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

Outline1. 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 TriplesSobolev/Traj. SpacesReproducing 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 iff Left-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)

image orientation 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: