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 • What effect does (remaining) differential bias have on nonrigid registration?

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

efluid SSD -n -400 nreg SSD -ds 2.5

efluid NMI

Displ. Magnitude

black = 0 white = 2mm nreg NMI

efluid SSD nreg SSD efluid NMI

Jacobian

black = 0.8

white = 1.2

nreg NMI

A second opinion, courtesy of Marc Modat

F3D (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

MI and Regional MI

• 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

• Studholme et al use simple boxcar kernels, overlapping by 50% • Each voxel contributes to 2 d d-dimensions bins in • 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)

• p r (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 

f

(

g

(

h

(

x

))) 

x

 

f



g f

(

g

) 

g



h g

(

h

) 

h



x h

(

x

)

Analytic derivatives

Only term depending on transformation

Analytic derivatives

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

Analytic derivatives

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

Results

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

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) registration Optimization of mutual information for multiresolution image . 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 aging . 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 . 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