Registration and Models in Depth

February 2005

The Problem

• Given n images (or volumes)
• Obtain the following
  • Warps that define correspondence
  • Model that faithfully describes the variation
• Several methods exist to solve the former task
• Weak solutions to the latter (e.g. SDM's)

Yet to Follow

• Registration:
  • Introduction of methods for obtaining correspondence
  • Comparison of methods
  • Proposal of alternatives

Yet to Follow - ctd.

• Model building:
  • Explanation of the process
  • Relation to registration
  • Framework for model construction

To Follow: Current Algorithms

• Introduction
• Goals
• Flaws, gaps and problems

Registration: Introduction

• What is needed:
  • Set of warps for each image (or volume)
  • The warps lead to overlap
  • Warps can be aggregated to form a single composite grid
  • Can use a list of warps to be applied in a sequence
• All warps are expected to have a pattern
• This implies a small and controlled distribution

Registration: Methods

• Warp; then measure similarity
• Warping involves
  • Setting up a grid, e.g. equally-spaced grid
  • Change (transform) grid
  • Resample data to fit deformed grid
• Similarity involves
  • Computing a number
  • The number decreases/increases as function of similarity

Registration: Problems

• Cannot guarantee correct structures are aligned
• Infinite number of ways to align data
• Noisy/corrupted data - need robustness

Registration: Comparison of Methods

• Approach in selection
  • Picking image pairs
  • Fixed reference
  • Changing reference
  • Mean/Median
  • Handling entire set
  • Handling subsets
• Solution depends on selection

Registration: Comparison of Methods

• Transformation types
  • Rigid
  • Affine
  • Non-rigid

Registration: Comparison of Methods - ctd.

• Transformation functions/mechanisms
  • B-splines
  • Clamped-plate splines
  • Cosine-based or bi-linear grid-based
• Any function will do, but needs to be
  • (Easily) Invertible
  • Quick to compute
  • Diffeomorphism

Registration: Alternative Approaches

• Measure similarity more analytically, e.g.
  • MDL-based
  • Model-based
• Compute similarity in different frame of reference
• Interlace data

Registration: Discussion

• Registration can be solved in many ways
• The ground truth is based on the anatomy
• Cannot place infinite number of markers in subject
• Hence registration can only ever approximate
• Problems when structures appear/disappear

Models: Explanation

• A means of
  • Capturing variation in data
  • Interpreting data
  • Synthesising data

Models: Relation to Registration

• Models require correspondence to be known
• Both entities involve transformation/deformation
• Models deform to fit/synthesise data
• Registration applies deformations to data

Models: Framework

• Take data set
• Identify correspondences
• Vectorise
• Apply PCA
• Use the model
  • For fitting
  • For interpretation
  • More...

Algorithms: Introduction

• Registration treated by different (yet related) methods/strands:
  • Carole - Minimum description length
  • Myself - Appearance model (similar idea to MDL)
  • Tim - All conventional methods and the ones above
  • Vlad - Works with Tim
  • Kola - Towards building of compartmentalised brain models

Algorithms: Goals

• Unifying registration and model construction
• Showing the powers of group-wise registration
• Devising a principled similarity measure
• Discovering faster registration methodologies, e.g.
  • Multi-resolution
  • Cunning point selection for triangulation
  • Localising warps

Algorithms: Piece-wise Affine

• Piece-wise affine representation of deformation fields
• In 2-D: triangulation of RoI
• In 3-D: e.g. tetrahedra of volume
• Position of nodes control deformation
• Heavily-based on existing algorithms

Algorithms: Piece-wise Affine - ctd.

• Advantages
  • Easy to invert
  • Efficiency in moving from frame to frame
  • Freedom in placement of nodes in image/volume
• Disadvantages
  • Derivatives of mapping are non-continuous
  • Triangulation might fold

Algorithms: Flaws and Problems

• Need annotated data
• Larger set size needed for group-wise optimisation
• Reduce load on memory at any given time
• Values for registration parameters are unknown

Notes on Group-wise versus Pair-wise

• Must care to warp to a sensible 'target'
• Reference image is an arbitrary choice
• Group-wise method should 'treat' all images equally
• Group-wise aspires to become a data-driven method
• Registration is dependent on the whole data

Notes on Frame of Reference

• Group-wise versus pair-wise
  • Similarity, for example, can be calculated in different frames of reference
    • Can warp the reference to fit target
    • Or warp target to fit reference
  • Also possible to make multiple warps
  • Transform one target to fit another target
  • Requires 'going through' reference
  • Warp is cheaper to compute than its inverse
  • Hence 'pull-back' is expensive

Summary

• Registration:
  • Warping methods
  • Similarity measures
  • Group-wise/pair-wise
• Models:
  • Require correspondence
  • Related to registration

Some Conclusions and Ways Forward

• Use deformation fields for model construction
• Fast transformations are essential
• Test and validate results