Shape Analysis Lasse Riis stergaard Dept. of Health Science and - - PowerPoint PPT Presentation

Shape Analysis

Lasse Riis Østergaard

• Dept. of Health Science and Technology

Shape Analysis

• Active contour models - snakes

– Theory – Applications

• Active shape models

– Theory – Applications

Active Contour Example

Active Contours

• Representation

– How the contour appears – Geometry

• Energy function

– A function determining the behavior of the contour – Physics

• Energy minimization

– The scheme to minimize the energy function – Approximation theory

• 2D representation: a line

– r(s) =x(s),y(s)

• 3D representation

– Various extensions of 2D, e.g. plane, closed surface (sphere), generalized cylinder

Representation

Energy Function

ds (s) E (s) E E

external internal snake

1

− ∫

=

Internal Energy

) (

rigidity tension internal

s E (s) E E +

=

Tension

2 2 tension

) ( ) ( ) ( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = ds s dy ds s dx ds s dr (s) E

• I.E, the square of how much the snake is

stretched at a given point

Rigidity

2 2 2 2 2 2 2 2 rigidity

) ( ) ( ) ( ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = ds s y d ds s x d ds s dr (s) E

• I.E, the square of how much the snake is bent at a

given point

Behavior of Rigidity

)) ( ), ( ( )) ( ), ( (

line

s y s x I s y s x F (s) E ± =

=

2 edge

) ( ), ( )) ( ), ( ( ) ( Ι ∇ =

=

s y s x s y s x F (s) E

Lines Edges

External energy

I x y x y x I x

Energy Minimization

• Finite Differences Method (FDM)
• Finite Element Method (FEM)
• Dynamic programming
• Greedy
• B-snakes

Greedy

• The search space (m) for each control node is
• searched. Complexity O(Nm)
• Fast greedy
• Complexity O(N ½m)
• Takes more iterations to converge

Dynamic Programming

• Treat the the minimization process as a

path planning problem.

• Values from combinations are stored in a
• matrix. The matrix is backtraced
• Complexity O(Nm^3)

Method Overview

15

3D data Vessel centre-line Vessel surface Diameter quantification

Centre-Line Extraction - MRA

Original MRA scan Vessel enhancement filtering Fuzzy c-means clustering Largest connected components Topology preserving thinning 3D vessel centre-line

16

Surface Deformation

• Parametric deformable model (discrete model)
• Circular homogenous generalized cylinder
• Energy function
• E_total = E_int + E_ext
• Non-uniformly expanding cylinder
• Iterative global optimization (Dynamic Programming)
• Surface reparameterization
• Avoid twisting (inflection points – curvature changes sign)
• Centre-line segments are processed individually
• End points are fixated to preserve connectivity

17

Surface Deformation – energy terms

• Internal energy
• Tension (elasticity) - uniform distribution of centre-line points
• Rigidity - smoothness of centre-line
• Radius – smooth variation of the radius
• Expansion – expanding behaviour
• Constant weight parameters for all internal energy terms
• External energy
• Edge – attracts circular contour towards vessel wall (image gradients)
• Weight parameter estimated based on a ”force balancing” strategy
• E_total = α · L(V_i)
• E_total = E_tension + E_rigidity + E_radius + E_expansion + E_edge

18

Surface Deformation – iterative approach

19

20

MRI Data Initial surface Inner surface Outer surface Measurements

Min Max

Method Overview

FACE

• Fast Accurate Cortex Extraction

DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY

21

Initial Surface

Native scan Registration and intensity correction Brain extraction WM classification Cerebrum WM segmentation Filling of ventricles and subcortex Hemisphere separation T

• pology correction

Surface tessellation Cortical reconstruction – method

DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY

22

Every 5th iteration Cortical reconstruction – method

Outer Surface Deformation

DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY

23

DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY

24

Cortical reconstruction – evaluation

Qualitative Evaluations – intersections with images