a modular registration algorithm for medical images

7/27/2019 A Modular Registration Algorithm for Medical Images

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S. Bertoluzza, G. Maggi, S. Tomatis


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Problem: individuate anatomical structures inmedical images

Atlas based segmentation Image R (to be segmented)

Atlas: image T already segmented

Register R and T R inherits the segmentation of T

Reduce segmentation problem to registrationproblem


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Grey scale images real valued functions R: R -> R

+, T: T -> R+

Find mapping : R -> T such that T0 and Rare as close as possible

Huge number of approaches Huge number of algorithms

See [Brown 92; Zitova,Flusser 03]


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Rigid transformations (translations, rotations) Affine transformation

Non rigid transformations

Splines, B-splines [Thevenaz, Unser 98]

Thin plate splines [Bookstein, 89] Interpolating wavelets [S.B., G. Maggi 13]

...

Parametric transformations


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N-dimensional parameter space Parameter -> mapping

(Ex: = i ei )

= (x;)

x : spatial coordinate (in R2 )

: parameter (in RN)

: R2 x RN function (represents the selected class

of transformations)


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Im: space of images

d: Im x Im -> R functional measuring thediscrepancy between the two images

: Im -> R defined as (X) = D(X,R)

c : RN -> R cost functional c() = d(T0 ,R) = (T0 )


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Many different possibilities Least square error

Human Visual System model distance [Mannos, 74] Structural similarity index SSI

[Wang et al, 04, Brunet, Vrscay, Wang 11]

Besov functional norm

Besov norm + divisive renormalization (generalisedSSI) [S.B., G. Maggi, 13]

Mutual information [Viola, Wells 97; Thevenaz, Unser 98]

...


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Find in RN = arg min c()

Unconstrained optimization problem

Several possible optimization algorithms Choice depends on the characteristics of

Transformation

Distance functional


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Need rules to

Evaluate images out of their domain of definition

(extrapolation) Compute values of (deformed) images at pixels

(interpolation)

Compute the values of derivative of images at

pixels (numerical differentiation)

Many different possibilities


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Huge number of algorithms obtained bycombining Different distance functionals

Different transformation classes

Different image models

Different optimization algorithms

Develop an environment allowing to easilytest different combinations


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dxxJxTxTDc );())(();()(

Most optimization algorithms need tocompute

Chain-rule

I II III

c


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I. Frchet derivative of the functional

II. Gradient of the image

III. Jacobian (w.r. to the parameter) of thetransformation

);( xXD

)(yT

);(

xJ


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Image module

Transformation module

Distance module

Optimization is dealt with by the MatlabOptimization Toolbox


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Input:

The parameter

Output:

The grid Gobtained by deforming the reference

grid via the mapping

The values assumed by the jacobian (w.r. to theparameter ) of the function at the points of the

reference grid


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Input:

A sampled image X

A grid G of arbitrary points (representing thecenter of the deformed pixels)

Output:

An interpolated (or extrapolated) value of X at thepoints of G

A value of X at the points of G


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Input

An arbitrary image X sampled at the points of the

reference grid

Output:

The value of the cost function c(X)

The value at the points of the reference grid of(the Frchet derivative of the cost functional at X)


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To evaluate c()

Call the mapping module to obtain the deformed

grid G Call the image model module to sample T at the

grid G -> T = deformed image sampled at the

reference grid Call the distance module to evaluate c() = d(T;R).


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To evaluate c

Call the mapping module and evaluate J(x; )at the points of the reference grid

Call the image module and evaluate T at the

points of the deformed grid

Call the distance module and evaluate D (T,x) atthe points of the reference grid

Evaluate the integral by a chosen quadrature rule


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Flexible algorithm allowing to switch one ofthe ingredient without modifying the others

New distance functionals / image models /transformation classes can be incorporatedeasily

Idea can be extended to the computation ofthe Hessian


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Computational cost: evaluation of the costfunctional

(*) computed on a machine twice as fast

64 x 64 128x128 256x256 512x512

LS 0.2815 0.35938 0,73438 3.0156

SSIM 0.23438 0.67188 1.8906 6.5625

G-SSIM 1.125 1.5313 2.9531 12.875

MI(*) 2.106 8.0965 31.902 134.41


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REFERENCE IMAGE TEMPLATE IMAGE

CT studies, of the same patient by different CT machines


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Quality of the results

Unreg. LS SSIM G-SSIMPearson 0.245 0.858 0.872 0.926

Overlap 0.492 0.992 0.993 0.992

M1 0.47 0.833 0.846 0.85

M2 0.253 0.868 0.861 0.933

a modular registration algorithm for medical images

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