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Derivation of Invariants by Tensor Methods

Derivation of Invariants by Tensor Methods. Tomáš Suk. Motivation. Invariants to geometric transformations of 2D and 3D images. Tensor Calculus. William Rowan Hamilton, On some extensions of Quaternions, Philosophical Magazine (4th series): vol. vii (1854), pp. 492 - 499,

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Derivation of Invariants by Tensor Methods

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  1. Derivation of Invariants by Tensor Methods Tomáš Suk

  2. Motivation Invariants to geometric transformations of 2D and 3D images

  3. Tensor Calculus William Rowan Hamilton, On some extensions of Quaternions, Philosophical Magazine (4th series): vol. vii (1854), pp. 492-499, vol. viii (1854), pp. 125-137, 261-9, vol. ix (1855), pp. 46-51, 280-290. GregorioRicci, Tullio Levi-Civita, Méthodes de calcul différentiel absolu et leurs applications, Mathematische Annalen (Springer) 54 (1–2): pp. 125–201, March 1900.

  4. Tensor Calculus Grigorii Borisovich Gurevich, Osnovy teorii algebraicheskikh invariantov, (OGIZ, Moskva), 1937. Foundations of the Theory of Algebraic Invariants, (Nordhoff, Groningen),1964. David Cyganski, John A. Orr, Object Recognition and Orientation Determination by Tensor Methods, in Advances in Computer Vision and Image Processing, JAI Press, pp. 101—144, 1988.

  5. Tensor Calculus Tensor = multidimensional array + rules of multiplication • Enumeration • Sum of products(dot product) 2 rules: Example: product of 2 vectors Sum of products:

  6. Tensor Calculus Example: product of 2 vectors aibk=cik Enumeration:

  7. Tensor Calculus Example: product of 2 matrices aijbkl Ienumeration, j=ksum of products, lenumeration

  8. Einstein Notation Instead of we write other options vectors: matrices:

  9. Other Notations Other symbols Penrose graphical notation

  10. Tensors in Affine Space Contravariant vector Affine transform in 2D Covariant vector

  11. Tensors in Affine Space Contravariant tensor Mixed tensor Covariant tensor

  12. Tensors in Affine Space Multiplication Adition

  13. Geometric Meaning Point - contravariant vector Angle of straight line – covariant vector Conic section – covariant tensor

  14. Properties Symmetric tensor • To two indices • To several indices • To all indices Antisymmetric tensor (skew-symmetric) • To two indices • To several indices • To all indices

  15. Other Tensor Operations Symmetrization Alternation (antisymmetrization)

  16. Other Tensor Operations Contraction Total contraction → Affine invariant

  17. Unit polyvector Also Levi-Civita symbol Completely antisymmetric tensor - covariant - contravariant n=2:

  18. Unit polyvector n=3: nn components n! non-zero note: vector cross product

  19. Affine Invariants • Products with total contraction • Using unit polyvectors

  20. Example - moments 2D Moment tensor if p indices equals 1 and q indices equals 2 Geometric moment

  21. Example - moments Relative oriented contravariant tensor with weight -1

  22. Example - moments

  23. Example – 3D moments

  24. Invariants to other transformations • Non-linear transforms Jacobi matrix

  25. Invariants to other transformations • Projective transform Homogeneous coordinates

  26. Rotation Invariants • Cartesian tensors Affine tensor Cartesian tensor Orthonormal matrix

  27. Cartesian Tensors • Rotation invariants by total contraction

  28. Rotation Invariants 2D: 3D:

  29. Rotation Invariants 2D: 3D:

  30. Rotation Invariants 2D: 3D:

  31. Alternative Methods • Method of geometric primitives • Solution of Equations Transformation decomposition System of linear equations for unknown coefficients

  32. Alternative Methods • Complex moments • Normalization 2D: 3D: Transformation decomposition

  33. Thank you for your attention

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