Perceptual Organization A Mathematical Approach Based on scale, orientation and curvature Remco Duits

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

3 Outline 1. Scale Space Repr. 2. Orientation Score

4 Part 1. Scale Space The notion of scale :

5 Scale Space Theory Axioms Gaussian scale space ? - NO ! Axioms

6 Scale Space Theory Generator: Gaussian SS: Poisson SS:

7 Scale Space Theory

8 Deep Structure: Top-points:

9 Top-point Reconstruction Gelfand Triples Sobolev/Traj. Spaces Reproducing Kernels

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

11 Fast Top-point Reconstruction on large images, cooperation with Frans Kanters, Bart Janssen : Cooperation with Frans Kanters, Bart Janssen

12 Part 2. Orientation Scores

13 Reducible Generalization wavelet theory : Stability Transformations :

14 Perceptual organization via orientation scores:

15 Processing on Orientation Scores image score processed score processed image Euclidean-invariant iffLeft-invariant

16 Left-invariant Evolutions on :

17 Application Automatic Closure of Gaps: Collision probability of forward and backward proces:

18 Orientation scores of 3D images (Euler angles) imageorientation score

19 3D Proper Wavelets allowing stable inverse transform :

20 Line enhancement in 3D via invertible orientation scores Application: Enhancement Adam-Kiewitz vessel

21 Robust orientation estimation (steerable) Tensor voting : Channel representations: