Lecture 8: Registration

Transcription

1 ME 328: Medical Robotics Winter 2019 Lecture 8: Registration Allison Okamura Stanford University

2 Updates Assignment 4 Sign up for teams/ultrasound by noon today at: Main parts: Readings/questions Image acquisition and analysis (ultrasound lab) Image registration Needle insertion (robot control) Remainder of assignment will be posted this morning

3 registration the alignment of multiple data sets into a single coordinate system such that the spatial locations of corresponding points coincide i.e., establish a quantitative relationship between different reference frames essential for the quantitative integration of various sources of information: 1. pre-, intra-, and post-operative images 2. trackers (tracking data not useful unless registered) 3. robots (cannot use images unless registered) Image-Guided Procedures: A Review, by Ziv Yaniv and Kevin Cleary (2006)

4 examples L. Joskowicz c 2011

5 suitability for image-guided procedures accuracy: target registration error (TRE) quantifies how far the predicted position of the anatomical target is from its actual position ~1 mm typically satisfactory speed: how long does it take the algorithm to produce the solution intraoperatively, need on the order of seconds to minutes robustness: how well does the algorithm deal with noise and outliers depends on how noisy the data is

6 navigation and registration example

7 a typical robot registration problem Preoperative model from a CT scan gives the location of a tumor in the brain... Now we are in the OR and want to operate with a robot based on this preoperative registration example and graphics provided by Russell H. Taylor, JHU image

8 a typical robot registration problem preoperative model intraoperative reality T reg v CT = T reg v ptr v ptr What is? T reg c Russell H. Taylor, 2011

9 coordinate systems and transformations y B T {B} y A z B x B {A} x A e.g., image or patient coordinate system z A e.g., robot base T 1 coordinate system

10 coordinate systems and transformations v A = F (v B ) v A = Tv B y B v A = Rv B + p T {B} v B y A z B x B {A} v A x A e.g., image or patient coordinate system z A e.g., robot base coordinate system T 1 T (R, p) = apple R 3 3 p

11 coordinate systems and transformations v A = F (v B ) first rotate v A = Tv B v A = Rv B + p {B} v B {A} v A T (R, p) = apple R 3 3 p

12 v A = F (v B ) v A = Tv B coordinate systems and transformations v A = Rv B + p {B} v B then translate {A} v A T (R, p) = apple R 3 3 p

13 interesting properties of T and R forward transformation v A = Tv B v A = Rv B + p inverse transformation T 1 v A = v B v B = R 1 (v A p) v B = R 1 v A R 1 p composition of transformations T 3 = T 1 T 2 R 3 = R 1 R 2 p 3 = R 1 p 2 + p 1 rotation matrices are orthogonal matrices R T = R 1 R T R = RR T = I and members of the group SO(3) det(r) =+1

14 composition of transformations F CH F CA F BE F BD F C F D F E F GC F G F DU F U F EH F BG F UA F H F B F GH? F GH = F 1 BG F BEF EH c Russell H. Taylor, 2011

15 feature-based registration preoperative model intraoperative reality tracked pointer/robot end-effector c Russell H. Taylor, 2011

16 features used for feature-based registration Point fiducials (markers) Point anatomical landmarks Ridge curves Contours Surfaces Line fiducials given two features (either the same features in two different reference frames, of two different features in the same reference frame), we need to know the distance between them

17 what the computer knows preoperative model intraoperative reality c Russell H. Taylor, 2011

18 identify corresponding points preoperative model intraoperative reality c Russell H. Taylor, 2011

19 how to identify correspondences { b } i manual vs. part of the optimization/ a k? minimization process state of the art is the iterative closest point (ICP) algorithm... but we will assume manual is possible

20 find the best rigid transformation preoperative model intraoperative reality X min T reg i w i D(T reg a i,b i ) minimize over all possible values of T reg weighting depending on which points are being considered c a disparity function Russell H. Taylor, 2011

21 disparity function, D a metric for the error between two feature sets minimize over all possible values of T reg X min T reg i w i D(T reg a i,b i ) weighting depending on which points are being considered sum of squares of residuals is common: X min T reg i other D possibilities include: maximum distance median distance a disparity function w i (T reg a i b i ) 2 cardinality depending in threshold

22 how to do the optimization for rigid registration Given points in two different coordinate systems (e.g., a set of points and a set of points ) {a i } {b i } Find the transformation matrix T (R, p) X That minimizes e T i e i i Where e i =(Ra i + p) b i this is tricky because of R Options: global vs. local numerical vs. direct (analytical) ways of dealing with local minima

23 minimizing registration errors Step 1: compute means and residuals of known points ā = 1 NX NX 1 b a i = b i N N i=1 i=1 ã i = a i ā bi = b b i X Step 2: Find R that minimizes (Rã bi i ) 2 Step 3: Find p p = b Rā i How??? There are 4 substeps within Step 4: Desired transformation is apple R p T (R, p) = this step

24 minimization substeps: direct method (there is also an iterative method) Step 1: Compute H = i!a i,x! bi,x!a i,x! bi,y!a i,x! bi,z!a i,y! bi,x!a i,y! bi,y!a i,y! bi,z!a i,z! bi,x!a i,z! bi,y!a i,z! bi,z Step 2: Compute the SVD of H = USV t Step 3: R = VU t outer product between a and b Step 4: Verify Det(R) = 1. If not, then algorithm may fail. K. S. Arun, T. S. Huang, S. D. Blostein. Least-Squares Fitting of Two 3-D Point Sets. IEEE Transactions on Pattern Analysis and Machine Intelligence, 9(5): , 1987.

25 Extra Slides: An Iterative Method for Finding the Rotation Matrix (I don t recommend using this for your assignment, though. The SVD method is much simpler.)

26 iterative method: solving for R Goal: given paired point sets {a i } and {b i }, find R = arg min X i (Rã i bi ) 2 Step 0: Make an initial guess R 0 Step 1: Given R, compute ˆbi k = R 1 k b i X Step 2: Compute R that minimizes ( Rã ˆbi i ) 2 Step 3: Set R k+1 = R k R Step 4: Iterate Steps 1-3 until residual error is sufficiently small (or other termination condition) i

27 more mathematical preliminaries matrix representation of cross product 2 T a = a x a y a z skew(a) = 4 a v =skew(a)v 0 a z a y a z 0 a x a y a x representations/constructions of rotation matrices angle-axis: rot(a, ) is a rotation about axis a Rodrigues formula: c = rot(a, )b = b cos( )+a b sin( )+a(a T b)(1 cos( )) exponential: rot(a, ) =e skew(a) = I + skew(a)+ 2 2! skew(a) rot(a, ) I + skew(a) more notation: rot(a, ) =R a ( ) R(a) = rot(a, a )

28 small transformations useful for linear approximations to represent small pose shifts Tv = Rv + p R a small rotation R a ( ) a rotation by a small angle about axis a rot(a, a )b a b + b for a sufficiently small R(a) a rotation that is small enough so that any error introduced by this approximation is negligible R( a) R(µb) R( a + µb) linearity for small rotations exercise: work out the linearity proposition by substitution

29 small transformations Tv = R(a)v + p Tv v + a v + p a v =skew(a)v = a z a y a z 0 a x a y a x v x v y v z 3 5 skew(a)a =0 R(a) I +skew(a) R(a) 1 I skew(a) =I +skew( a)

30 iterative method: solving for Goal: given paired point sets {a i } and {b i }, find R = arg min X i R (Rã i bi ) 2 Step 0: Make an initial guess R 0 Step 1: Given R, compute ˆbi k = R 1 k b i X Step 2: Compute R that minimizes ( Rã ˆbi i ) 2 Step 3: Set R k+1 = R k R Step 4: Iterate Steps 1-3 until residual error is sufficiently small (or other termination condition) i

31 iterative method: solving for R Approximate R as (I +skew( )) Which is equivalent to Rv v + v remember: multiplying by a skew-symmetric matrix is equivalent to taking cross product Then our least squares problem becomes X min ( Rã ˆbi i ) 2 X min (ã ˆbi i + ã i ) 2 R i This is a linear least squares problem in i Then compute R( )