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Minimization of the approximation error for describing of an image by piecewise constant approximations

Authors:

Abstract

An exact definition of superpixels (elementary sets of pixels) is given.
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 1
Minimization of the approximation error for describing
Minimization of the approximation error for describing
of an image by piecewise constant approximations
of an image by piecewise constant approximations
M. Kharinov
St. Petersburg Institute for Informatics and Automation of
the Russian Academy of Sciences
khar@iias.spb.su
28 November 2019
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 2
Optimal image approximations
etc 216 optimal approximations in total. Program Z_OptHstSegSuF.exe free access
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 3
etc 216 hierarchical approximations by means of superpixels.
Superpixels
Superpixels are the maximum sets of pixels from which k ~ 100 optimal
image approximations can be composed without distortion.
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 4
Model of hierarchical approaching of optimal image approximations
E=32
g
E1
g0 the number of
objects in the image
N
g1 limiting
cluster number
Optimal
approximations
Image
262144 piecewise-
constant image
approximations
dg
dE
H
Heterogeneity
parameter
2
11
gg
g
EE
E
approxima-
tion error
s(g1 ) the number of
superpixels
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 5
5 levels, 14132 segments
13151 colors
Tuning parameters in the model
1) The number of objects in the image g0=262144
2) The number of superpixels s=262144
3) Heterogeneity threshold |Esplit| = 1% (onlineparameter)
Online conversion of the hierarchy of approximations into an “object map”
Object map
The averaged
map
Image
1% H
Threshold
262144 approximations
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 6
min,
merge
E
0min
correct
E
,maxmin
splitmerge
EE
The system of modernized methods of cluster analysis to minimize E
0)2()1()21(
2
21
21
21
II
nn
nn
EEEE
merge
)2,1()21(
mergesplit
EEH
2,121
 
NNNNNN
t
t
.......
1
255
256
15
16
3
4
2
1) Recursive piecewise Ward's clustering. Criterion:
where
Computational complexity ~
2) СI (Clustering Improvement) split/merge method of structured pixel clusters
Criterion:
where , cluster merging is reversible:
3) K-meanless method of hierarchical reclassification of pixel sets from one to
another cluster (Dvoenko S.D., 2014), Criterion:
where
Designations: are the pixel numbers in the clusters;
are the averaged pixel intensities in clusters.
2
1
1
1
2
2
2
2
kkcorrect
II
kn
n
II
kn
n
kE
21
,, nnk
21
,, III
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 7
Merging of Sleator-Tarjan trees Merging of cyclic graphs
An example of kernel network for a 4-pixel image
Sleator-Tarjan dynamic tree Cyclic graph
Reversible algebraic multilayer network (AMN) formation
All calculations are performed in terms of the algebraic multilayer network formed
by Sleator-Tarjan dynamic trees (acyclic graphs), cyclic graphs and pointer systems. The
arcs of the tree are assigned the values of heterogeneity H as weights, and the nodes are
supplied with additive and other characteristics. The network connects the source pixels.
It is supported the reversibility of calculations that remain available for modifica-
tion and optimization owing to the reversibility of the merging operation of pixel clusters.
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 8
Static Dynamic
(for storage/transmitting) (for processing and optimization)
Kernel
network
Image
R
G
B
Sleator-Tarjan dynamic tree
Cyclic graph
R
G
B
Metadata to speed up
computation and optimize the
kernel network
Algebraic Multilayer Network (AMN) data structure
Sleator-Tarjan dynamic tree
Cyclic graph
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 9
Experimental results
M. Kharinov «Mathematical Metods for Pattern Recognition» (MMPR-2019), Моscow, November 26-29, 2019 10
Conclusion
Cluster Analysis Modernization for detection of object hierarchy
Content Remark
Image segmentation Pixel clustering Hierarchical
Split/merge Split/merge of strucured pixel clusterrs
Reversible operation&
&computing
Ward’s clustering Recursive piecewise
Ward’s clustering
Hierarchy of millions
of approximations is
supported
Kmeans method Kmeanless method of hierarchical
reclassification of pixel sets
Dvoenko S.D.
(PRIP’2014)
Dendrograms → Network = Sleator-Tarjan Trees +
+ Cyclic graphs
Algebraic Multilayer
Network (AMN)
Table
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