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CSci 6971: Image Registration Lecture 5: Feature-Base Regisration January 27, 2004

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## CSci 6971: Image Registration Lecture 5: Feature-Base Regisration January 27, 2004

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CSci 6971: Image Registration Lecture 5: Feature-Base RegisrationJanuary 27, 2004

Prof. Chuck Stewart, RPI

Dr. Luis Ibanez, Kitware

Overview

- What is feature-based (point-based) registration?
- Feature points
- The correspondence problem
- Solving for the transformation estimate
- Putting it all together: ICP
- Discussion and conclusion

Lecture 5

What is Feature-Based Registration?

- Images are described as discrete sets of point locations associated with a geometric measurement
- Locations may have additional properties such as intensities and orientations
- Registration problem involves two parts:
- Finding correspondences between features
- Estimating the transformation parameters based on these correspondences

Lecture 5

Feature Examples: Range Data

- Range image points:
- (x,y,z) values
- Triangulated mesh
- Surface normals are sometimes computed
- Notice:
- Some information (locations) is determined directly by the sensor (“raw data”)
- Some information is inferred from the data

Lecture 5

Feature Examples: Vascular Landmarks

- Branching points pulmonary images:
- Lung vessels
- Airway branches
- Retinal image branches and cross-over points
- Typically augmented (at least) with orientations of vessels meeting to form landmarks

Lecture 5

Points Along Centers of Vessels and Airways

- Airways and vessels modeled as tubular structures
- Sample points spaced along center of tubes
- Note that the entire tube is rarely used as a unit
- Augmented descriptions:
- Orientation
- Radius

Lecture 5

“Interest” Points

- Locations of strong intensity variation in all directions
- Augmented with summary descriptions (moments) of surrounding intensity structures
- Recent work in making these invariant to viewpoint and illumination.
- We’ll discuss interest points during Lectures 16 and 17

Brown and Lowe, Int. Conf. On Computer Vision, 2003

Lecture 5

Feature Points: Discussion

- Many different possible features
- Problem is reliably extracting features in all images
- This is why more sophisticated features are not used
- Feature extraction methods do not use all intensity values
- Use of features dominates range-image registration techniques where “features” are provided by the sensor

Lecture 5

Preamble to Feature-Based Registration: Notation

- Set of moving image features
- Set of fixed image features
- Each feature must include a point location in the coordinate system of its image. It may include more
- Set of correspondences

Lecture 5

Mathematical Formulation

- Error objective function depends on unknown transformation parameters and unknown feature correspondences
- Each may depend on the other!
- Transformation may include mapping of more than just locations
- Distance function, D, could be as simple as the Euclidean distance between location vectors.
- We are using the forward transformation model.

Lecture 5

Correspondence Problem

- Determine correspondences before estimating transformation parameters
- Based on rich description of features
- Error prone
- Determine correspondences at the same time as estimation of parameters
- “Chicken-and-egg” problem
- For the next few minutes we will assume a set of correspondences is given and proceed to the estimation of parameters
- Then we will return to the correspondence problem

Lecture 5

Example: Estimating Parameters

- 2d point locations:
- Similarity transformation:
- Euclidean distance:

Lecture 5

Putting This Together

Lecture 5

What Do We Have?

- Least-squares objective function
- Quadratic function of each parameter
- We can
- Take the derivative with respect to each parameter
- Set the resulting gradient to 0 (vector)
- Solve for the parameters through matrix inversion
- We’ll do this in two forms: component and matrix/vector

Lecture 5

Component Derivative (a)

Lecture 5

Component Derivative (b)

At this point, we’ve dropped the leading factor of 2. It will be eliminated when this is set to 0.

Lecture 5

Component Derivatives tx and ty

Lecture 5

Gathering

- Setting each of these equal to 0 we obtain a set of 4 linear equations in 4 unknowns. Gathering into a matrix we have:

Lecture 5

Solving

- This is a simple equation of the form
- Provided the 4x4 matrix X is full-rank (evaluate SVD) we easily solve as

Lecture 5

Matrix Version

- We can do this in a less painful way by rewriting the following intermediate expression in terms of vectors and matrices:

Lecture 5

Matrix Version (continued)

- Taking the derivative of this wrt the transformation parameters (we didn’t cover vector derivatives, but this is fairly straightforward):
- Setting this equal to 0 and solving yields:

Lecture 5

Comparing the Two Versions

- Final equations are identical (if you expand the symbols)
- Matrix version is easier (once you have practice) and less error prone
- Sometimes efficiency requires hand-calculation and coding of individual terms

Lecture 5

Resetting the Stage

- What we have done:
- Features
- Error function of transformation parameters and correspondences
- Least-squares estimate of transformation parameters for fixed set of correspondences
- Next:
- ICP: joint estimation of correspondences and parameters

Lecture 5

Iterative Closest Points (ICP) Algorithm

- Given an initial transformation estimate 0
- t = 0
- Iterate until convergence:
- Establish correspondences:
- For fixed transformation parameter estimate, t, apply the transformation to each moving image feature and find the closest fixed image feature
- Estimate the new transformation parameters,
- For the resulting correspondences, estimate t+1

ICP algorithm was developed almost simultaneous by at least 5 research groups in the early 1990’s.

Lecture 5

Finding Correspondences (continued)

- Enforce unique correspondences
- Avoid trivial minima of objective function due to having no correspondences
- Spatial data structures needed to make search for correspondences efficient
- K-d trees
- Digital distance maps
- More during lectures 11-15…

Lecture 5

Initialization and Convergence

- Initial estimate of transformation is again crucial because this is a minimization technique
- Determining correspondences and estimating the transformation parameters are two separate processes
- With Euclidean distance metrics you can show they are working toward the same minimum
- In general this is not true
- Convergence in practice is sometimes problematic and the correspondences oscillate between points.

Lecture 5

2d Retinal Example

- White = vessel centerline points from one image
- Black = vessel centerline points from second image
- Yellow line segments drawn between corresponding points
- Because of the complexity of the structure, initialization must be fairly accurate

Lecture 5

For a given transformation estimate, we can only find a new, better estimate, not the best estimate, based on the gradient step.

We then need to update the constraints and re-estimate

For given set of correspondences, we can directly (least-squares) estimate the best transformation

BUT, the transformation depends on the correspondences, so we generally need to re-establish the correspondences.

ComparisonIntensity-Based

Feature-Based

Lecture 5

Summary

- Feature-based registration
- Feature types and properties
- Correspondences
- Least-squares estimate of parameters based on correspondences
- ICP
- Comparison

Lecture 5

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