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.