1. Derivation of Invariants by Tensor Methods Tomáš Suk

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

3. Tensor Calculus Gregorio Ricci, Tullio Levi-Civita, Méthodes de calcul différentiel absolu et leurs applications, Mathematische Annalen (Springer) 54 (1–2): pp. 125–201, March 1900. 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.

4. Tensor Calculus 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. Grigorii Borisovich Gurevich, Osnovy teorii algebraicheskikh invariantov, (OGIZ, Moskva), 1937. Foundations of the Theory of Algebraic Invariants, (Nordhoff, Groningen), 1964.

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

6. Tensor Calculus Example: product of 2 vectors a i b k =c ik Enumeration:

7. Tensor Calculus Example: product of 2 matrices a ij b kl I enumeration, j=k sum of products, l enumeration

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

9. Other Notations Other symbols Penrose graphical notation

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

11. Tensors in Affine Space Mixed tensor Contravariant 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 Completely antisymmetric tensor - covariant n=2: - contravariant Also Levi-Civita symbol

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

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 Cartesian tensor Affine 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