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

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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?

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

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

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

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

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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

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Summary of paper (Spatially) Conditional Mutual Information proposed –An improvement over Studholme et als 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

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

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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 MI and Regional MI

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Studholme et al use simple boxcar kernels, overlapping by 50% Each voxel contributes to 2 d 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

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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

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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)

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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 CorrelationConditional MIYe Olde Traditional MI

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RMI vs cMI (not Studholme vs Loeckx) p r (m1,m2) = p(m1, m2 | r) –The following seem equivalent to me…

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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…

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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

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Analytic derivatives Only term depending on transformation

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

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

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Results

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Dice Similarity Coefficient DSC = volume of intersection / avg vol. higher is better centroid distance cD = distance between centres of mass of segmentations lower is better

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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 dont like local histogram estimation methods… –John Ashburner

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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 SPM5s Unified Segmentation algorithm Bias relative to unbiased tissue priors from atlas is modelled –Also done in FSLs not-yet-released FNIRT (Jesper Andersson) Directly correct differential bias –E.g. filter difference or ratio image (Lewis and Fox) –Less principled?

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References Loeckx, D.; Slagmolen, P.; Maes, F.; Vandermeulen, D. & Suetens, P. (2007) Nonrigid image registration using conditional mutual information. IPMI 20: Nonrigid image registration using conditional mutual information 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: Deformation-based mapping of volume change from serial brain MRI in the presence of local tissue contrast change Thevenaz, P. & Unser, M. (2000) Optimization of mutual information for multiresolution image registration. IEEE Trans. Image Proc. 9: Optimization of mutual information for multiresolution image registration

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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 Comparison Between Parzen Window Interpolation and Generalised Partial Volume Estimation for Nonrigid Image Registration Using Mutual Information Kybic, J. & Unser, M. (2003) Fast parametric elastic image registration. IEEE Trans. Image Proc.12: Fast parametric elastic image registration 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 23:99-110Accurate template-based correction of brain MRI intensity distortion with application to dementia and aging Lewis, E. B. & Fox, N. C. (2004) Correction of differential intensity inhomogeneity in longitudinal MR images. Neuroimage 23:75-83Correction of differential intensity inhomogeneity in longitudinal MR images

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