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

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Outline1. Scale Space

Repr.

2. Orientation Score

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Part 1. Scale Space

The notion of scale :

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Scale Space TheoryAxioms Gaussian scale space ? - NO !

Axioms

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Scale Space TheoryGenerator:

Gaussian SS:

Poisson SS:

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Scale Space Theory

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Deep Structure:Top-points:

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Top-point Reconstruction

Gelfand TriplesSobolev/Traj. SpacesReproducing Kernels

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If energy is norm on Hilbert space exact

solution is obtained by orthogonal projection :

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Fast Top-point Reconstruction on large images,

cooperation with Frans Kanters, Bart Janssen :

Cooperation with Frans Kanters, Bart Janssen

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Part 2. Orientation Scores

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Reducible Generalization wavelet theory :

Stability Transformations :

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Perceptual organization via orientation scores:

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Processing on Orientation Scores

image score

processed score

processed image

Euclidean-invariant iff Left-invariant

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Left-invariant Evolutions on :

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

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Orientation scores of 3D images

(Euler angles)

image orientation score

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3D Proper Wavelets allowing stable inverse transform :

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Line enhancement in 3D via invertible orientation scores

Application: Enhancement Adam-Kiewitz vessel

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Robust orientation estimation

• (steerable) Tensor voting :

• Channel representations: