Non-Rigid Registration


Non-Rigid Registration

Why Non-Rigid Registration
In many applications a rigid transformation is sufficient. (Brain) Other applications: Intra-subject: tissue deformation Inter-subject: anatomical variability across
individuals

Fast-Moving area: Non-rigid

Registration Framework
In terms of L.Brown.(1992)
– – – – Feature Space Transformation Similarity Measure Search Strategy (Optimization)

Rigid vs. Non-rigid in the framework

Feature Space
Geometric landmarks: Points Edges Contours Surfaces, etc. Intensities: 23 35 Raw pixel values 56 24

Transformation

Transformation
Rigid transformation: 3DOF (2D) 6 DOF (3D) Affine transformation: 12 DOF
? ? ? T ( x, y, z) = ? ? ? ? x ' ? ? a 00 ? ? y ' ? ? a 10 = z ' ? ? a 20 ? ? 1 ? ? 0 ? ? a 01 a 11 a 21 0 a 02 a 12 a 22 0

a 03 ? ? x ? ?? ? a 13 ? ? y ? a 23 ? ? z ? ?? ? 1 ?? 1 ? ?? ?

Transformation
Additional DOF. Second order polynomial-30 DOF
? ? ? T ( x, y, z) = ? ? ? ? x ' ? ? a 00 ? ? y ' ? ? a 10 = z ' ? ? a 20 ? ? ? ? 1 ? ? 0 ... ... ... ... a 08 a 18 a 28 0 a 09 ? ? ?? a 19 ? ? a 29 ? ? ?? ? 1 ?? ? x2 ? ? 2 y ? ? M ? 1 ? ?

Higher order: third-60, fourth-105,fifth-168 Model only global shape changes

Transformation
For each pixel (voxel), one 2d(3d) vector to describe local deformation. Parameters of non-rigid >> that of rigid

Similarity Measure
Point based ---The distance between features, such as points,curves,or surfaces of corresponding anatomical structure. --- Feature extraction. Voxel based ---Absolute Difference, Sum of squared differences, Cross correlation, or Mutual information

Search Strategy
Registration can be formulated as an optimization problem whose goal is to minimize an associated energy or cost function. General form of cost function: C = -Csimilarity+Cdeformation

Search Strategy
Powell’s direction set method Downhill simplex method Dynamic programming Relaxation matching Combined with Multi-resolution techniques

Registration Scheme

Non-rigid Registration
Feature-based
– Control Points: TPS – Curve/Edge/Contour – Surface

Intensity-based
– Elastic model – Viscous fluid model – Others

Thin-plate splines (TPS)
Come from Physics: TPS has the property of minimizing the bending energy.

TPS (cont.)
Splines based on radial basis functions
t(x, y, z) = a1 + a2 x + a3 y + a4 z + ∑bjθ (φ j ? (x, y, z) )
j =1 n

Surface interpolation of scattered data
T = ( t1 , t 2 , t 3 ) T

Description of the Approach
1. Select the control points in the

images. 2. Calculate the coefficients for the TPS. 3. Apply the TPS transformation on the whole image.

Synthetic Images

T1

T2

TPS-Results(1)

TPS-Results(2)

Rigid and non-rigid registration
Rigid Registration as pre-processing (global alignment) Non-rigid registration for local alignment

Next time
Affine-mapping technique


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