Topology-Preserving Ordering of the RGB Space with an Evolutionary Algorithm

Topology-Preserving Ordering

of the RGB Space with an

Evolutionary AlgorithmF. Flórez-Revuelta

Some concepts

Colour mathematical morphology

Current orderings of the RGB space

Topology preservation

Evolutionary ordering of the RGB space

Experimentation

Conclusions and current work

Summary

Morphology: description of the properties of shape and structure of objects

In CV: description of the properties of shapes of areas in the image

Operations on MM were originally defined as operations on sets, but it soon

became clear that they were useful for image processing

Two basic operations:

Erosion:

Dilation:

Combining these two basic operations, many others can be created

Mathematical morphology

Other operations:

Morphological gradient:

Internal gradient:

External gradient:

White top-hat:

Black top-hat:

Watershed

Granulometry

Morphological skeleton


It can also be applied to greyscale images

A usual application is noise reduction


How can colours be ordered?

Different methods:

Marginal ordering:

apply MM to each channel and combine the results

new colours are incorporated to the image



Different methods:

Marginal ordering

Lexicographical ordering:

order by the first component, if equal by the second, then by the third



Different methods:

Marginal ordering

Lexicographical ordering

Bit-interlacing ordering


A curve that fills an n-dimensional space

Discovered by Peano

Space-filling curves

Peano curve Hilbert curve

The different orderings can be seen as space-filling curves

But, which ordering is better?

Space-filling curves

This is a well-known problem with self-organising neural networks

These networks try to adapt their topology to the input space

There are different measures of topology preservation: topographic product,

topographic function, Kaski-Lagus measure

Measuring topology preservation

12

1

1 1 1

, ,1log

( 1 ) ,,

Al

Vl

kV A AkN N j n j l

A VVj k l lj n j

d w w d j n jP

N N d j n jd w w

Topology preservation of the

colour orderings

Use of a 1D GNG to learn the 3D RGB space

Extremes are fixed to (0,0,0) and (1,1,1)

Flórez-Revuelta, F. Ordering of the RGB space with a growing self-organizing network.

Application to color mathematical morphology. In Proceedings of the 15th International

Conference on Artificial Neural Networks: Biological Inspirations - Volume Part I (pp.

385-390), Springer-Verlag, 2005

First approach

Separate clustering and ordering

Clustering is required as it is currently unfeasible to order all the elements in

the colour space.

Then, a P-order is obtained, not a total order

Clustering: k-means

Ordering: evolutionary algorithm to minimise P

Current approach

Path representation

As the extrema of the ordering are fixed, they are not included in the

chromosome

Representation of individuals

Linear Order Crossover (LOX)

Crossover

The algorithm uses several mutation operators that are usually employed in

combinatorial optimisation problems:

Exchange: Selects at random two elements in the path and exchanges them

Swap: Selects at random two consecutive elements and swaps them

Insertion: Selects at random one element, removes it from the current position

and inserts it in a random place

Simple inversion: Selects at random two elements and reverses the

subsequence between them

Scramble: Selects at random two elements and scrambles the subsequence

between them

Displacement: Selects a subsequence at random, removes it and inserts it in a

random place

The algorithm also includes a Subsequence ordering mutation, which selects a random

subsequence and orders it using the same EA, following a recursive approach

Mutation

Evolutionary orderings of the RGB

space

Image-specific ordering

Ordering of colours

Given a pair of colours 1, 20,13

1 is lower than 2 if centre(1) is lower tan centre(2)

However, both colours can be mapped onto the same cluster

A criterion needs to be chosen to select the order in that case. This work

compares the relation between the distances from each of those colours to

the neighbouring cluster centres

Experimentation

Size of population: 25 individuals

Parents selected by fitness ranking

All the mutations have the same probability to be selected

Next generation’s population is selected by ranking with elitism for the best

The evolution finishes if the best individual does not change for 1,000

generations

Special parameters for the Subsequence ordering mutation:

Length of the subsequence in the interval [8,24]

Size of the population: 10 individuals

Number of generations without changes: 100 generations

Experimentation

Current work

Genetic algorithm to obtain an ordering of n clusters, n > 64

Extension to other colour spaces

Selection of the extremes of the ordering as foreground and background

Extension to other n-dimensional spaces: histograms, skeletons, bags of

words

This would allow the application of MM to these multivariate spaces

Topology-Preserving Ordering

of the RGB Space with an

Evolutionary AlgorithmF. Flórez-Revuelta

[email protected]

Topology-Preserving Ordering of the RGB Space with an Evolutionary Algorithm

Technology

Transcript of Topology-Preserving Ordering of the RGB Space with an Evolutionary Algorithm