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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
Mathematical morphology
It can also be applied to greyscale images
A usual application is noise reduction
Mathematical morphology
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
Colour mathematical morphology
How can colours be ordered?
Different methods:
Marginal ordering
Lexicographical ordering:
order by the first component, if equal by the second, then by the third
Colour mathematical morphology
How can colours be ordered?
Different methods:
Marginal ordering
Lexicographical ordering
Bit-interlacing ordering
Colour mathematical morphology
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
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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
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
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
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