A Log-Euclidean Polyaffine Framework for Locally Rigid or

Affine Registration

Vincent ARSIGNY.

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Why locally rigid or affine deformations?

• Well-adapted to certain situations:

– histological slices [Pitiot, MedIA, 2005]– articulated structures [Papademitris, Miccai’05]

• After global alignment, allow finer registration with very smooth deformations. Alternatives:

– B-Splines [Rueckert, TMI, 99]– RBFs [Rohde, TMI, 2003]– Vortex particules [Cuzol, IPMI’05]– Etc.

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Example: Head and Neck

Reference image Resampled floating image

Images obtained with SuperBaloo algorithm. Thanks Olivier.

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Overview

1. Presentation

2. (Previous) Polyaffine framework

3. Log-Euclidean Polyaffine framework

4. 3D Registration Results

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Locally Rigid or AffineDeformations

• Parameters: N rigid or affine components where:

– : rigid or affine transformations– : weights, modeling spatial

influence/anchoring or components.

• Example:

(Ti ;wi )Ti = (M i ;ti )wi (x) > 0

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Locally Affine Deformations

• Direct fusion into global deformations(with normalized weights):

• Advange: simplicity, smoothness

• Drawback: non-invertibility in general

T(x) =P

i wi (x):Ti (x)

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Polyaffine Framework

• Idea: use infinitesimal fusion,via integration of an ODE during 1 unit of time.

• How? In [Arsigny, MedIA,05], [Arsigny, Miccai’03]:

– Use logarithms of linear parts

– ODE: _x =P

i wi (x):(ti +log(M i )(x ¡ sti )) for s 2 [0;1]:

Vincent Arsigny, Xavier Pennec, and Nicholas Ayache. Polyrigid and Polyaffine Transformations: A Novel Geometrical Tool to Deal with Non-Rigid Deformations - Application to the registration of histological slices .

Medical Image Analysis, 9(6):507-523, December 2005.

M i

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Example: Mixing Two Rotations

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Examples

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Histology: Correction of Artifacts

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Registration Results

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Overview

1. Presentation

2. (Previous) Polyaffine framework

3. Log-Euclidean Polyaffine framework

4. 3D Registration Results

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Why a Novel Polyaffine Framework?

• Previous framework has severaldefects:

– Inverse of polyaffine transformation not polyaffine

– Fusion: depends on the coordinate system(not affine-invariant)

• ODE integration: high computational cost.

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Log-Euclidean Framework

• Idea: use logarithms of transformations (vs. only linear parts previously)

• New polyaffine ODE (homog. coord.):

log(T) = logµM t0 1

¶

=

µL v0 0

¶:

_x = V(x) def=P

i wi (x) log(Ti ) :x

• Novel framework: [Arsigny,WBIR’06], INRIA research report pending.

Vincent Arsigny, Olivier Commowick, Xavier Pennec, and Nicholas Ayache. A Log-Euclidean Polyaffine Framework for Locally Rigid or Affine Registration. In Proc. of WBIR’06 (to appear).

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Log-Euclidean Framework

• Remarkable novel properties:– ODE is stationary (or autonomous)– Integration of ODE: yields

one-parameter subgroup of LEPTs– In particular: inverse (or square root):

is a LEPT with identical weights and inversed (or square rooted) transfos

– affine-invariance

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Log-Euclidean Framework

• Why called Log-Euclidean?– Based on logarithms of transfos– If weights do not depend on x:

Log-Euclidean mean of transfos:

– Properties: geometric interpolation of determinants, inverse-invariance.

T = exp(P

i wi log(Ti ))

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Fast Polyaffine Transform (FPT)

• Fast computation of LEPTs on regular grids possible, efficiency somehow comparable to FFT.

• Idea: generalize ‘Scaling and Squaring’method for matrix exponential.Basic property:

exp(M ) = exp( M2N)2N

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Fast Polyaffine Transform

• Integrating the ODE:

• Integration time-step doubles ateach interation.

Scaling Step: choose N, so that V(x)/2^N is close enough to zero (depending on accuracy desired).

Exponentiation step: compute deformations at time 1/2^N with a numerical scheme.

Squaring step: small deformations squared recursively N times (N compositions of mappings) to obtain the LEPT

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Fast Polyaffine Transform

Typical

example:

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Fast Polyaffine Transform

• Relative accuracy, 50x40 grid.

• When N varies,convergence toward:

– Average error: 0.2 %

– Error max: 2%

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Inversion with the FPT

Inversion of LEPTs:

Just compute FPT

with

inverted transfos.

Accuracy: 0.2%.

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Fast Polyaffine Transform

• In the following: used for post-processing of already estimated components.

• Interest:– Singularities removed (vs. direct fusion)– Fast computation of inverse – No artifact introduced in fusion

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Fast Polyaffine Transform

Direct fusionLEPT fusion

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Overview

1. Presentation

2. (Previous) Polyaffine framework

3. Log-Euclidean Polyaffine framework

4. 3D Registration Results

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Robust Block-Matching Algorithm

Olivier Commowick, Vincent Arsigny, Jimena Costa, Nicholas Ayache, and Grégroire Malandain. An Efficient Locally Affine Framework for the Registration of Anatomical Structures. In Proc. of ISBI’06 (to appear).

² Presented in [Commowick, ISBI06]² a±ne/ rigid components chosen before registration² weight functions de ned viamasks of structures of interest² global a±nealignement ¯rst² two levels of resolution.² 10 block-matching iterations at each resolution² similarity criterion: local correlation ratio² components estimated with direct fusion, with weighted LTS² visco-elastic Log-Euclidean regularization step² post-processing: FPT (8 squarings) of transfo and inverse² typical CPU time: < 10minutes (8 components, 200x200x200 voxels)

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Example: Head and Neck

Reference image Resampled floating image

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Brain Critical Structures Segmentation

• Method: – Atlas-to-Subject

registration– Affine component

areas defined on the atlas

– Apply transformation to contours

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Results on Brain Structures Segmentation

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Results on Brain Structures Segmentation

→ Comparable accuracy, locally affine much smoother.

Patient # 1 2 3 4 5 6Sensitivity ( Runa ) 0.88 0.86 0.84 0.79 0.85 0.78Speci¯city ( Runa ) 0.86 0.91 0.79 0.94 0.89 0.91Distance to (1,1) ( Runa ) 0.19 0.17 0.27 0.22 0.19 0.24Sensitivity ( MAF) 0.86 0.85 0.82 0.84 0.84 0.76Speci¯city ( MAF) 0.89 0.91 0.83 0.92 0.90 0.93Distance to (1,1) ( MAF) 0.18 0.17 0.25 0.18 0.19 0.25

Table 1: Registration results on brainstem with dense method (Runa)[Stephanescu, MedIA, 2005] and locally a±ne. Sensitivity and speci¯city ob-tained: STAPLE algorithm, with seven experts

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Conclusions

• Current locally-affine framework:– Well-adapted in a number of situations– Efficient estimation of local deformations

(vs. [Arsigny, MedIA,05])– Invertibility of global deformations and

intuitive properties of fusion:Log-Euclidean polyaffine framework

– Efficient computation of global transformation as well as inverse: FPT

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• Estimation of geometry components:ideas of [Pitiot, MedIA,05] to integrate,necessary for histology in particular.

• Statistics on deformability in low-dimensional spaces (vs. dense deformations),easier to embed in registration?

• Locally rigid/affine deformations between global alignment and dense deformations?Goal: comparable or better accuracy with smoother deformations

Perspectives

Thank you for your

attention!

Any questions?