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

Prof. Chuck Stewart, RPI

Dr. Luis Ibanez, Kitware

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

Lecture 5

what do we have
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
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
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
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
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
Matrix Version (continued)
  • This becomes
  • Manipulating:

Lecture 5

matrix version continued1
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
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
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
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
Finding Correspondences
  • Map feature into coordinate system of If
  • Find closest point

Lecture 5

finding correspondences continued
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
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
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

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

Lecture 5

looking ahead to lecture 6
Looking Ahead to Lecture 6
  • Introduction to ITK and the ITK registration framework.

Lecture 5