Algebraic Invariance of Image Characteristics
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.
KeywordsImage Characteristic Optical Flow Scalar Invariant Algebraic Expression Symmetric Polynomial
Unable to display preview. Download preview PDF.
- C. Truesdell, R. A. Toupin: Classical Field Theories, Handbuch der Physik, Group 2, Vol. 3, Part (Springer, Berlin, Heidelberg 1960)Google Scholar
- A. C. Eringen: Nonlinear Theory of Continuous Media (McGraw-Hill, New York 1962)Google Scholar
- C. Truesdell, W. Noll: Nonlinear Field Theories of Mechanics, Handbuch der Physik, Group 2, Vol. 3, Part 3 (Springer, Berlin, Heidelberg 1965)Google Scholar
- A. C. Eringen (ed.): Continuum Physics, Vols. I-IV (Academic, New York 1971, 1975, 1976, 1977)Google Scholar
- A. J. M. Spencer: “Theory of Invariants”, in Continuum Physics, Vol. 1, ed. by A. C. Eringen (Academic, New York 1971) pp. 239–353Google Scholar
- C.-C. Wang: A new representation theorem for isotropic functions: An answer to Professor G. F. Smith’s criticism of my papers on representations for isotropic functions, Part 1, Scalar-valued isotropic functions. Arch. Ration. Mech. Anal. 36, 166–197Google Scholar
- C.-C. Wang: A new representation theorem for isotropic functions: An answer to Professor G. F. Smith’s criticism of my papers on representations for isotropic functions, Part 2, Vector-valued isotropic functions, symmetric tensor-valued isotropic functions, and skew-symmetric tensor-valued isotropic functions. Arch. Ration. Mech. Anal. 36, 198–223 (1970)CrossRefGoogle Scholar