1 / 32

Image Registration Lecture 16: View-Based Registration May 3, 2005

Image Registration Lecture 16: View-Based Registration May 3, 2005. Prof. Charlene Tsai. Overview. Retinal image registration The Dual-Bootstrap ICP algorithm Covariance matrix Covariance propagation Model selection View-based registration Software design.

Download Presentation

Image Registration Lecture 16: View-Based Registration May 3, 2005

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Image Registration Lecture 16: View-Based RegistrationMay 3, 2005 Prof. Charlene Tsai

  2. Overview • Retinal image registration • The Dual-Bootstrap ICP algorithm • Covariance matrix • Covariance propagation • Model selection • View-based registration • Software design Lecture 16

  3. Retinal Image Registration: Applications • Mosaics • Multimodal integration • Change detection Lecture 16

  4. Mosaics Lecture 16

  5. Multimodal Integration Lecture 16

  6. Change Visualization Lecture 16

  7. Retinal Image Registration - Preliminaries • Features • Transformation models • Initialization Lecture 16

  8. Features • Vascular centerline points • Discrete locations along the vessel contours • Described in terms of pixel locations, orientations, and widths • Vascular landmarks • Pixel locations, orientations and width of vessels that meet to form landmarks landmarks vascular centerlines Lecture 16

  9. Transformation Models Lecture 16

  10. Initializing Registration • Form list of landmarks in each image • Form matches of one landmark from each image • The selection of these matches will be discussed in Lectures 18 and 19 • Choose matches, one at a time • For each match: • Compute an initial similarity transformation in the small image region surrounding the landmarks • Apply Dual-Bootstrap ICP procedure to see if the initial alignment can be successfully grown into an accurate, image-wide alignment • End when one match leads to success, or all matches are exhausted Lecture 16

  11. Dual-Bootstrap - Overview Iterate until convergence • Match and refine estimate in each region • Bootstrap the model: • Low-order for small regions; • High-order for large • Automatic selection • Bootstrap the region: • Covariance propagation gives uncertainty Lecture 16

  12. Matching and Estimation in Each Region • Matching - standard stuff: • Vascular centerline points from within current region of moving image • Mapped using current transform estimate • Find closest point using Borgefors digital distance map • Estimation: • Fix scale estimate • Run IRLS Lecture 16

  13. Covariance Matrix of Estimate • Measures uncertainty in estimate of transformation parameters • Basis for region growth and model selection • The next few slides will give an overview of computing an approximate covariance matrix • We’ll start with linear regression Lecture 16

  14. Problem Formulation in Linear Regression • Independent (non-random) variable values: • Dependent (random) variable values • Linear relationship based on k+1 dimensional parameter vector a: Lecture 16

  15. Least-Squares Formulation • Least-squares error term: • Here: Lecture 16

  16. Estimate and Covariance Matrix • Estimate: • Residual error variance (square of “scale”): • Parameter estimate covariance scatter matrix for 1st order Equation #1 Shape of the cov matrix magnitude Lecture 16

  17. Aside: Equation #1 • How to obtain • Assuming that y is the only random variable, independent and identically distributed, with covariance • The covariance matrix of a is Lecture 16

  18. Aside: Line Fitting in 2D • Form of the equation: • If the xi values are centered: • Then the parameters are independent with variances for the linear and constant terms, respectively Lecture 16

  19. Hessians and Covariances • Back to k dimensions, re-consider the objective function: • Compute the Hessian matrix: • Observe the relationship Lecture 16

  20. Hessians and Covariances • This is exact for linear regression, but serves as a good approximation for non-linear least-squares • In general, the Hessian will depend on the estimate (in regression it doesn’t because the problem is quadratic), so the approximate relationship is Lecture 16

  21. Hessian in Registration • Recall the weighted least-squares objective function: • Keeping the correspondences and the weights fixed, where Dk gives the error of the k-th correspondence • Inverting this gives the covariance approximation. • This approximation is only good when the estimate is fairly accurate Lecture 16

  22. Back to Dual-Bootstrap ICP • Covariance is used in two ways in each DB-ICP iteration: • Determining the region incorporates enough constraints to switch to a more complex model • Similarity => Affine => Reduced Quadratic => Quadratic • Determining the growth of the dual-bootstrap region: • More stable transformation estimates lead to faster growth Lecture 16

  23. Model Selection • What model should be used to describe a given set of data? • Classic problem in statistics, and many methods have been proposed • Most trade-off the fitting accuracy of higher-order models with the stability (or lower complexity) of lower-order models Lecture 16

  24. Model Selection in DB-ICP • Use correspondence set • Estimate the IRLS parameters and covariance matrices for each model in current set • For each model (with dmparameters) this generates a set of weights and errors and a covariance matrix: • Choose the model maximizing the model selection equation (derived from Bayesian modeling): • The first two terms increase with increasingly complex models; the last term decreases accuracy stability Lecture 16

  25. Region Growth in DB-ICP • Grow each side independently • Grow is inversely proportional to uncertainty in mapping of boundary point on the center of each side • New rectangular region found from the new positions of each of the boundary points Lecture 16

  26. Aside: Covariance Propagation and Transfer Error • Given mapping function: • We will treat Q as a random variable, but not gk • Uncertainty in Qmakes gk’ a random variable. • What then is the covariance of gk’? • We solve this using standard covariance propagation techniques: • Compute the Jacobian of the transformation, evaluated at gk: • Pre- and post-multiply to obtain the covariance of gk’ • In computer vision, this is called the “transfer error” Lecture 16

  27. Outward Growth of a Side • Let hk be the outward normal of the side, and let rk be the distance of the side from the center of the region • Project the transfer error covariance onto hk to obtain a scalar variance sk • The outward growth (along normal hk) is • where b controls the maximum growth rate, which occurs when sk < 1 Lecture 16

  28. Putting It All Together - The Example, Revisited Lecture 16

  29. Turning to the Software • A “view” is a definition or snapshot of the registration problem. • A “view” contains: • An image region (current region, plus goal region) • A current transformation estimate and estimator • A current stage (resolution) of registration • Views work in conjunction with multistage registration Lecture 16

  30. View-Based Registration - Procedural • The following is repeated for each initial estimate • For each stage: • Do • Match • Compute weights • Estimate scale • For each model • Run IRLS to estimate parameters and covariances • Re-estimate scale • Generate next view • Choose best model • Grow region • Until region has converged and highest order model used • Prepare for next stage Lecture 16

  31. Implementation • rgrl_view • Store the information about the view • rgrl_view_generator • Generate the next view • rgrl_view_based_registration • Mirrors rgrl_feature_based_registration with modifications based on the outline on previous slide • Example • rgrl/example/registration_retina.cxx Lecture 16

  32. Summary • Retina registration: • Models, features and initialization • DB-ICP: • Matching, estimation and covariances • Model selection • Region growing • Generalization to view-based registration and its implementation in the toolkit. Lecture 16

More Related