Perceptual Organization

A Mathematical Approach Based on scale and orientation

Remco Duits

Perceptual Organization

:

The organization of

local image fragments into

the global structure

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Part1: Scale

Painting by Dali

• Objects exist at certain ranges of scale.

• It is not known a priory at what scale to look.

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Scale Space:

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Critical Points and Paths

Maxima

Minimum

Saddles

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Scale Space Theory

Axioms Gaussian scale space ?

Axioms

NO!

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Differences with

Different

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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

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Part2: Orientation, Invertible Orientation scores

image orientation score

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Perceptual organization via orientation scores:Simple modelling of visual

illusions

Enhancement elongated structures in medical images

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

process:

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Original ImageNoisy Image Improved MapGround TruthOrientation MapOrientation Map

Robust Orientation Estimation

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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.

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Scale Space Theory

Generator:

Gaussian SS:

Poisson SS:

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If energy is norm on Hilbert space exact

solution is obtained by orthogonal projection :