1. Perceptual Organization A Mathematical Approach Based on scale and orientation Remco Duits

2. Perceptual Organization: The organization of local image fragments into the global structure

3. 3 Part1: Scale Painting by Dali Objects exist at certain ranges of scale. It is not known a priory at what scale to look.

4. 4 Scale Space:

5. 5 Critical Points and Paths Maxima Minimum Saddles

6. 6 Scale Space Theory Axioms Gaussian scale space ? Axioms NO!

7. 7 Differences with Different 

8. 8 Reconstruction It is possible to reconstruct the image by using a limited number of points from the Deep Structure. Drastically Improved Reconstruction by Reproducing Kernel Theory and Gelfand Triples

9. 9 Part2: Orientation, Invertible Orientation scores image orientation score

10. 10 Perceptual organization via orientation scores: Simple modelling of visual illusions Enhancement elongated structures in medical images

11. 11 Application Automatic Closure of Gaps: Collision probability of forward and backward process:

12. 12 Original ImageNoisy ImageImproved MapGround Truth Orientation Map Robust Orientation Estimation

13. 13 1.New scale space theory obeying scale space axioms 2.Improved reconstruction by means of Gelfand-Triples 3.New wavelet theory for invertible orientation scores 4.New theory/applications for stochastic completion fields.

14. 14

15. 15 Scale Space Theory Generator: Gaussian SS: Poisson SS:

16. 16 If energy is norm on Hilbert space exact solution is obtained by orthogonal projection :