Collective Sensing: a Fixed-Point Approach in the Metric Space1Xin LiLDCSEE, WVU1This work is partially supported by NSF ECCS-0968730

Unreasonable Effectiveness of Mathematics in EngineeringUnreasonable effectiveness of mathematics in natural sciences Wigner1960To understand how nature works, you need to grasp the tool of mathematics firstThe tension between mathematicians and engineersWavelets vs. filter banksthe hype that would arise around wavelets caused surprise and some understandable resentment in the subband filtering community in Where do wavelets come from? I. Daubechies1996Compressed sensing is another example of how mathematicians have stolen the show from engineers

Mathematical Structures are Double-Bladed SwordsHilbert-space: a completeInner-product spaceQuantum mechanics

Fourier/waveletanalysis

Learning theory

PDE(e.g., Total-Variation)Mathematical formalism(Hilbert, Ackermann, Von Neumann )Metric space: a set witha notion of distanceGeneral relativity

Fixed-point theorems

Game theory

Dynamic systemsMathematical constructivism(Poincare, Brouwer, Weyl )

Criticism of Compressed SensingWhere does sparsity come from?

Nonlinear processing of wavelet coefficientsNonlinear diffusion minimizing TVWhat 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)

basisfunctionsapproximation of l0signalof interest

A Physical View of SparsityHow nature works? (e.g., variational principle)Reaction-diffusion systems A. Turing1952More is Different. P.W. Anderson1972Self-organizing systems I. Prigogine1977Fractals and Chaos Mandelbrot1977Complex networks 1990s-Implications into image processing Hilbert space might not be a proper mathematical framework for characterizing the complexity of natural images?

From Hilbert-space to Metric-spaceImages are viewed as the fixed-points in the metric space f=PfNonlinear mapping P characterizes the organizational principle underlying imagesExample (nonlocal filter):Non-expansiveness of PNLF guarantees the existence of fixed-pointsBilateral , nonlocal mean and BM3D filters are special cases of PNLF

Phase Space of Image SignalsSA-DCTTVBM3DNonlocal-TVLocal filtersNonlocal filters

Nonlocal Regularization MagicBM3DNonlocal-TVKey Observation: As the temperature (regularization) parameter varies, nonlocalmodels can traverse different phases corresponding to coarse/fine structures

From Compressed Sensing to Collective SensingKey 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

Variational InterpretationsTV:Nonlocal TV:BM3D:

Application (I): Collective Sensingl1-magicPSNR=68.53dBOursPSNR=84.47dBl1-magicPSNR=19.53dBOursPSNR=40.97dB

Application (II): Lossy CompressionHouse (256256)Barbara (512512)JPEG-decodedNL-enhancedNL-enhancedSPIHT-decodedMATLAB codes accompanying this work are available at my homepage:http://www.csee.wvu.edu/~xinl/ or Google Xin Li WVU

Image Comparison ResultsJPEG-decoded at rate of 0.32bpp(PSNR=32.07dB)

NL-enhanced at rate of 0.32bpp(PSNR=33.22dB)

SPIHT-decoded at rate of 0.20bpp(PSNR=26.18dB)

NL-enhanced at rate of 0.20bpp(PSNR=27.33dB)

Maximum-Likelihood (ML) DecodingMaximum a Posterior (MAP) Decoding

Application (III): Image DeblurringISNR(dB) comparison among competing deblurring schemes for cameraman image: uniform 99 blurring kernel and noise level of BSNR=40dB

Image Comparison ResultsoriginaldegradedTVMMOursISTSADCT

Unexpected ConnectionsSpectral clusteringEigenvectors of graph Laplacian determine a provably optimal embeddingNonlinear dynamical systemsRegularization implemented by the joint force of excitation and inhibition in a neuron networkStatistical physicsVariational principle underlying Ising model, spin glass and Hopfield network

Summary and ConclusionsOne way of competing with mathematicians is to think like physicistsBasis construction/pursuit is only one (local and suboptimal) way of understanding sparsityNonlocal regularization can more effectively handle complexity of natural imagesThe distinction between signals and systems is artificial and a holistic (collective) view is preferred

Ongoing WorksDuality between similarity and dissimilarityThe implication of sensory inhibition into image processingFrom graphical models to complex networksThe role of complex network topology Unification of signal reconstruction and object recognitionTo remove the artificial boundary between low-level and high-level vision

Dont be fooled by the fancy name name is a surface thing. Just like compressed sensing is also called compressed sampling, or compressive sensing, or compressive sampling, name is only for the convenience of communication.From compressed sensing to collective sensing: toward a better understanding of the relationship between mathematics and engineering.*A historical view toward compressed sensing. Poor engineers! mathematics make people subtle *At the foundation of mathematics: formalism vs. intuitionism. Which one is more fundamental? No one knows.*Two attack points: locality and inner-product.*Nonlocal phenomenon in natureMathematics serves as a language*