Some thoughts on regularization for vector-valued inverse problems

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

2. 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

3. 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

4. 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

5. GE Tomosynthesis Optical measurement done under mammographic compression Optical Imager 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

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

7. 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

8. De-Mosaicing Bayer pattern • 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 • yred = observed red pixels over sub-sampled grid. 9 vector for example • frwd= red pixels values over all pixels in image. 30 vector in example • Kred = selection matrix with a single “1” in each row, all others 0. 9x30 matrix for example

9. 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

10. 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 • f1 = sound speed • f2= absorption •  = frequency dependent“filters” for each parameter

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

12. 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

13. Priors (cont) • What about co-variations between f1 and f2? • 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

14. i+1,j i,j i,j-1 i,j+1 i-1,j Option 1: Gauss-Markov Random Field-Type Prior • Natural generalization of the smoothness prior that correlates the two functions f2 f1 i+1,j i,j i,j-1 i,j+1 i-1,j w1 i,j

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

16. GMRF: Middle Pixel Correlation Lag y Lag x

17. 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

18. 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 f1 and f2

19. 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

20. 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)

21. Mutual Information: Example f1(x,y) y Peak when overlap is perfect Mutual Information x f2(x,y)= f2(x+,y) 

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

23. Gradient Correlation • Idea is simple: gradients should be similar • Certainly where there are physical edges, one would expect jumps in both f1 and f2 • Also would think that monotonic trends would be similar OK OK Not OK

24. A Correlative Approach • A correlation coefficient based metric

25. Let’s See How They Behave 5 f1(x,y) -5 f2(x,y)= f2(x+,y)

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

27. DOT Reconstructions GMRF Tikhonov Truth Corr. Coeff MI

28. X Ray Reconstructions Tikhonov GMRF Truth Corr. Coeff MI

29. DOT Reconstructions Tikhonov GMRF Truth Corr. Coeff MI

30. X-ray Reconstructions GMRF Tikhonov Truth Corr. Coeff MI

31. Mean Normalized Square Error

32. source receiver 5 cm 6 cm Example 2: Sensor Diffusion • 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

33. Sound Speed Reconstructions GMRF Tikhonov Truth Corr. Coeff MI

34. Absorption Reconstructions GMRF Tikhonov Truth Corr. Coeff MI

35. Sound Speed Reconstructions Tikhonov GMRF Truth Corr. Coeff

36. Absorption Reconstructions GMRF Tikhonov Truth Corr. Coeff

37. Mean Normalized Square Error

38. Demosaicing

39. Eye Region: Red Original Tikhonov Corr. Coeff.

40. Eye Region: Green Original Tikhonov Corr. Coeff.

41. Chair Region: Red Original Tikhonov Corr. Coeff.

42. Chair Region: Green Original Tikhonov Corr. Coeff.

43. Normalized Square Error

44. 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