1. University of Ioannina
Registration
T-11 Computer Vision
University of Ioannina
Christophoros Nikou
Images and slides from:
James Hayes, Brown University, Computer Vision course
D. Forsyth and J. Ponce. Computer Vision: A Modern Approach, Prentice Hall, 2003.
Computer Vision course by Svetlana Lazebnik, University of North Carolina at Chapel Hill.
Computer Vision course by Kristen Grauman, University of Texas at Austin.

2. How do we fit the best alignment?

3. Alignment-Registration
Estimate parameters of model that maps one set of points to another
Typically want to solve for a global transformation that accounts for *most* true correspondences
Difficulties
Noise (typically 1-3 pixels)
Outliers (often 50%)
Many-to-one matches or multiple objects

4. Parametric (global) warping
p = (x,y)
p’ = (x’,y’)
Transformation T is a coordinate-changing machine: p’ = T(p)
What does it mean that T is global?
It is the same for any point p
It can be described by just a few numbers (parameters)
For linear transformations, we can represent T as a matrix p’ = Tp

5. Common transformations
original
scaling
translation
rotation
affine
perspective
Slide credit (next few slides): A. Efros and/or S. Seitz

6. Scaling
Scaling a coordinate means multiplying each of its components by a scalar
Uniform scaling means this scalar is the same for all components:
 2

7. Scaling
Non-uniform scaling: different scalars per component:
x  2, y 0.5

8. Scaling
Scaling operation:
Or, in matrix form:
scaling matrix S

9. θ 2-D Rotation (x’, y’) (x, y) x’ = x cosθ - y sinθ
y’ = x sinθ + y cosθ

10. 2-D Rotation R This is easy to capture in matrix form:
Even though sinθ and cosθ are nonlinear functions of θ,
x’ is a linear combination of x and y
y’ is a linear combination of x and y
What is the inverse transformation?
Rotation by θ
For rotation matrices R

11. Basic 2D transformations
Scaling
Shearing
Rotation
Translation
Affine is any combination of translation, scale, rotation, shear
Affine

12. Affine Transformations
Affine transformations are combinations of
Linear transformations, and
Translations
Properties of affine transformations:
Lines map to lines
Parallel lines remain parallel
Ratios are preserved
Closed under composition or
Ratios of distances along straight lines are preserved

13. Projective Transformations
Projective transformations are combos of
Affine transformations, and
Projective warps
Properties of projective transformations:
Lines map to lines
Parallel lines do not necessarily remain parallel
Ratios are not preserved
Closed under composition
Models change of basis
Projective matrix is defined up to a scale (8 DOF)

14. 2D image transformations (reference table)
Szeliski 2.1

15. Example: solving for translationB1 B2 B3
Given matched points in {A} and {B}, estimate the translation of the object

16. Example: solving for translation
(tx, ty) B1 B2 B3
Least squares solution
Write down objective function
Derived solution
Compute derivative
Compute solution
Computational solution
a) Write in form Ax=b
b) Solve using pseudo-inverse

17. Example: solving for translationB4 B1 B2 B3
(tx, ty) A4 B5
Problem: outliers
RANSAC solution
Sample a set of matching points (1 pair)
Solve for transformation parameters
Score parameters with number of inliers
Repeat steps 1-3 N times

18. Example: solving for translationB4 B5 B6 A1 A2 A3 B1 B2 B3
(tx, ty) A4 A5 A6
Problem: outliers, multiple objects, and/or many-to-one matches
Hough transform solution
Initialize a grid of parameter values
Each matched pair casts a vote for consistent values
Find the parameters with the most votes
Solve using least squares with inliers

19. Example: solving for translation
(tx, ty)
Problem: no initial guesses for correspondence

20. What if you want to align but have no prior matched pairs?
Hough transform and RANSAC not applicable
Important applications
Medical imaging: match brain scans or contours
Robotics: match point clouds

21. Iterative Closest Points (ICP) Algorithm
Goal: estimate the transformation between two dense sets of points
Initialize transformation (e.g., compute difference in means and scale)
Assign each point in {Set 1} to its nearest neighbor in {Set 2}
Estimate transformation parameters
e.g., least squares or robust least squares
Transform the points in {Set 1} using estimated parameters
Repeat steps 2-4 until change is very small

22. Example: aligning boundariesq p

23. Example: solving for translation
(tx, ty)
Problem: no initial guesses for correspondence
ICP solution
Find nearest neighbors for each point
Compute transform using matches
Move points using transform
Repeat steps 1-3 until convergence

24. Registration of multimodal images
What if the images are not acquired by the same acquisition system (e.g. medical images MRI, CT, PET, SPECT,…)
Hypothesis: each tissue has a distinct gray value in each image modality (not necessarily the same)
When the images are correctly registered, knowing the value of the tissue in one image reveals the value of the tissue in the second image

25. Mutual information
The joint probability of intensities of the two images should be high
This may be expressed by the mutual information of the two images considered as random variables:
Recall that the entropy is given by:

26. Mutual information
You can view this as the extent to which knowing the value of Y reduces the uncertainty about the value of X
In other words, what does Y convey about X
Registration may be achieved by maximizing the mutual information

27. Algorithm Summary Least Squares Fit Robust Least Squares
closed form solution
robust to noise
not robust to outliers
Robust Least Squares
improves robustness to noise
requires iterative optimization
Hough transform
robust to noise and outliers
can fit multiple models
only works for a few parameters (1-4 typically)
RANSAC
works with a moderate number of parameters (e.g, 1-8)
Iterative Closest Point (ICP)
For local alignment only: does not require initial correspondences