Level Set Based Segmentation with Intensity and Curvature Priors

1
Level Set Based Segmentation with Intensity and
Curvature Priors

• IEEE 2000
• Michael E. Leventon1, Olivier Faugeras1,2, W.
  Eric L. Grimson1, William M. Wells III1,3
• 1MIT AI Lab Cambridge, MA 02139,
• 2INRIA Sophia Antipolis, France,
• 3Surgical Planning Lab, Brigham and Womens
  Hospital
• Sang Ho Lee
• Sang Joon Park

2
Abstract

• A method is presented for segmentation of
  anatomical structures
• Intensity
• Curvature
• from a training set of images and boundaries
• Intensity distribution
• Function of signed distance from the object
  boundary
• Curvature profile
• Boundary regularization term
• Demonstrated on
• Synthetic data
• Magnetic resonance imagery.

3
Introduction

• Medical image processing applications all
  benefit from segmentation of anatomical
  structures from medical images.
• Surgical planning.
• Navigation.
• Simulation.
• Diagnosis.
• And therapy evaluation.

4
Introduction (Cont.)

• Segmentation is typically performed
• Automated techniques.
• Semi-automated techniques.
• Intensity thresholding
• the distribution of intensity values
  corresponding to one structure may vary
  throughout the structure.
• overlap those of another structure.
• Gradient-based boundary detection methods
• The strength of an edge at the boundary of the
  structure may vary or be weak relative to the
  texture inside the object.
• So, Segmentation is challenging and requires more
  sophisticated algorithms and significant human
  input.

5
Introduction (Cont.)

• Boundary finding segmentation methods (Snakes)
• local algorithms
• may be sensitive to the starting position.
• may leak through the boundary of the object.
• Level set segmentation
• involves solving the energy based active contours
  minimization problem.
• computation of geodesics
• minimal distance curves

6
Introduction (Cont.)

• Segmentation is performed by
• evolving a curve to maximally separate
  predetermined statistics inside and outside the
  curve.
• Paragios and Deriche-Geodesic active regions for
  supervided texture segmentation, IEEE 1999- in,
  build prior texture models and perform
  segmentation by combining boundary and region
  information.
• include both global and local information.
• adding robustness to noise and weak boundaries
• providing the advantages of numerical stability
  and topological flexibility

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Introduction (Cont.)
surface
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Introduction (Cont.)
9
Introduction (Cont.)

• Two common segmentation techniques
• Pixel classification based methods
• Boundary localization techniques
• image term
• regularization term
• Markov prior expressing
• dependent on the shape properties of the object
  and noise in the image.

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Level Set Based Segmentation with Intensity and
Curvature Priors

• Method of segmentation incorporates prior
  information about the intensity and curvature of
  the structure based on previously segmented
  training data.
• Model the distribution of intensity over the
  entire image as a function of the signed distance
  from the boundary of the structure.
• A distribution of the curvature of the structure
  is also modeled from the training data to
  determine the degree of regularization of the
  underlying level sets.

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The Prior Image-Surface Model
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Definition of parameters
The Prior Image Surface Model

• X(t) position vector
• I(x) image
• C closed surface
• U(x) signed distance function to the nearest
  point on the curve C (the height of the
  surface U)
• The boundary of the object is found by extracting
  the zero level set of U

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Definition of Statistical Dependency
The Prior Image Surface Model

• Define a statistical dependency network over U
  and I
• Maximize
• I entire image
• the rest of surface

Difficult to model, given all the dependencies !!
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Simplification to Local Dependency
The Prior Image Surface Model

• Assumption
• Ux depends only on the intensity value (Ix) of
  the image at that point, and the neighboring
  heights of the surface
• N(x) the neighborhood of x

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Derivation for Estimation
The Prior Image Surface Model
Estimate these functions based on prior training
data!!
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Intensity Model
The Prior Image Surface Model

• Define a joint distribution model P(i, d) (i
  intensity value, d signed distance), based on a
  set of training images and training segmentations
• Let I1, I2, , In be a set of images of the
  same modality containing the same anatomical
  structure
• Let C1, C2, , Cn be the set of corresponding
  segmented boundary curves of the anatomical
  structure
• Let Uj be the signed distance map to the closed
  curve Cj
• Let the training set ? ltI1, U1gt, , ltIn,
  Ungt

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Intensity Model (contd)
The Prior Image Surface Model
P(i, d) the mean PDF of many 2D Gaussian
centered at the positions ltIj(x), Uj(x)gt
for every pixel in every training image
? the set of all positions in the image
si , sd the variations
Z (the normalization
factor)
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Intensity Model (contd)
The Prior Image Surface Model
Higher Level Knowledge
Gradient
Thresholding
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Intensity Model (contd)
The Prior Image Surface Model
A example to require prior knowledge of the
intensity distribution
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Curvature Model
The Prior Image Surface Model

• Goal
• Modeling P(UN(x) Ux) (a local regularizer)
• In order to uniformly penalize regions of high
  curvature
• One Method
• Creating a 5D joint PDF over Ux and its 4
  neighbors

1. This approach requires many training samples !!
2. The neighborhood depends on the embedded
coordinate system, and would give different
results based on the pose of the object !!
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Curvature Model (contd)
The Prior Image Surface Model

• Use 4 neighbors in the direction of the local
  normal (n) and tangent (t) to the embedded curve

Assumption the neighbors in the normal direction
and those in the tangent direction are
conditionally independent given Ux
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Curvature Model (contd)
The Prior Image Surface Model

• Define
• U a finite network of nodes
• Ux a particular node
• u(x) the analogous particular surface over x
• The unit normal and unit tangent to the
  underlying level set of u

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Curvature Model (contd)
The Prior Image Surface Model

• To compute the directional derivatives, evaluate
  the second-order Taylor expansion of u at x in
  the arbitrary direction w

Computing a Directional Derivative Dnf(x, y)
?f(x, y) u
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Curvature Model (contd)
The Prior Image Surface Model

• Evaluating the first partial in both the
  directions of n and t
• If u is a signed distance map

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Curvature Model (contd)
The Prior Image Surface Model

• Compute expressions for the second derivatives
• If u is a signed distance function

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Curvature Model (contd)
The Prior Image Surface Model

• The curvature of the underlying level set, ?
• In summary, under the constraint that the surface
  u is a continuous signed distance map

We can use this information about the ideal
distance map and underlying curvatures to
regularize U at x !!
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Curvature Model (contd)
The Prior Image Surface Model

• Consider the formulas for centered finite
  difference of an arbitrary 1D function

Changing f(x) does directly affect the second
derivative, So model the relationship between Ux
and its neighbors to adhere to the second
derivative constraints
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Curvature Model (contd)
The Prior Image Surface Model

• The likelihood of the neighbors in the normal
  direction

Z1 a normalization factor
s1 determines the strength of the
constant-normal-direction constraint
1. This has the effect of keeping the gradient
magnitude of the surface constant,
preventing the surface from evolving arbitrarily
2. This direction of regularization reduces the
need to reinitialize the surface
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Curvature Model (contd)
The Prior Image Surface Model

• The likelihood of the neighbors in the tangential
  direction

?j(x) curvatures observed in the training data
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Curvature Model (contd)
The Prior Image Surface Model
The curvature profiles of training sets of
various objects
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Curvature Model (contd)
The Prior Image Surface Model

• In summary,
• The regularization of the surface is broken down
  into a regularization in the local normal and
  tangent directions
• The second derivative in the normal direction is
  modeled as a zero mean, low variance Gaussian to
  keep the surface linear in that direction
• A distribution over the second derivative in the
  tangent direction is derived from training data
  and used as an object-specific curvature
  regularization term

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Surface Estimation
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Surface Estimation
Surface Estimation

• Given the prior model and the image, the height
  of the surface at location x is related to the
  image intensity at x and the local neighborhood
  of the surface

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Re-estimation
Surface Estimation

• Re-estimate each surface point independently,
  maximizing its log probability
• Adjust Ux in the direction of increasing
  probability towards the local maximum by
  differentiating the log probability

This update is repeated until there is little
change in the surface !!
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Result
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Result (Cont.)
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• Gracious