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Where dimensions come from?
- dynamical approach to scaling
Milan Jovovic, Belgrade Univ.
18. Jun 2008.
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• Modeling approach based on free energy and
distortion energy
• Estimation of dynamical parameters of
clustering by statistical inference
• Multi-spectral decomposition, in hierarchy of
scales
• Application: scale analysis of complex systems
Analysis of signal distortion by multi-
scale decomposition
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Introduction (1 of 2)
• Cluster parameters:
o Selected spatial window: Wr
o Computed cluster vector within Wr
• Statistical inference defines PDF, with the associated
distortion energies, F and V
• Energy functions are generally multi-dimensional and
non-convex
• Non-linear map defines dynamical scale-space
clustering
• Clustering is important optimization problem
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Model of signal distortion:
- definitions
• Distortion measure:
1. d = z2 = (Ir-r)2 + (Ig-g)2 + (Ib-b)2 2. d = z2 =
• Partition functions:
• Distortion energies -
free energy, and variance:
• PDF:
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Scale-space computing
• Series of convex min/max of free energy F- /F+:
upscale melting & downscale cooling:
• Evolution scheme – path integrals:
• Way to move through the scale-space ?
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Motion through the scale-space:
- wave equation
• The same potential level difference the equilibrium point moves by (2)
and (3)
12
,vF
Vv grad
(1)
(2)
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Cluster Bindings
• Motion binding:
• Determinant of the map:
• Criteria of splitting a cluster at the “scale equilibrium”:
• Spatial coherency of information:
• Coupled domains of computation:
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Scalable coding
• Coupled data structure of the hierarchy of
binary images
• Efficient coding, control, data transfer
• Parallelization: computing and control by
parallel computing architectures
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Reconstruction from singular data sets:
Scale hierarchy of data manifolds
• Multi-scale manifolds data decomposition – scalable coding,
transmission and data reconstruction
• Reconstruction of manifold spanning spaces
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Summary presentation of current work
• Images: multi-spectral decomposition and clusters
coupling, spectral signature recognition
• Movements: trajectory analysis, learning, coding and
control by scale-space computing
• Bio/chemical informatics: data-mining and knowledge
discovery
• Scalable data decomposition: coding, control, and
transmission
• Synchronous computing scheme: upscale melting &
downscale cooling
• Parallel computing implementation
• Perspective and future directions
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• Scale singularity of data sets is used in detecting rain
patterns
Still images decomposition
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Spectral filtering
Original image, -> decomposition by: d = |I - c|2 + |I - sig|2 > 0
• Resolution of the pattern [rsig gsig bsig] = [207 50 83] increases with
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Sequence of 2 images: 2 clusters
decomposition
• 2D ball expansion expansion and diagonal
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Sequence of 3 images: 2 groups decomposition
• To right 2 groups V_x residual V_y residual
• To right and z 2 groups V_x residual V_y residual
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