Group-Theoretical Methods in Image Understanding pp 103-146 | Cite as

# 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.

## Keywords

Image Characteristic Optical Flow Scalar Invariant Algebraic Expression Symmetric Polynomial## Preview

Unable to display preview. Download preview PDF.

## Bibliography

- H. Weyl: The Classical Groups, Their Invariants and Representations (Princeton University Press, Princeton, NJ 1946)zbMATHGoogle Scholar
- 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
- E. Kroner (ed.): Mechanics of Generalized Continua (Springer, Berlin, Heidelberg 1968)zbMATHGoogle 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: On a general representation theorem for constitutive relations. Arch. Ration. Mech. Anal. 33, 1–25 (1969)MathSciNetCrossRefGoogle Scholar
- C.-C. Wang: On representations for isotropic functions, Part II. Isotropic functions of skew-symmetric tensors, symmetric tensors, and vectors. Arch. Ration. Mech. Anal. 33, 249–267 (1969)CrossRefGoogle Scholar
- G. F. Smith: On a fundamental error in two papers of C.-C. Wang “On representations for isotropic functions, Parts I and II”, Arch. Ration. Mech. Anal. 36, 161–165 (1970)CrossRefGoogle 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
- G. F. Smith: On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int. J. Eng. Sci. 9, 899–916 (1971)MathSciNetCrossRefGoogle Scholar
- C.-C. Wang: Corrigendum to my recent papers on “Representations for isotropic functions”. Arch. Ration. Mech. Anal. 43, 392–395 (1971)CrossRefGoogle Scholar
- K. Kanatani: Camera rotation invariance of image characteristics. Comput. Vision, Graphics Image Process. 39, 328–354 (1987)CrossRefGoogle Scholar