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Collective Sensing: a Fixed-Point Approach in the Metric Space 1

Collective Sensing: a Fixed-Point Approach in the Metric Space 1. Xin Li LDCSEE, WVU. 1 This work is partially supported by NSF ECCS -0968730. Unreasonable Effectiveness of Mathematics in Engineering. “Unreasonable effectiveness of mathematics in natural sciences” Wigner’1960

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Collective Sensing: a Fixed-Point Approach in the Metric Space 1

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  1. 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” Wigner’1960 • 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. Daubechies’1996 • Compressed sensing is another example of how mathematicians have stolen the show from engineers

  3. 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 Learning theory PDE(e.g., Total-Variation) Mathematical constructivism (Poincare, Brouwer, Weyl …) Mathematical formalism (Hilbert, Ackermann, Von Neumann …)

  4. 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) Criticism of Compressed Sensing signal of interest basis functions approximation of l0

  5. A Physical View of Sparsity • How nature works? (e.g., variational principle) • Reaction-diffusion systems A. Turing’1952 • “More is Different.” P.W. Anderson’1972 • Self-organizing systems I. Prigogine’1977 • Fractals and Chaos Mandelbrot’1977 • Complex networks 1990s- • Implications into image processing • Hilbert space might not be a proper mathematical framework for characterizing the complexity of natural images?

  6. Images are viewed as the fixed-points in the metric space f=Pf Nonlinear mapping P characterizes the organizational principle underlying images Example (nonlocal filter): From Hilbert-space to Metric-space Bilateral , nonlocal mean and BM3D filters are special cases of PNLF Non-expansiveness of PNLF guarantees the existence of fixed-points

  7. “Phase Space” of Image Signals SA-DCT TV BM3D Nonlocal-TV Nonlocal filters Local filters

  8. 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

  9. 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

  10. Variational Interpretations TV: Nonlocal TV: BM3D:

  11. Application (I): Collective Sensing l1-magic PSNR=68.53dB Ours PSNR=84.47dB l1-magic PSNR=19.53dB Ours PSNR=40.97dB

  12. Application (II): Lossy Compression NL-enhanced NL-enhanced JPEG-decoded SPIHT-decoded House (256×256) Barbara (512×512) MATLAB codes accompanying this work are available at my homepage: http://www.csee.wvu.edu/~xinl/ or Google “Xin Li WVU”

  13. Image Comparison Results NL-enhanced at rate of 0.32bpp (PSNR=33.22dB) JPEG-decoded at rate of 0.32bpp (PSNR=32.07dB) SPIHT-decoded at rate of 0.20bpp (PSNR=26.18dB) NL-enhanced at rate of 0.20bpp (PSNR=27.33dB) Maximum-Likelihood (ML) Decoding Maximum a Posterior (MAP) Decoding

  14. Application (III): Image Deblurring ISNR(dB) comparison among competing deblurring schemes for cameraman image: uniform 9×9 blurring kernel and noise level of BSNR=40dB

  15. Image Comparison Results original degraded TVMM SADCT IST Ours

  16. 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

  17. 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

  18. 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

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