Collective Sensing: a Fixed-Point Approach in the Metric Space1

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Collective Sensing a Fixed-Point Approach in the
Metric Space1

• Xin Li
• LDCSEE, WVU

1This work is partially supported by NSF
ECCS-0968730
2
Unreasonable Effectiveness of Mathematics in
Engineering

• Unreasonable effectiveness of mathematics in
  natural sciences Wigner1960
• To understand how nature works, you need to grasp
  the tool of mathematics first
• The tension between mathematicians and engineers
• Wavelets vs. filter banks
• the hype that would arise around wavelets caused
  surprise and some understandable resentment in
  the subband filtering community in Where do
  wavelets come from? I. Daubechies1996
• Compressed sensing is another example of how
  mathematicians have stolen the show from engineers

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Mathematical Structures are Double-Bladed Swords
Metric space a set with a notion of distance
Hilbert-space a complete Inner-product space
General relativity Fixed-point theorems Game
theory Dynamic systems
Quantum mechanics Fourier/wavelet analysis Learn
ing theory PDE(e.g., Total-Variation)
Mathematical constructivism (Poincare, Brouwer,
Weyl )
Mathematical formalism (Hilbert, Ackermann, Von
Neumann )
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Criticism of Compressed Sensing

• Where does sparsity come from?
• Nonlinear processing of wavelet coefficients
• Nonlinear diffusion minimizing TV
• What is wrong?
• Over-emphasize the role of locality (it does not
  hold in complex systems)
• Inner-product is an artificial structure (it
  carries little insight about how patterns form in
  nature)

basis functions
signal of interest
approximation of l0
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A Physical View of Sparsity

• How nature works? (e.g., variational principle)
• Reaction-diffusion systems A. Turing1952
• More is Different. P.W. Anderson1972
• Self-organizing systems I. Prigogine1977
• Fractals and Chaos Mandelbrot1977
• Complex networks 1990s-
• Implications into image processing
• Hilbert space might not be a proper mathematical
  framework for characterizing the complexity of
  natural images?

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From Hilbert-space to Metric-space

• Images are viewed as the fixed-points in the
  metric space fPf
• Nonlinear mapping P characterizes the
  organizational principle underlying images
• Example (nonlocal filter)

Bilateral , nonlocal mean and BM3D filters are
special cases of PNLF
Non-expansiveness of PNLF guarantees the
existence of fixed-points
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Phase Space of Image Signals
SA-DCT
TV
BM3D
Nonlocal-TV
Nonlocal filters
Local filters
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Nonlocal Regularization Magic
BM3D
Nonlocal-TV
Key Observation As the temperature
(regularization) parameter varies,
nonlocal models can traverse different phases
corresponding to coarse/fine structures
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From Compressed Sensing to Collective Sensing

• Key messages
• From local to nonlocal regularization thanks to
  the fixed-point formulation in the metric space
  (PNLF depends on the clustering result or
  similarity matrix)
• From convex to nonconvex optimization
  deterministic annealing (also-called graduated
  nonconvexity) is the black magic behind

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Variational Interpretations
TV
Nonlocal TV
BM3D
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Application (I) Collective Sensing
l1-magic PSNR68.53dB
Ours PSNR84.47dB
l1-magic PSNR19.53dB
Ours PSNR40.97dB
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Application (II) Lossy Compression
NL-enhanced
NL-enhanced
JPEG-decoded
SPIHT-decoded
House (256256)
Barbara (512512)
MATLAB codes accompanying this work are available
at my homepage http//www.csee.wvu.edu/xinl/ or
Google Xin Li WVU
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Image Comparison Results
NL-enhanced at rate of 0.32bpp (PSNR33.22dB)
JPEG-decoded at rate of 0.32bpp (PSNR32.07dB)
SPIHT-decoded at rate of 0.20bpp (PSNR26.18dB)
NL-enhanced at rate of 0.20bpp (PSNR27.33dB)
Maximum-Likelihood (ML) Decoding
Maximum a Posterior (MAP) Decoding
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Application (III) Image Deblurring
ISNR(dB) comparison among competing deblurring
schemes for cameraman image uniform 99 blurring
kernel and noise level of BSNR40dB
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Image Comparison Results
original
degraded
TVMM
IST
SADCT
Ours
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Unexpected Connections

• Spectral clustering
• Eigenvectors of graph Laplacian determine a
  provably optimal embedding
• Nonlinear dynamical systems
• Regularization implemented by the joint force of
  excitation and inhibition in a neuron network
• Statistical physics
• Variational principle underlying Ising model,
  spin glass and Hopfield network

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Summary and Conclusions

• One way of competing with mathematicians is to
  think like physicists
• Basis construction/pursuit is only one (local and
  suboptimal) way of understanding sparsity
• Nonlocal regularization can more effectively
  handle complexity of natural images
• The distinction between signals and systems is
  artificial and a holistic (collective) view is
  preferred

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Ongoing Works

• Duality between similarity and dissimilarity
• The implication of sensory inhibition into image
  processing
• From graphical models to complex networks
• The role of complex network topology
• Unification of signal reconstruction and object
  recognition
• To remove the artificial boundary between
  low-level and high-level vision