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Nonrigid Image Registration Using Conditional Mutual Information Loeckx et al. IPMI 2007

Nonrigid Image Registration Using Conditional Mutual Information Loeckx et al. IPMI 2007. CMIC Journal Club 14/04/08 Ged Ridgway. Motivation – differential bias. MRI typically corrupted by smooth intensity bias field Worse at higher field strengths Approximate correction is possible

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Nonrigid Image Registration Using Conditional Mutual Information Loeckx et al. IPMI 2007

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  1. Nonrigid Image Registration Using Conditional Mutual InformationLoeckx et al. IPMI 2007 CMIC Journal Club 14/04/08 Ged Ridgway

  2. Motivation – differential bias • MRI typically corrupted by smooth intensity bias field • Worse at higher field strengths • Approximate correction is possible • What effect does (remaining) differential bias have on nonrigid registration?

  3. BrainWeb T1, 3% noise 0 and 40% bias Difference img Rato img Ratio of smooth images (10 mm stdev Gaussian) Applied BW bias

  4. Displ. Magnitude black = 0 white = 2mm efluid SSD -n -400 efluid NMI nreg SSD -ds 2.5 nreg NMI

  5. Jacobian black = 0.8 white = 1.2 efluid SSD efluid NMI nreg SSD nreg NMI

  6. A second opinion, courtesy of Marc Modat F3D (GPU Fast FFD), 2.5mm spacing, Mutual Information

  7. Conclusions • Clear problem • Also for (N)MI – possibly even worse • Particularly important for Jacobian Tensor Based Morph • Caveats • Large (+/- 40%) bias (though not that large…) • No attempt at prior correction

  8. Summary of paper • (Spatially) Conditional Mutual Information proposed • An improvement over Studholme et al’s Regional MI... • Implementation • B-spline (quadratic) Free Form Deformation Model • Same for image interp. (continuously differentiable) • Parzen Window or Partial Volume histogram estimation • Analytical derivatives in limited mem quasi-Newton optimizer • Comparisons • Artificial “multi-modal” data • Lena with strong bias field • CT/MR with clinical segmentation

  9. MI and Regional MI • Studholme et al. (2006) proposed regional mutual information (mathematically, “total correlation”) treating spatial location as a third “channel” of info

  10. MI and Regional MI • The RMI objective is equivalent to optimising a weighted sum of the regional MI estimates • P(r) is simply the relative volume of the region with respect to the whole image

  11. MI and Regional MI • Studholme et al use simple boxcar kernels, overlapping by 50% • Each voxel contributes to 2d bins in d-dimensions • This choice simplifies the computation of the gradient • Studholme et al implement a symmetric large deformation fluid algorithm, with analytical derivatives

  12. Conditional MI • Conditional entropies given the spatial distribution • MI expresses reduction of uncertainty in R from knowing F (and vice-versa) • cMI: reduction in uncertainty when the spatial location is known • “cMI corresponds to the actual situation in image registration”

  13. RMI vs cMI (not Studholme vs Loeckx) • C(R, F, X) = H(R) + H(F) + H(X) - H(R, F, X) • I(R, F | X) = H(R | X) + H(F | X) - H(R, F | X) • Generally, H(A, B) = H(A | B) + H(B) • I(R, F | X) = H(R, X) + H(F, X) - H(R, F, X) - H(X)

  14. Figure 1 revisited • Similar to probabilistic Venn diagram • However, p(A, B) gives intersection; H(A, B) gives union • C(R, F, X) = H(R) + H(F) + H(X) - H(R, F, X) • I(R, F | X) = H(R, X) + H(F, X) - H(R, F, X) - H(X) Total Correlation Conditional MI Ye Olde Traditional MI

  15. RMI’ vs cMI (not Studholme vs Loeckx) • pr(m1,m2) = p(m1, m2 | r) • The following seem equivalent to me…

  16. Studholme vs Loeckx • Fluid vs FFD • Large deformation (velocity regularised) vs small • Symmetric vs standard (displacement in target space) • Boxcar vs B-spline spatial Parzen window • Loeckx more principled (?) • “same settings for knot-spacing in both formulas – local transformation guided by local joint histogram, both using the same concept and scale of locality” • but means finer FFD levels have fewer samples…

  17. Analytic derivatives • “Our” FFD algorithm estimates the derivative of the cost function with respect to a particular control-point by finite differencing (moving one control point) • Loeckx (and Studholme) show that expressions for the derivative can be obtained in closed form • Spline interpolation means the image is differentiable • The (multivariate) chain rule lets us decompose the cost-function Jacobian into constituent parts

  18. Analytic derivatives Only term depending on transformation

  19. Analytic derivatives Analytic derivatives of B-splines known, e.g. Thevenaz and Unser (2000)

  20. Analytic derivatives But we want cMI = The paper is incomplete – see Thevenaz and Unser for more…

  21. Results

  22. Dice Similarity Coefficient DSC = volume of intersection / avg vol.higher is better centroid distance cD = distance betweencentres of mass of segmentationslower is better

  23. Objections to cMI • Worse histogram estimation • Effectively, fewer samples • Even (unnecessarily) in homogeneous regions • Ten times slower (!?) • Not yet clear how much re-implementation could help • “I don’t like local histogram estimation methods…” • John Ashburner

  24. Alternative approaches • Reduce bias (in both images separately) • Different acquisition techniques (Ordidge) • Better correction algorithms • Use derived information, e.g. segmentations, features • Model differential bias • Effectively part of SPM5’s Unified Segmentation algorithm • Bias relative to unbiased tissue priors from atlas is modelled • Also done in FSL’s not-yet-released FNIRT (Jesper Andersson) • Directly correct differential bias • E.g. filter difference or ratio image (Lewis and Fox) • Less principled?

  25. References • Loeckx, D.; Slagmolen, P.; Maes, F.; Vandermeulen, D. & Suetens, P. (2007) Nonrigid image registration using conditional mutual information. IPMI 20:725-737 • Studholme, C.; Drapaca, C.; Iordanova, B. & Cardenas, V. (2006) Deformation-based mapping of volume change from serial brain MRI in the presence of local tissue contrast change. IEEE TMI 25:626-639 • Thevenaz, P. & Unser, M. (2000) Optimization of mutual information for multiresolution image registration. IEEE Trans. Image Proc. 9:2083-2099

  26. Other related papers • Loeckx, D.; Maes, F.; Vandermeulen, D. & Suetens, P. (2006) Comparison Between Parzen Window Interpolation and Generalised Partial Volume Estimation for Nonrigid Image Registration Using Mutual Information. Workshop on Biomedical Image Registration • Kybic, J. & Unser, M. (2003) Fast parametric elastic image registration. IEEE Trans. Image Proc.12:1427-1442 • Studholme, C.; Cardenas, V.; Song, E.; Ezekiel, F.; Maudsley, A. & Weiner, M. (2004) Accurate template-based correction of brain MRI intensity distortion with application to dementia and aging. IEEE TMI 23:99-110 • Lewis, E. B. & Fox, N. C. (2004) Correction of differential intensity inhomogeneity in longitudinal MR images. Neuroimage 23:75-83

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