Matching a 3D Active Shape Model on sparse cardiac image data, a

comparison of two methodsMarleen Engels

Supervised by: dr. ir. H.C. van Assen

Committee:prof. dr. ir. B.M. ter Haar Romenydr. A. Vilanova Bartrolidr. ir. H.C. van Assendr. ir. H.M.M. ten Eikelder

June 2007

Outline Introduction Active Shape Model Optimization methods Method of Least Squares Cross Out method Experiments with phantoms Experiments with real data Results Conclusions and discussion Future work

IntroductionAnatomy of the heart

Supplying the entire body of blood

Introduction

Introduction

Increasing number image acquisitions Automate segmentation and diagnosis Reduce scanning time by reducing the number

of image slices per acquisition → sparse data

Motivation

IntroductionGoal of the project

To segment sparse cardiac image, using a 3D Active Shape Model, implementing and testing 2 different approaches

1) Optimization methods, like Lötjönen et al. did.2) Cross Out, newly developed in this project

Active Shape Model A Statistical Shape Model (SSM) contains

information about the mean shape and shape variations based on a representative training set.

x = xmean + Φbb = ΦT(x - xmean)

When a SSM is used to segment unseen data then it is called an Active Shape Model (ASM).

Active Shape Modelfirst mode third modesecond mode

Active Shape Model

Active Shape Model An ASM requires complete data sets Modify ASMs

• SPASM by van Assen et al.• Optimization Methods by Lötjönen et al.• Cross Out Method (new)

Optimization methods A different b vector generates a different shape x Finding a vector b which generates a shape that

fits the sparse data best→ using optimization methods

Optimization methods: finding an optimum (global minimum or maximum) of a (cost)function

M

ii sd

MYf

1,

1, byb

Optimization methods Steepest Descent method Conjugate Gradients method Space method …

It is application dependent which method works best

Optimization methodsSteepest Descent method

A new point, closer to the minimum, is found by searching for a minimum in the opposite direction of the gradient at the current point

Bad convergence if xo is badly chosen

Optimization methods

Uses non-interfering search directions, conjugate directions

A minimum can be found in a t-dimensional space in t iterations

Conjugate Gradients method

Optimization methodsConjugate Gradients method

x2

x2

x1

x1

Steepest descent

Conjugate gradients

Optimization methods

Repetitive search to find the optimal vector bopt

Each element of b, bi for i = 1,…,t, is separately optimized

The initial b is bopt = 0, bi,opt = 0

Space method

Optimization methodsSpace method

f(b)

bibi,opt-3√λi 3√λi

Method of Least Squares

2

1

N

iii xfy

xf

Method of Least Squares Can be applied to solve a linear system

Ax = b x* = (ATA)-1ATb is the least squares

solution of the linear system Ax = b, the distance between Ax* and b minimized

A is the coefficient matrix, x are the unknown variables, and b are the known variables

A shape can be generated with:x = xmean + Φb

Linear system: Φb = (x – xmean),Φ the coefficient matrix, b the unknown

variables, (x – xmean) the known variables Least squares solution is:

b* = (ΦTΦ)-1ΦT(x – xmean) In literature:

b* = ΦT(x – xmean)

Method of Least SquaresApplication to ASM’s

Method of Least SquaresApplication to ASM’s

A shape x0 is generatedwith b0

b*calc,1 =

ΦT(x0 – xmean)

b*calc,2 =

(ΦTΦ)-1ΦT(x0 – xmean)

Cross Out methodWhen x is not complete (sparse data) the equation

Φb = (x – xmean) = dx

still holds, when corresponding rows of dx and Φ

are crossed regarding the dimensions

[3N x t][t x 1] = [3N x 1] →

[3N – 3R x t][t x 1] = [3N – 3R x 1]

Cross Out Method Now a sparse linear system is created

Φsparseb = dxsparse = xsparse – xmean,sparse

Using the method of least squares to calculate b*

sparse

b*sparse=(Φsparse

TΦsparse)-1ΦsparseT(xsparse–xmean,sparse)

Experiments

Error: average point to point distance between the point of calculated shape and the original shape

ptosError: average point to surface distance between the points of the calculated shape and the surface of the original shape

The performance of the cross out method and the optimization methods can be determined by:

Experiments with phantoms Per experiment a set of 15 shapes is used 15 different b vectors Each element of b is randomly chosen

with the restriction that the generated shape resembles the shapes of the training set.

Experiments with phantoms

1) Deleting 500 points with the most variation with the least variation randomly

2) Deleting points in slices and vary the number of deleted slices

3) Using 60 and 89 modes

Testing the Cross Out method

Experiments with phantomsTesting the Cross Out method (1),

deleting 500 points

Complete shape Shape without points with least variation

Shape without points with most variation

Shape without 500 random points

Experiments with phantomsTesting the Cross Out method (1),

deleting 500 points

Experiments with phantomsTesting the Cross Out method (2), vary

the number of slices to deletenumber of slices removed

slice number 0 1 2 3 4 5 6 7 8 9 101 X X X X X2 X X X X X X3 X X X X X4 X X X X5 X X X X X6 X X X X X7 X X X X8 X X X X X9 X X X X X X

10 X X X X X11 X X X X X

X = deleted

Experiments with phantomsTesting the Cross Out method (2), vary

the number of slices to delete

Experiments with phantoms

The complete model has 89 modes of variations, 100 % of all the variation present in the training set

60 modes contains about 97 % of the variation present in the training set

15 shapes in 5 configurations

Testing the Cross Out method (3), using 60 and 89 modes

Experiments with phantoms

configurationslice number 11 slices 9 slices 7 slices 5 slices 2 slices

1 X2 X3 X4 X X5 X X X6 X7 X X X X89 X X

10 X X X11 X X X

Testing the Cross Out method (3), using 60 and 89 modes

X = deleted

Experiments with phantomsTesting the Cross Out method (3),

using 60 and 89 modes

Experiments with phantoms

It does matter which points are deleted, deleting points with least variation gives the best result

Up till 8 slices can be deleted and still a good shape is found

Using 89 modes gives a better result than 60 modes

Testing the Cross Out method, conclusions

Experiments with phantoms

Implemented in C by dr. J. Lötjönen using 60 modes Optimization method Step size of the gradient Range of the parameter space

15 shapes in 4 different configurations Conjugate gradients method with step size 0.1 for

Error Steepest descent method with step size 0.1 for

ptosError

Optimization methods

Experiments with phantoms

15 shapes in 4 configurations Cross Out method with 60 modes Cross Out method with 89 modes Conjugate gradients with step size 0.1 Steepest Descent with step size 0.1

Optimization versus Cross Out

Experiments with phantomsOptimization versus Cross Out

11 slices 9 slices 7 slices 5 slices

ResultsOptimization versus Cross Out, using

phantoms

ResultsOptimization versus Cross Out, using

phantoms

Experiments with real data 15 shapes in 4 configurations

Cross Out 60 modes, Cross Out 89 modes, Conjugate gradients step size 0.1

11 slices 6 slices 4 slices8 slices

ResultsReal data

Conclusions and Discussion When using a ASM it is better to use the

least squares method The Cross Out method gives better results

than the optimization methods The performance of ASM depends on how

well the training set represents the entire population

Future work Test the robustness of the Cross Out

method Cross Out method should implemented as

iterative procedure Designing a smart scanning protocol

Questions?

Special thanks to

Hans van Assen

Bart ter Haar Romeny