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Laboratory of Mathematics in ImagingHarvard Medical School Brigham and Women’s Hospital
Fast and Accurate Redistancing for Level Set Methods
&Multiscale Segmentation of the Aorta in
3D Ultrasound Images
Karl Krissian
McGill Montreal 2003
I. Level Sets
Introduction
• Narrow Band
– initialization
– distancing
• Experiments
– MRA
– SPGR white matter
– RGB white matter
• Discussion
McGill Montreal 2003
Level Sets: principle
– Implicit representation of the evolving surface.
– Natural topology changes.
uFt
u
McGill Montreal 2003
Level Sets: forces
1. Smoothing:
– mean curvature [Sethian 96, Caselles 97]
– minimal curvature [Ambrioso and Soner 98, Lorigo et al. 00]
2. Advection or Contour attachment.
3. Balloon or expansion:
– constant.
– based on intensity statistics [Zeng et al. 98,Paragios and Deriche
00, Barillot et al. 00].
BAs FFFF
McGill Montreal 2003
CURVES
Lorigo et al. , Medical Image Analysis, 2001.
Ambrioso, Soner, Journal Differential Geometry, 1996.
CURve evolution for VESsel segmentation
Codimension-2 Active Contours
Provided by L. Lorigo MIT AI Lab.
• v is (positive) distance to curve (v, 2v) is smaller principal curvature of tube• d is some vector field in R3
vt = |v|(v, 2v) + v · d
McGill Montreal 2003
CURVES Example
co-dim 2
co-dim 1
Provided by L. Lorigo MIT AI Lab.
McGill Montreal 2003
Fast implementation
• Numerical stability and reinitialization: distance map– Fast Marching Method [Sethian, 99], computes geodesic
distances with complexity n.log(n).
• Speed improvement: itkNarrowBandCurvesLevelSetImageFilter
Sub-voxel reinitialization: itkIsoContourDistanceImageFilter
Fast Distance Transform: itkFastChamferDistanceImageFilter
McGill Montreal 2003
I. Level Sets
Introduction
Narrow Band
initialization
– distancing
• Experiments
– MRA
– SPGR white matter
– RGB white matter
• Discussion
McGill Montreal 2003
SubVoxel Reinitialization
For the voxels neighbors to the isosurface,keep the same linearly interpolated surface:
Remarks:• Points with several neighbors crossing the surface.• Regions of high curvature.
McGill Montreal 2003
Subvoxel versus Binary
binary (+/- 0.5) subvoxel
Evolution of a sphere of radius 3 voxels under constantpropagation force.
McGill Montreal 2003
binary sub-voxel
Sub-Voxel vs Binary
McGill Montreal 2003
I. Level Sets
Introduction
Narrow Band
initialization
distancing
• Experiments
– MRA
– SPGR white matter
– RGB white matter
• Discussion
McGill Montreal 2003
Chamfer Distance Transform
< a, b, c > = < 0.93, 1.34, 1.66 > Relative maximal error 7.356% [Borgefors, On Digital Distance Transforms in Three Dimensions, CVIU, 1996]
McGill Montreal 2003
Narrow Banded Fast DT
McGill Montreal 2003
Narrow-Banded Fast DT
Main speed improvements:
1. Linear complexity.
2. Don’t compute voxels out of the narrow band.
3. Factorize the additions for each kind of neighbor.
4. Keep track of a bounding box.
5. Get positive and negative distances at the same time.
McGill Montreal 2003
Interpretation
Binary Subvoxel
McGill Montreal 2003
I. Level Sets
Introduction
Narrow Band
initialization
distancing
Experiments
– MRA
– SPGR white matter
– RGB white matter
• Discussion
McGill Montreal 2003
Experiments
Image 2003, Euclidian distance up to 5, Pentium III 1.1 GHz.
tim
e in
sec
onds
radius in voxels
Spheres of increasing radii
McGill Montreal 2003
Accuracy experiments
Constant evolution Curvature evolution
2D disk
radius=30
3D sphere
radius=30
McGill Montreal 2003
White matter from SPGR image
McGill Montreal 2003
Fast implementation
Computation time (in sec.) for segmenting White Matter on a 256x256x124 Spoiled Gradient Recall MR.
McGill Montreal 2003
Magnetic Resonance Angiography
initial iso-surface
Fast Marching result Fast Chamfer result
• Resampling• Minimal curvatureSpeed up:• Narrow Band Dist x7• Total Processing x2• Multi-Threading x6
McGill Montreal 2003
Applications MR Angiography
Segmentation resultMaximum Intensity Projection
McGill Montreal 2003
Applications MR Angiography
Segmentation resultIso-surfaces 112, 60 and 40
McGill Montreal 2003
High res. RGB White Matter
Color Image:
-cropped: 1056x1211
-10 seed points
-2D level set
data provided by Peter Ratiu
McGill Montreal 2003
High res 3D RGB White Matter
• 800x1056x1211 sub-volume– Pyramidal multiscale
– 2 seed points
200x264x302100x132x15150x76x75
McGill Montreal 2003
Interface Integration
VTK-tcl ITK-tcl VTK-tcl
Generic slicer module (tcl/tk)
ConnectVTKToITK
ConnectITKToVTKvtkSlicerITK module
Input image volume
Output image volume
McGill Montreal 2003
Graphical Interface
Open Source:www.slicer2.org
Insight Toolkit:www.itk.org
McGill Montreal 2003
Conclusion
• Exact linear Euclidian Distance [Danielsson, 80]
– Propagation, Parallel (multi-threaded)
• Multi-Channel images (RGB, blood flow, multi-
modalities, diffusion tensor)
• Skeleton
• Shape constraints
• Bayesian approach with several level sets.
McGill Montreal 2003 Outline
II. Multiscale Segmentation
Introduction
Methodology
Results
Conclusion and future work
McGill Montreal 2003 Introduction
Medical Interest
• 3D Ultrasound for vascular and gastrointestinal surgery.
• low cost and no radiation exposure.
• with or without preoperative CT.
• need of automatic segmentation of the aorta.
McGill Montreal 2003 Introduction
Introduction
• non homogeneous intensity
• close vessels
McGill Montreal 2003
Koller, Gerig et al. 95 Lorenz et al., 97 Sato et al., 98 Frangi et al., 98Krissian, Malandain,
Ayache, 98
Dimension 2D (3D) 3D 3D 3D 3D
Purpose Line extraction Line extraction Visualization Visualization Line extraction
Response offset central central central offset
Hessian matrix Eigenvectors gradient Eigenvectors Eigenvectors Eigenvectors Eigenvectors gradient
Multiscale methods
Linear multiscale analysis
– Robustness– Accuracy– Optimization
McGill Montreal 2003
H Hessian matrix eigenvalues 321
eigenvectors 321 vvv
Hessian matrix and local structure
Linear structures [Lorenz et al.]
0, 21 321 ,
VOLUMINAL MODELS
)()(2
.)()( 32
hOvIHvh
vIhMIvhMI t
Taylor expansion:
McGill Montreal 2003
yxGCzyxI ,,,00
0:radius initial 20
2: scale aat radius
Cylindrical model
• Analytic analysis– Radius estimation– Optimization of the response– Behavior of the Hessian matrix
McGill Montreal 2003
cylindrical
Cylindrical model20
01
I
2
020
02 1
CMI
03
toroidal
Toroidal model Rxx curvatureR
1
x
xI
120
03
elliptical
20
1y
I
2
2
1
y
x
Elliptical model 20
2x
I
Hessian matrix
McGill Montreal 2003 Methodology
– Hessian Matrix– Structure Tensor
– New Second Order Orientation Descriptor
– 3 parameters
Extraction of local orientations
McGill Montreal 2003 Methodology
– Properties:• zoom invariance.• Symmetric positive.• Continuity of the eigenvectors.• Orientation extraction for both contours (1st order
derivatives) and lines (2nd order derivatives).
Extraction of local orientations
McGill Montreal 2003
Single scale response computation
• Pre-selection of candidates• Plan of the cross-section
21,, vvx
),( x 1v
2v v
dvvxuxR
)(2
1),(
2
0
direction radialv
• Response
pointcurrent x
alityproportion oft coefficien
McGill Montreal 2003 Methodology
• Homogeneity constraint
• Eccentricity constraint
Tubular constraints
McGill Montreal 2003
Normalization of the response function
= 1.0 = 1.28
=1.65
= 2.12
= 2.72 = 3.5
),( xRN
),(),( xRxRN
-normalization [Lindeberg, 96]
• Estimation of the vessel radius
20
2max ),( h
Zoom invariance1
3
Maximization of the maximal response
McGill Montreal 2003
• « height ridge » [Furst et al, 97]
• « marching lines » [Thirion et Gourdon, 97; Fidrich, 97; Lindeberg, 96, Furst et al, 96]
),( ix is a local maximum
),(),( 1 iN
iN xRxR
Extraction of local maxima
),( ix 1v
2v
McGill Montreal 2003
• Tangent vessels• Junctions
• Curvature
153 Rr 103 Rr 53 Rr 35.1 Rr
• Images of variable width
Tests on synthetic images
McGill Montreal 2003 Methodology
• Scales– 10 scales, logarithmic discretization
• Extraction of local maxima
Multiscale analysis
McGill Montreal 2003 Methodology
Results
McGill Montreal 2003 Outline
Results
McGill Montreal 2003 Outline
Conclusion and Future Work
• Conclusion• Semi-automatic segmentation of aorta in 3D
Ultrasound.
• Model Based multiscale linear approach.
• Second Order Orientation Descriptor.
• Homogeneity and eccentricity constraints.
• Future work• Active contours.
• Validation.