Nonrigid Image Registration Using Conditional Mutual

InformationLoeckx et al. IPMI 2007

CMIC Journal Club 14/04/08Ged Ridgway

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?

BrainWebT1, 3% noise0 and 40% bias

Difference imgRato img

Ratio of smooth images (10 mm stdev Gaussian)

Applied BW bias

Displ. Magnitudeblack = 0white = 2mmefluid

SSD-n -400

nregSSD-ds 2.5

efluidNMI

nregNMI

Jacobianblack = 0.8white = 1.2efluid

SSD

nregSSD

efluidNMI

nregNMI

A second opinion, courtesy of Marc ModatF3D (GPU Fast FFD), 2.5mm spacing, Mutual Information

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

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

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

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

MI and Regional MI

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

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”

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)

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

RMI’ vs cMI (not Studholme vs Loeckx)• pr(m1,m2) = p(m1, m2 | r)

– The following seem equivalent to me…

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…

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

)()()(

)))(((

xhhggf xh

hg

gf

xxhgf

Analytic derivatives Only term depending on transformation

Analytic derivatives

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

Analytic derivatives

The paper is incomplete – see Thevenaz and Unser for more…

But we want cMI =

Results

Dice Similarity CoefficientDSC = volume of intersection / avg vol.higher is better

centroid distance cD = distance betweencentres of mass of segmentationslower is better

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

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?

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

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