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. 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

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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

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

Lecture 5

Example: Estimating Parameters
• 2d point locations:
• Similarity transformation:
• Euclidean distance:

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 (b)

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

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)
• This becomes
• Manipulating:

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
• Map feature into coordinate system of If
• Find closest point

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.

Comparison

Intensity-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