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Some thoughts on regularization for vector- valued inverse problems Eric Miller Dept. of ECE Northeastern University

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Outline Caveats Motivating examples –Sensor fusion: multiple sensors, multiple objects –Sensor diffusion: single modality, multiple objects Problem formulation Regularization ideas –Markov-random fields –Mutual information –Gradient correlation Examples Conclusions

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Caveats My objective here is to examine some initial ideas regarding multi-parameter inverse problems Models will be kept simple –Linear and 2D Consider two unknowns. –Case of 3 or more can wait Regularization parameters chosen by hand. Results numerical. Whatever theory there may be can wait for later

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Motivating Applications Sensor fusion –Multiple modalities each looking at the same region of interest –Each modality sensitive to a different physical property of the medium Sensor diffusion –Single modality influenced by multiple physical properties of the medium

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Sensor Fusion Example Multi-modal breast imaging Limited view CT –Sensitive to attenuation –High resolution, limited data Diffuse optical tomography –Sensitive to many things. Optical absorption and scattering or chromophore concentrations –Here assume just absorption is of interest –Low resolution, fairly dense data Electrical impedance tomography coming on line GE Tomosynthesis Optical Imager Optical measurement done under mammographic compression

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Linear Physical Models Tomosynthesis Source Detector Region of interest Diffuse optical Source Detector

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Sensor Fusion (cont) Overall model relating data to objects Assume uncorrelated, additive Gaussian noise. Possibly different variances for different modalities All sorts of caveats –DOT really nonlinear –Tomosynthesis really Poisson –Everything really 3D –Deal with these later

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De-Mosaicing Color cameras sub- sample red, green and blue on different pixels in the image Issues: filling in all of the pixels with all three colors Bayer pattern y red = observed red pixels over sub-sampled grid. 9 vector for example f rwd = red pixels values over all pixels in image. 30 vector in example K red = selection matrix with a single “1” in each row, all others 0. 9x30 matrix for example

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Sensor Diffusion Example Diagnostic ultrasound guidance for hyperthermia cancer treatment Use high intensity focused ultrasound to cook tissue Need to monitor treatment progress MRI state of the art but it is expensive Ultrasound a possibility –Absorption monotonic w/ temperature –Also sensitive to sound speed variations –Traditional SAR-type processing cannot resolve regions of interest –Try physics-based approach Thanks to Prof. Ron Roy of BU

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Ultrasound model As with diffuse optical, exact model is based on Helmholtz-type equation and is non-linear Here we use a Born approximation even in practice because problem size quite large (10’s of wavelengths on a side) Model f 1 = sound speed f 2 = absorption = frequency dependent “filters” for each parameter

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Estimation of parameters Variational formulation/penalized likelihood approach Issue of interest here is the prior Gaussian log likelihood Prior information, regularizer

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Prior Models Typical priors based on smoothness of the functions = regularization parameter p = 1 gives total variation reconstruction with edges well preserved p = 2 gives smooth reconstructions

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Priors (cont) What about co-variations between f 1 and f 2 ? Physically, these quantities are not independent –Tumors, lesions, etc. should appear in all unknowns –Speculate that spatial variations in one correlate with such variations in the other Looking to supplement existing prior with mathematical measure of similarity between the two functions or their gradients Three possibilities examined today

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Option 1: Gauss-Markov Random Field-Type Prior Natural generalization of the smoothness prior that correlates the two functions i,j i+1,j i-1,j i,j+1 i,j-1 i,j i+1,j i-1,j i,j+1 i,j-1 f1f1 f2f2 i,j w1w1

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GMRF (cont) Matrix form The GMRF regularizer Implies that covariance of f is equal to What does this “look” like?

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GMRF: Middle Pixel Correlation Lag x Lag y

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GMRF: Comments Motivated by / similar to use of such models in hyperspectral processing Lots of things one could do –One line parameter estimation –Appropriate neighborhood structures –Generalized GMRF a la Bouman and Sauer –More than two functions

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Option 2: Mutual Information An information theoretic measure of similarity between distributions Great success as a cost function for image registration (Viola and Wells) Try a variant of it here to express similarity between f 1 and f 2

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Mutual Information: Details Suppose 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

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Mutual Information: Details Mutual information registration used not the images but their histograms Estimate histograms using simple kernel density methods and similarly for p(y) and p(x,y)

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Mutual Information: Example x y f 1 (x,y) f 2 (x,y)= f 2 (x+ ,y) Mutual Information Peak when overlap is perfect

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Mutual Information: Regularizer For simplicity, we use a decreasing function of MI as a regularizer Larger the MI implies smaller the cost

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Gradient Correlation Idea is simple: gradients should be similar –Certainly where there are physical edges, one would expect jumps in both f 1 and f 2 –Also would think that monotonic trends would be similar OK Not OK OK

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A Correlative Approach A correlation coefficient based metric

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Let’s See How They Behave f 1 (x,y) f 2 (x,y)= f 2 (x+ ,y) 5 -5

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Example 1: Sensor Fusion 5 cm 6 cm X-ray source DOT source/detector DOT detectors X-ray detector Noisy, high resolution X ray. 15 dB Cleaner, low resolution DOT, 35 dB

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DOT Reconstructions Truth Tikhonov GMRF Corr. Coeff MI

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X Ray Reconstructions Truth Tikhonov GMRF Corr. Coeff MI

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DOT Reconstructions Truth Tikhonov GMRF Corr. Coeff MI

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X-ray Reconstructions Truth Tikhonov GMRF Corr. Coeff MI

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Mean Normalized Square Error

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Example 2: Sensor Diffusion 5 cm 6 cm source receiver Ultrasound problem Tissue-like properties 5 frequencies between 5kHz and 100 kHz Wavelengths between 1 cm and 30 cm Image sound speed and attenuation High SNR (70 dB), but sound speed about 20x absorption and both in cluttered backgrounds

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Sound Speed Reconstructions Truth Tikhonov GMRF Corr. Coeff MI

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Absorption Reconstructions Truth Tikhonov GMRF Corr. Coeff MI

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Sound Speed Reconstructions Truth Tikhonov GMRF Corr. Coeff

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Absorption Reconstructions Truth Tikhonov GMRF Corr. Coeff

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Mean Normalized Square Error

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Demosaicing

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Eye Region: Red OriginalTikhonovCorr. Coeff.

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Eye Region: Green OriginalTikhonovCorr. Coeff.

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Chair Region: Red OriginalTikhonovCorr. Coeff.

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Chair Region: Green OriginalTikhonovCorr. Coeff.

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Normalized Square Error

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Conclusions etc. Examined a number of methods for building similarity into inverse problem involving multiple unknowns Lots of things that could be done –Objective performance analysis. Uniform CRB perhaps –Parameter selection, parameter selection, parameter selection –3+ unknowns –Other measures of similarity

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