# 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.

• Easy to invert
• Efficiency in moving from frame to frame
• Freedom in placement of nodes in image/volume
• 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