Some thoughts on regularization for vector-valued inverse problems

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Some thoughts on regularization for vector-valued inverse problems. Eric Miller Dept. of ECE Northeastern University. Outline. Caveats Motivating examples Sensor fusion: multiple sensors, multiple objects Sensor diffusion: single modality, multiple objects Problem formulation - PowerPoint PPT Presentation

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  • Some thoughts on regularization for vector-valued inverse problemsEric MillerDept. of ECENortheastern University

  • OutlineCaveatsMotivating examplesSensor fusion: multiple sensors, multiple objectsSensor diffusion: single modality, multiple objectsProblem formulationRegularization ideasMarkov-random fields Mutual informationGradient correlationExamplesConclusions

  • CaveatsMy objective here is to examine some initial ideas regarding multi-parameter inverse problemsModels will be kept simpleLinear and 2DConsider two unknowns. Case of 3 or more can waitRegularization parameters chosen by hand.Results numerical. Whatever theory there may be can wait for later

  • Motivating ApplicationsSensor fusionMultiple modalities each looking at the same region of interestEach modality sensitive to a different physical property of the mediumSensor diffusionSingle modality influenced by multiple physical properties of the medium

  • Sensor Fusion ExampleMulti-modal breast imagingLimited view CTSensitive to attenuationHigh resolution, limited dataDiffuse optical tomographySensitive to many things. Optical absorption and scattering or chromophore concentrationsHere assume just absorption is of interestLow resolution, fairly dense dataElectrical impedance tomography coming on line

  • Linear Physical ModelsTomosynthesisSourceDetectorRegion of interestDiffuse opticalSourceDetector

  • Sensor Fusion (cont)Overall model relating data to objects

    Assume uncorrelated, additive Gaussian noise. Possibly different variances for different modalitiesAll sorts of caveatsDOT really nonlinearTomosynthesis really PoissonEverything really 3DDeal with these later

  • De-MosaicingColor cameras sub-sample red, green and blue on different pixels in the imageIssues: filling in all of the pixels with all three colorsBayer patternyred = observed red pixels over sub-sampled grid. 9 vector for examplefrwd = red pixels values over all pixels in image. 30 vector in exampleKred = selection matrix with a single 1 in each row, all others 0. 9x30 matrix for example

  • Sensor Diffusion ExampleDiagnostic ultrasound guidance for hyperthermia cancer treatmentUse high intensity focused ultrasound to cook tissueNeed to monitor treatment progressMRI state of the art but it is expensiveUltrasound a possibility Absorption monotonic w/ temperatureAlso sensitive to sound speed variationsTraditional SAR-type processing cannot resolve regions of interestTry physics-based approachThanks to Prof. Ron Roy of BU

  • Ultrasound modelAs with diffuse optical, exact model is based on Helmholtz-type equation and is non-linearHere we use a Born approximation even in practice because problem size quite large (10s of wavelengths on a side)Modelf1 = sound speedf2 = absorption = frequency dependent filters for each parameter

  • Estimation of parametersVariational formulation/penalized likelihood approach

    Issue of interest here is the prior

    Gaussian log likelihoodPrior information, regularizer

  • Prior ModelsTypical priors based on smoothness of the functions

    = regularization parameterp = 1 gives total variation reconstruction with edges well preservedp = 2 gives smooth reconstructions

  • Priors (cont)What about co-variations between f1 and f2?Physically, these quantities are not independentTumors, lesions, etc. should appear in all unknownsSpeculate that spatial variations in one correlate with such variations in the otherLooking to supplement existing prior with mathematical measure of similarity between the two functions or their gradientsThree possibilities examined today

  • Option 1: Gauss-Markov Random Field-Type PriorNatural generalization of the smoothness prior that correlates the two functionsi,ji+1,ji-1,ji,j+1i,j-1f1f2i,jw1

  • GMRF (cont)Matrix form

    The GMRF regularizer

    Implies that covariance of f is equal toWhat does this look like?

  • GMRF: Middle Pixel CorrelationLag xLag y

  • GMRF: CommentsMotivated by / similar to use of such models in hyperspectral processingLots of things one could doOne line parameter estimationAppropriate neighborhood structuresGeneralized GMRF a la Bouman and SauerMore than two functions

  • Option 2: Mutual InformationAn information theoretic measure of similarity between distributionsGreat success as a cost function for image registration (Viola and Wells)Try a variant of it here to express similarity between f1 and f2

  • Mutual Information: DetailsSuppose we had two probability distributions p(x) and p(y)Mutual information is

    Maximization of mutual information (basically) minimizes joint entropy, -H(x,y), while also accounting for structure of the marginals

  • Mutual Information: DetailsMutual information registration used not the images but their histogramsEstimate histograms using simple kernel density methods

    and similarly for p(y) and p(x,y)

  • Mutual Information: Examplexyf1(x,y)f2(x,y)= f2(x+,y)Mutual InformationPeak when overlap is perfect

  • Mutual Information: RegularizerFor simplicity, we use a decreasing function of MI as a regularizerLarger the MI implies smaller the cost

  • Gradient CorrelationIdea is simple: gradients should be similarCertainly where there are physical edges, one would expect jumps in both f1 and f2Also would think that monotonic trends would be similarOKNot OKOK

  • A Correlative ApproachA correlation coefficient based metric

  • Lets See How They Behavef1(x,y)f2(x,y)= f2(x+,y) 5-5

  • Example 1: Sensor Fusion5 cm6 cmX-ray sourceDOT source/detectorDOT detectorsX-ray detectorNoisy, high resolution X ray. 15 dBCleaner, low resolution DOT, 35 dB

  • DOT ReconstructionsTruthTikhonovGMRFCorr. CoeffMI

  • X Ray ReconstructionsTruthTikhonovGMRFCorr. CoeffMI

  • DOT ReconstructionsTruthTikhonovGMRFCorr. CoeffMI

  • X-ray ReconstructionsTruthTikhonovGMRFCorr. CoeffMI

  • Mean Normalized Square Error

    Sheet1

    DOT

    TikhonovGMRFCorr. CoeffMI

    First ExampleWhole region0.841.270.301.05

    Anomaly only0.250.180.110.12

    X Ray

    TikhonovGMRFCorr. CoeffMI

    Whole region0.280.540.270.46

    Anomaly only0.080.120.080.10

    DOT

    TikhonovGMRFCorr. CoeffMI

    Second ExampleWhole region0.170.420.090.09

    Anomaly only0.060.140.030.03

    X Ray

    TikhonovGMRFCorr. CoeffMI

    Whole region0.130.330.120.13

    Anomaly only0.040.070.040.04

    Sound Speed

    TikhonovGMRFCorr. CoeffMI

    First ExampleWhole region0.290.250.301.97

    Anomaly only0.160.170.170.33

    Absorption

    TikhonovGMRFCorr. CoeffMI

    Whole region0.470.350.6346.57

    Anomaly only0.300.110.134.65

    Sound Speed

    TikhonovGMRFCorr. Coeff

    Second ExampleWhole region0.200.180.20

    Anomaly only0.120.130.12

    Absorption

    TikhonovGMRFCorr. Coeff

    Whole region0.680.490.79

    Anomaly only0.410.180.20

    Sheet2

    Sheet3

  • Example 2: Sensor DiffusionUltrasound problemTissue-like properties 5 frequencies between 5kHz and 100 kHzWavelengths between 1 cm and 30 cmImage sound speed and attenuationHigh SNR (70 dB), but sound speed about 20x absorption and both in cluttered backgrounds

  • Sound Speed ReconstructionsTruthTikhonovGMRFCorr. CoeffMI

  • Absorption ReconstructionsTruthTikhonovGMRFCorr. CoeffMI

  • Sound Speed ReconstructionsTruthTikhonovGMRFCorr. Coeff

  • Absorption ReconstructionsTruthTikhonovGMRFCorr. Coeff

  • Mean Normalized Square Error

    Sheet1

    DOT

    TikhonovGMRFCorr. CoeffMI

    First ExampleWhole region0.841.270.301.05

    Anomaly only0.250.180.110.12

    X Ray

    TikhonovGMRFCorr. CoeffMI

    Whole region0.280.540.270.46

    Anomaly only0.080.120.080.10

    DOT

    TikhonovGMRFCorr. CoeffMI

    Second ExampleWhole region0.170.420.090.09

    Anomaly only0.060.140.030.03

    X Ray

    TikhonovGMRFCorr. CoeffMI

    Whole region0.130.330.120.13

    Anomaly only0.040.070.040.04

    Sound Speed

    TikhonovGMRFCorr. CoeffMI

    First ExampleWhole region0.290.250.301.97

    Anomaly only0.160.170.170.33

    Absorption

    TikhonovGMRFCorr. CoeffMI

    Whole region0.470.350.6346.57

    Anomaly only0.300.110.134.65

    Sound Speed

    TikhonovGMRFCorr. Coeff

    Second ExampleWhole region0.200.180.20

    Anomaly only0.120.130.12

    Absorption

    TikhonovGMRFCorr. Coeff

    Whole region0.680.490.79

    Anomaly only0.410.180.20

    Sheet2

    Sheet3

  • Demosaicing

  • Eye Region: RedOriginalTikhonovCorr. Coeff.

  • Eye Region: GreenOriginalTikhonovCorr. Coeff.

  • Chair Region: RedOriginalTikhonovCorr. Coeff.

  • Chair Region: GreenOriginalTikhonovCorr. Coeff.

  • Normalized Square Error

    Sheet1

    DOT

    TikhonovGMRFCorr. CoeffMI

    First ExampleWhole region0.841.270.301.05

    Anomaly only0.250.180.110.12

    X Ray

    TikhonovGMRFCorr. CoeffMI

    Whole region0.280.540.270.46

    Anomaly only0.080.120.080.10

    DOT

    TikhonovGMRFCorr. CoeffMI

    Second ExampleWhole region0.170.420.090.09

    Anomaly only0.060.140.030.03

    X Ray

    TikhonovGMRFCorr. CoeffMI

    Whole region0.130.330.120.13

    Anomaly only0.040.070.040.04

    Sound Speed

    TikhonovGMRFCorr. CoeffMI

    First ExampleWhole region0.290.250.301.97

    Anomaly only0.160.170.170.33

    Absorption

    TikhonovGMRFCorr. CoeffMI

    Whole region0.470.350.6346.57

    Anomaly only0.300.110.134.65

    Sound Speed

    TikhonovGMRFCorr. Coeff

    Second ExampleWhole region0.200.180.20

    Anomaly only0.120.13