Image Segmentation and Registration

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Image Segmentation and Registration

• Rachel Jiang
• Department of Computer Science
• Ryerson University
• 2006

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Rigid registration

• Intensity-based methods
• Have been very successful
• Monomodal
• Multimodal
• Assuming a global relationship between the
  images to register
• Deriving a suitable similarity measure
• Correlation coefficient
• Correlation ratio
• Mutual information
• Block matching

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Methods for fusing images

• There are three general methods for fusing images
  from different (or the same) image modalities
• landmark matching
• include external fiducial landmarks or anatomic
  landmarks.
• surface matching
• uses an algorithm that matches different images
  of the same patient surface.
• intensity matching.

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intensity matching

• uses mutual intensity information to co-register
  different images.
• The matched intensities may come from the same
  scanner
• two different MRI scans acquired on different
  days
• from different modalities such as MRI and PET.

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Rigid Body Model

• Most constraint model for medical imaging
• asserts that the distance and internal angles
  within images can not be changed during the
  registration

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Non rigid model

• can detect and correct discrepancies of small
  spatial extent, by deforming one of the images
  (source) to match the other (reference).
• Spatial deformation model can be based on
  different physical properties like elasticity or
  viscosity, or their generalizations and
  simplifications.
• Deformation is driven by external forces, which
  tend to minimize image differences, measured by
  image similarity measures.

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Mutual information

• The mutual information of two images is the
  amount of information that one image contains
  about the other or vice versa.
• Transforming one image with respect to the other
  such that their mutual information is maximized.
  The images are assumed to be registered.

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Mutual information

• A function of transformation between the images
• an algorithm that searches maximum value for a
  function that gives the alignment information
  between images
• different transformation estimates are evaluated
• (those transformation estimates will result in
  varying degrees of overlap between the two
  images)
• (MI as a registration criteria is not invariant
  to the overlap between images)

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MI
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MI in 2D/3D space

• In 3D space
• We attempt to find the registration by maximizing
  the information that one volumetric image
  provides about the other.
• In 2D space
• Two curve that are to be matched as the reference
  curve and the current active evolving curve.
• We seek an estimate of the transformation that
  registers the reference curve and the active
  curve by maximizing their mutual information

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Degree of freedom

• Six degree of freedom charactering the rigid
  movements
• 3 describe the Rotation
• 3 describe Translation

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Similarity measures

• Common registration methods can be grouped as
• Feature based techniques
• Rely on the presence and identification of
  natural landmarks or fiducial marks in the input
  dataset to determine the best alignment
• Intensity-based measures
• Operate on the pixel/voxel intensities directly
• Varies statistics are calculated by using the raw
  intensity values

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Taking geometric constraints

• Special land marks on human body
• Manually embed land marks
• Spacial correlations

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Correlation ratio

• Given two images I and J, the basic principle of
  the CR method is to search spatial transformation
  T and an intensity mapping f such that, by
  displacing J and remapping its intensities, the
  resulting image f(JoT) be as similar as possible
  to I.

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Example of Image Segmentation

• Bone fracture detection

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Image Segmentation Techniques

• threshold techniques
• make decisions based on local pixel information
• edge-based methods
• Weakness broken contour lines causes failure
• region-based techniques
• partitioning the image into connected regions by
  grouping neighbouring pixels of similar intensity
  levels.
• Adjacent regions are then merged under certain
  criterion. Criteria create fragmentation or
  overlook blurred boundaries and overmerge.
• Active contour models

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Deformable models

• Snakes/Balloons/Deformable Templates
• provide a curve as a compromise between
  regularity of the curve and high gradient values
  among the curve points.
• (Kass et al., 1988 Cohen, 1991 Terzopoulos,
  1992)
• Level set methods
• Level Set Methods are numerical techniques which
  can follow the evolution of interfaces. These
  interfaces can develop sharp corners, break
  apart, and merge together.
• (Osher and Sethian, 1988 Sethian, 2001)
• Geodesic Active Contour
• take the advantages of both Snake and Level set
  methods
• (Caselles et al., 1995 Malladi et al, 1995
  Sapiro, 2001)

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The Snake formula
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Snake image force
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Snake example
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Level Set Method

• Level Set Methods
• provide formulation of propagating interfaces, a
  mathematical formulation and numerical algorithm
  for tracking the motion of curve and surfaces
• (Osher and Sethian, 1988 Sethian, 2001)
• For segmenting several objects simultaneously or
  an objects with holes, it is possible to model
  the contour as a level set of a surface, allow it
  to change its topology in a nature way
• (Cohen, 1997).

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Level Set
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Formula of Level Set method

• Curve evolving formula

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LSM example
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LSM More Example
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Geodesic Active Contour (GAC)

• presents some nice properties
• the initialization step does not impose any
  significant constraint
• can deal successfully with topological changes,
• finding the global minimum of energy minimizing
  curve can be solved by mapping the boundary
  detection problem into a single minimum problem.
• The new model mathematically inherit
• the way handling the topological changes from the
  Level Set
• the minimizing deformation energy function with
  internal and external energies along its
  boundary from the traditional Snake.
• This model inherit the advantages of LS and
  Snakes by transform mathematical formulation of
  Snake Lagrenge formula with PDEs
• The theory behind the GAC is the use of partial
  differential equations and curvature-driven
  flows.
• (Caselles et al., 1995 Malladi et al, 1995
  Sapiro, 2001)

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Segmentation result using ACM
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Surface reconstruction
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Mathematical Morphology

• provides the foundation for measuring
  topological shape, size, location.
• The theory behind mathematical morphology is
  defining computing operations by primitive shapes
• Georges Matheron, Jean Serra and their colleagues
  of Centre de Morphologie Mathematique
• G.Matheron 1975, Serra, 1982, Vicent, 1990
• offer several robust theories and algorithms
• to implement on digital images to extract complex
  features
• uses Set Theory as the foundation for its
  functions. The simplest functions to implement
  are Dilation and Erosion.

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Erosion and Dilation (1D)
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Erosion and Dilation (2D)
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Erosion Dilation example
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Opening Closing
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Shape Operators

• Shapes are usually combined by means of

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Dilation
B
A
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Dilation
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Extensitivity
A
B
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Erosion
A
B
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Erosion
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Erosion
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Opening and Closing

• Opening and closing are iteratively applied
  dilation and erosion
• Opening
• Closing

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Opening and Closing
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Opening and Closing

• They are idempotent. Their reapplication has not
  further effects to the previously transformed
  result

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Watershed
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Capturing the shape prior

• the curve C and the transformation S, R, T is
  calculated such that the curve Cnew SRC T and
  C are perfectly aligned.
• The minimization problem now can be solved by
  finding steady state solutions to the following
  system

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Minimization processing system
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Distance measure

• d(x,y) d(C ,(x,y)) is the distance of the
  point (x,y) from C
• The function d is evaluated at
• SRC(p) T

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Minimizing Energy function