Matching a 3D Active Shape Model on sparse cardiac image data, a comparison of two methods Marleen...
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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