Calibration. Reality. Error. Measuring device. Model of Reality Fall 2001 Copyright R. H. Taylor 1999, 2001

Transcription

1 Calibration Calibrate (vt) : 1. to determine the caliber of (as a thermometer tube); 2. to determine, rectify, or mark the gradations of (as a thermometer tube); 3. to standardize (as a measuring instrument) by determining the deviation from a standard so as to ascertain the proper correction factors; 4. ADJUST, TUNE

2 Calibration Reality Measuring device + - Error Model of Reality

3 Calibration Reality Actuation device + - Error Model of Reality

4 Basic Techniques Parameter Estimation Mapping the space

5 Parameter Estimation Compare observed system performance to reference standard ( ground truth ) Compute parameters of mathematical model that minimizes residual error. System Parameters + - Model of reality Reality Error Calibration process

6 Pointing device calibration p pivot p t (unknown) F = ( R, p ) i i i

7 Pointing device calibration p pivot p t (unknown) F = ( R, p ) i i i

8 Pointing device calibration p pivot p t F = ( R, p ) i i i

9 Pointing device calibration p pivot p t (unknown) Rp + p = p i t i pivot F = ( R, p ) i i i

10 Pointing device calibration p pivot p t (unknown) p t R I p p j pivot j F = ( R, p ) i i i

11 Linear Parameter Estimation p = f( q) where q= [ q,... q nom 1 n T ] are parameters * p = f( q+ q) L fi f( q) + ( q) q NM j f( q) + J ( q) q f O L QP NM q q 1 n O QP

12 Linear Parameter Estimation Given f(q), and a set of observations p k * corresponding to nominal parameter values q k *, solve the least squares problem L NM O QP L NM * f ( q ) q p f( q ) J k k k O QP

13 Example: 2 link robot calibration a 2 * p k a 1

14 Example: 2 link robot calibration a 2 * p k a 1

15 Example: 2 link robot calibration p = L NM a a sinθ + a sin( θ ) cosθ + a cos( θ ) O QP where θ = θ + θ p * = L NM ( a1+ a1)sin( θ1+ θ1) + ( a2 + a2)sin( θ12 + θ1+ θ2) 0 ( a + a cos( θ + θ ) + ( a + a ) cos( θ + θ + θ ) O QP

16 Example: 2 link robot calibration p = f( q ) = f( a, a, θ, θ ) = k k L NM a1sinθ1, k + a2sin( θ12, k) 0 a cosθ + a cos( θ ) 1 1, k 2 12, k O QP where θ = θ + θ so we solve the least squares problem L NM f a OL QP NM a1 f f f a 2 ( q ) ( q ) ( q ) ( q ) a θ θ θ 1P θ arot 1 ( y, θ 1 k ) + arot( y, θ ) * k k k k p, k , k 2 O Q L NM L M N O QP O P Q

17 Example: 2 link robot calibration Here so J f L NM ( q ) = k L NM sinθ sin θ a (cosθ + cos θ ) a cosθ , k 12, k 1 1, k 12, k 2 12, k cosθ cos θ a (sinθ + sin θ ) a cosθ 1, k 12, k 1 1, k 12, k 2 12, k a 1 sinθ, k sin θ, k a (cosθ, k + cos θ, k) a cosθ , k a cosθ, k cos θ, k a (sinθ, k + sin θ, k) a cosθ , k θ PM 1P θ OL QN 2 O Q O QP L NM x a sinθ + a sin( θ ) z a cos a cos( ) * k 1 1, k 2 12, k * k 1 θ1, k + 2 θ12, k O QP

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19 Example: Robodoc Wrist Calibration Basic robot had very accurate calibration Custom wrist was less accurate Crucial goal was to determine position of cutter tip Cutter Calibration post

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21 Kinematic Model p = p + R( z, θ + θ ) ( α x+ v ) tool wrist 4 4 distal v = R( x, β ) [ R( y, θ + θ )( v + v )] distal 5 5 c c

22 Linearization p p + [ R R ( v + v )] +... post wrist 4 5 c c

23 Linear Least Squares Most commonly used method for parameter estimation Many numerical libraries See the web site Here is a quick review Microsoft PowerPoint Presentation

24 Example: Undistorted fluoroscope calibration

25 Calibration if no distortion (version 1) Assume no distortion. For the moment also assume that you have N point calibration features (e.g., small steel balls) at known { } positions a0,, a N 1 relative to the detector. Assume further that the points { } create images at corresponding points d0,, d N 1 on the detector. Estimate the position s of the x-ray source relative to the detector

26 Approach F obj s Calibration object a k d k Detector

27 Projection of a point feature s s= λ(a-d)+d (a - d) (s - d) λ = (a - d) (a - d) a d d= µ ( a s) + s µ = (a - s) (d - s) (a - s) (a - s)

28 Approach s d a ( a d) ( s d) = 0 skew( a d) s = ( a d) d = a d d d = a d 0 dz az ay dy az dz 0 dx ax s = a d dy ay ax dx 0

29 Solve least squares problem Approach skew( a0 d0) sx a0 d0 s y s z skew( N 1 N 1) a d an 1 dn 1 s a d

30 What if pose of calibration object is imprecisely known? This is a hairier problem, but solvable In fact, it makes a great homework assignment.

31 Mapping the space Compare observed system performance to reference standard ( ground truth ) Interpolate residual errors System + - Model of reality Reality Error Model Correction Lookup Table

32 Example: Fluoroscope calibration

33 Projection of a point feature with distortion a s s= λ(a-d)+d (a-d) (s-d) λ = (a - d) (a - d) d u u = f( d, ν )

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36 C-Arm Detector and dewarp plate Experimental Setup Robot Arm Corkscrew Phantom Surgical Cutter

37 Dewarping Method

38 Intrinsic Image Calibration Intrinsic imaging parameters (Schreiner et. al.) Image Warping (Checkerboard Based Method) 1/16 4/16 3/16 4/16 Top View Side View

39 Step 0: Acquire Image

40 Step 1: Find groove points Find image points corresponding to the centerline of each vertical and horizontal groove

41 Step 2: Fit 5 th order Bernstein Fit least square smooth curve to each vertical and horizontal groove 5 th order Bernstein Polynomial Polynomial Curves k B( a0,, a5; v) = ak ( 1 v) v k = 0 k k

42 Step 3: Dewarp Employ a two pass scan line algorithm to dewarp the image

43 Advantages Fast 2 seconds on Pentium II 400 Robust works well even with overlaid objects Sub-pixel Accuracy mean error 0.12 mm on the central area Does not completely obscure the image trades off image contrast depth for image area

44 Two Plane Method Plane 1 pattern Plane 2 pattern E.g., Lavallee E.g., Helm

45 Two Plane Method Plane 1 pattern Plane 2 pattern Given q = a point in image coordinates, determine the points f f The desired ray in space passes through and * 1 * 2 f = the point on grid 1 corresponding to = the point on grid 2 corresponding to * 2. f * 1 q q

46 Photos: Sofamor Danek

47 Interpolation Ubiquitous throughout CIS research and applications Many techniques and methods Here are a few more notes Microsoft PowerPoint Presentation

48 Two plane calibration Again, the essential problem is to determine the coordinates in the two planes at which the source-to-detector ray passes through the plane. Many methods for this. E.g., Find the four surrounding bead locations on each plane and use bilinear interpolation Fit a general spline model for the distortion on each plane and then directly interpolate