1、,CSci 6971: Image Registration Lecture 5: Feature-Base Regisration January 27, 2004,Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware,Image Registration,Lecture 5,2,Overview,What is feature-based (point-based) registration? Feature points The correspondence problem Solving for the transformation est
2、imate Putting it all together: ICP Discussion and conclusion,Image Registration,Lecture 5,3,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 orientatio
3、ns Registration problem involves two parts: Finding correspondences between features Estimating the transformation parameters based on these correspondences,Image Registration,Lecture 5,4,Feature Examples: Range Data,Range image points:(x,y,z) values Triangulated mesh Surface normals are sometimes c
4、omputed Notice: Some information (locations) is determined directly by the sensor (“raw data”) Some information is inferred from the data,Image Registration,Lecture 5,5,Feature Examples: Vascular Landmarks,Branching points pulmonary images: Lung vessels Airway branches Retinal image branches and cro
5、ss-over points Typically augmented (at least) with orientations of vessels meeting to form landmarks,Image Registration,Lecture 5,6,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 ra
6、rely used as a unit Augmented descriptions: Orientation Radius,Image Registration,Lecture 5,7,“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 viewpoi
7、nt and illumination. Well discuss interest points during Lectures 16 and 17,Brown and Lowe, Int. Conf. On Computer Vision, 2003,Image Registration,Lecture 5,8,Feature Points: Discussion,Many different possible features Problem is reliably extracting features in all images This is why more sophistica
8、ted 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,Image Registration,Lecture 5,9,Preamble to Feature-Based Registration: Notation,Set of moving image featuresS
9、et of fixed image featuresEach feature must include a point location in the coordinate system of its image. It may include more Set of correspondences,Image Registration,Lecture 5,10,Error objective function depends on unknown transformation parameters and unknown feature correspondences Each may de
10、pend 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.,Mathematical Formulation,Image Registration,Lecture 5,11,Correspondence Proble
11、m,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 pro
12、ceed to the estimation of parameters Then we will return to the correspondence problem,Image Registration,Lecture 5,12,Example: Estimating Parameters,2d point locations:Similarity transformation:Euclidean distance:,Image Registration,Lecture 5,13,Putting This Together,Image Registration,Lecture 5,14
13、,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 Well do this in two forms: component and matrix/vector,Image Re
14、gistration,Lecture 5,15,Component Derivative (a),Image Registration,Lecture 5,16,Component Derivative (b),At this point, weve dropped the leading factor of 2. It will be eliminated when this is set to 0.,Image Registration,Lecture 5,17,Component Derivatives tx and ty,Image Registration,Lecture 5,18,
15、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:,Image Registration,Lecture 5,19,Solving,This is a simple equation of the formProvided the 4x4 matrix X is full-rank (evaluate SVD) we easily solve as,Image Registration,Le
16、cture 5,20,Matrix Version,We can do this in a less painful way by rewriting the following intermediate expression in terms of vectors and matrices:,Image Registration,Lecture 5,21,Matrix Version (continued),This becomesManipulating:,Image Registration,Lecture 5,22,Matrix Version (continued),Taking t
17、he derivative of this wrt the transformation parameters (we didnt cover vector derivatives, but this is fairly straightforward):Setting this equal to 0 and solving yields:,Image Registration,Lecture 5,23,Comparing the Two Versions,Final equations are identical (if you expand the symbols) Matrix vers
18、ion is easier (once you have practice) and less error prone Sometimes efficiency requires hand-calculation and coding of individual terms,Image Registration,Lecture 5,24,Resetting the Stage,What we have done: Features Error function of transformation parameters and correspondences Least-squares esti
19、mate of transformation parameters for fixed set of correspondences Next: ICP: joint estimation of correspondences and parameters,Image Registration,Lecture 5,25,Iterative Closest Points (ICP) Algorithm,Given an initial transformation estimate 0 t = 0 Iterate until convergence: Establish corresponden
20、ces: 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
21、5 research groups in the early 1990s.,Image Registration,Lecture 5,26,Finding Correspondences,Map feature into coordinate system of IfFind closest point,Image Registration,Lecture 5,27,Finding Correspondences (continued),Enforce unique correspondences Avoid trivial minima of objective function due t
22、o having no correspondences Spatial data structures needed to make search for correspondences efficient K-d trees Digital distance maps More during lectures 11-15,Image Registration,Lecture 5,28,Initialization and Convergence,Initial estimate of transformation is again crucial because this is a mini
23、mization 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 correspond
24、ences oscillate between points.,Image Registration,Lecture 5,29,2d 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 mus
25、t be fairly accurate,Image Registration,Lecture 5,30,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,Intensity-Based,Feature-Based,For given set of corre
26、spondences, 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.,Image Registration,Lecture 5,31,Summary,Feature-based registration Feature types and properties Correspondences Least-squares estimate of parameters based on correspondences ICP Comparison,Image Registration,Lecture 5,32,Looking Ahead to Lecture 6,Introduction to ITK and the ITK registration framework.,
