of the representation. In this chapter, we remove the restriction of linearity. We consider image characteristics whose new values are algebraic expressions in the original values. By taking algebraic combinations, we rearrange them into groups such that each has independent transformation properties. Then, we construct algebraic expressions that do not change their values under camera rotation. Such expressions are called scalar invariants. We will also show that if two images depict one and the same scene viewed from two different camera angles, the camera rotation that transforms one image into the other can be reconstructed from a small number of image characteristics.