Texture Symmetry

1. Texture Symmetry A lecture by Alexey Burshtein

2. Definitions • Regular texture is a periodic pattern containing translation symmetry and (possibly) rotation, reflection and glide-reflection symmetries. • Near-regular texture is a texture that is not strictly symmetrical. The difference may be in color, deformations, resolution.

3. Definitions – cont. • Perfect regularities are uncommon; • Near-regular textures are very common.

4. Definitions – cont. • Tile (also lattice item) – a smallest parallelogram (or hexagon), whose orbit produces a cover of the original pattern with no gaps or overlaps.

5. In a way, every picture as a distorted regular pattern. Thus, every picture may be generated by a near-regular pattern generator. We’ll see how to create this from this:

6. Theory of wallpaper-like regular structures exists for more then 100 years; it’s called Theory of Wallpaper Groups. There are only 5 possible shapes of tiles: • parallelogram; • rectangle; • rhomb; • square; • hexagonal.

7. You cannot just fill all the image with one tile. The result will be not naturalistic. • There are two contradictive objectives: • Preparing a realistic image (= giving up the regularity) • Preserving the regularity (= filling all image with same tile and giving up the naturalism) • The following algorithm avoid these traps.

8. Near-regularTexture Synthesis Part 1.

9. Algorithm explained • Input: a sample near-regular texture S • Output: a synthesized texture S0 statistically similar to S. • The color and intensity are captured from different tiles to give the output more natural appearance. • This will not affect the regularity of the synthesized pattern.

10. t’2 t’1 t2 t1 Algorithm explained – cont. • First of all, recognize the tile precisely. • This is performed with algorithm “regions of dominance” which is beyond our scope. • The found lattice is determined by two vectors t1 and t2. • Offset of the lattice from each side is NOT determined.

11. Algorithm explained – cont. • The vectors t1 and t2 may be also specified (or verified) by user. • The lattice has to be placed so that all original tiles are uniquely defined. This is usually controlled by the user. • Minimum tiles – the tiles carved by the lattice. • Maximum tile – smallest rectangular shape which circumscribes the minimum tile.

12. Algorithm explained – cont. Yellow rhombs – minimum tiles recognized by the algorithm. White rectangles – maximum tiles constructed over the minimum tiles.

13. Algorithm explained – cont. • After all minimum tiles had been defined, (and possibly verified by user), construct appropriate set of maximum tiles. • Start generation: • Start from top-left corner. • Take a random maximum tile. • Stamp it to create a row of tiles in direction of t1 + t2 with step of |(t1+t2)/2|. • After reaching the right bound, place one tile at t2 – t1 with step of |(t1+t2)/2| to the left of the generated row of tiles.

14. Algorithm explained – cont. • The tile to be used as a stamp is a random selection from tiles that fit most closely to generated color of the place. • Therefore, at each cell in the generated lattice the closest tile will be placed. • The chosen tile is allowed to move a bit around the calculated place to fit the structure even better. • Neighboring tiles are “stitched” together. To avoid conflicts, boundaries of tiles are blended. • Continue until the image is created.

15. Algorithm - conclusions • Maximum tiles are used instead of minimum tiles to allow smoother blending of tiles’ edges. Each tile has from ½ to ¾ of overlapped area. • The generated structure is smoother then the original – because of blending. • The result may be more regular then the source – because of too tight error threshold in selection of tiles to be used as stamps. • It’s either bigger choice of tiles to be used in every point of lattice, or smoother transition between neighbor tiles.

16. Algorithm - conclusions • Damaged tiles may never be used at all. • Examples:

17. Algorithm - conclusions • Question: will regularity be preserved if bigger size of tile will be taken? Answer:NO, not really. • The regularity will exist, but it will be some other regularity, not the one from the original tile set.

18. Algorithm – advantages • The tile is not necessarily a rectangle. • The orientation of tiles is not necessarily horizontal. • Perfect adaptation to the given sample. • We’ve covered here only translational symmetry; rotation, reflection and glide reflection can also be used.

19. What if the latticeitself is irregular? Part 2.

20. The Problem • The lattice may not be defined with straight lines. • Examples:

21. The Approach • Idea: beneath each successful man stands a more successful woman. • Idea: beneath each irregular lattice stands a distorted regular one. • Let’s find out the near-regular lattice which resembles the given irregular lattice most.The lattice is computed using energy minimization function • The difference between the lattices will allow us to compute the distortion field.

22. How do we compute the (near) regular lattice from the irregular one? • The function is: Where t1 and t2 are the vectors defining the near-regular lattice, li, lj, lk, lm are links in directions t1, t2, t1+t2, t1-t2 respectively; Ni, Nj, Nk, Nm are total number of such links.  is angle between t1 and t2. It may be computed from t1, t2, t1+t2.

23. Algorithm explained • Compute the near-regular lattice. • Compute the distortion field. • Rectify the input texture into near-regular structure. • Create 2D-image from near-regular texture as explained in part 1. • Create 2D-pattern of size equal to the requested image from the deformation field.More tight requirements for smoothness on edges!

24. Algorithm explained • Apply the resulting distortion field map built at step 5 to the image built at step 4.

25. Practical Usage –Generalized Symmetry Transform Part 3.

26. Generalized Symmetry Transform • Input: edge map. • Output: map of points of high symmetry, with intensity and orientation. • Usage: detection of symmetry; possibility to detect symmetric patterns. • The symmetry transform assigns continuous symmetry measure to each point. • Symmetry transform is local, not global!

27. How the symmetry is quantified? • Each point receives its symmetry value calculated as a sum of contributions from all other points. • We’ve already seen thisbefore. • Symmetry direction isdefined as direction ofmaximal product. rj j ri (i+j)/2 i

28. How the symmetry is quantified? • It is possible to detect radial symmetry as well as other types of symmetry. • Line is treated as a single object (reminds Hough transform). • The contributed weight of each symmetry is defined using the Gaussian. This Gaussian may be elliptic – for recognizing human eyes, for example.

29. Result of Symmetry Transform • From left to right – original image, the edges, symmetric points. • Symmetric transform generalizes most methods for finding interesting regions.

30. Usage of the transform • Using this transform, it is possible to recognize symmetry in patterns. Examples: Note: X`es are detected as +`es; circles are detected as points.

31. Conclusions • This transform can be used to detect symmetric figures on asymmetric background or (even better) vice versa.It’s easier to detect ellipses among circles. • The model may also be used as predictor of human behavior when recognizing symmetric and asymmetric patterns. • It should be a part of a bigger processing system rather then stand-alone application.

32. References • Y. Liu and W. Lin “Deformable Texture: the Irregular-Regular-Irregular Cycle”http://www.ri.cmu.edu/pubs/pub_4722.html • Y. Liu, Y. Tsin, and W. Lin ,”The Promise and Perils of Near-Regular Texture”http://www.ri.cmu.edu/pubs/pub_4406.html • J. Beck. “Textural segmentation.” In J. Beck, editor, Organization and Representation in Perception, pages 285-318. Hilldale,NJ: Lawrence Erlbaum, 1982. • www.google.com