Theory of Bouguet s MatLab Camera Calibration Toolbox

Transcription

1 Theory of Bouguet s MatLab Camera Calibration Toolbox Yuji Oyamada 1 HVRL, University 2 Chair for Computer Aided Medical Procedure (CAMP) Technische Universität München June 26, 2012

2 MatLab Camera Calibration Toolbox Calibration using a planar calibration object. Points correspondence requires manual click on object s corners. Core calibration part is fully automatic. C implementation is included in OpenCV. Web

3 Functions Extract grid corners: Points correspondence Calibration: Core calibration Show Extrinsic: Visualization of estimated extrinsic parameters Reproject on images: Visualization of reprojection error Analyse error: Recomp. corners: Recompute the corners using estimated parameters

4 Input: Extract grid corners n sq {x,y}: Number of square along {x,y} axis d{x,y}: Size of square in m unit. I: Input image Output: {x i}, i = 1,..., N: Set of detected 2D points. {X i}, i = 1,..., N: Set of known 3D points. where x i and X i have M i elements as [ ] x i(1) x i(j) x i(mi ) x i = and y i(1) y i(j) y i(m i ) X i(1) X i(j) X i(m i ) X i = Y i(1) Y i(j) Y i(m i ) Z i(1) Z i(j) Z i(m i )

5 Calibration Input: {x i}: Set of detected 2D points. {X i}: Set of known 3D points. Output: Intrinsic parameters Extrinsic parameters Method: 1. Initialize intrinsic and extrinsic params. 2. Non-linear optimization to refine all parameters.

6 Camera parameters Intrinsic parameters: fc R 2 : Camera focal length cc R 2 : Principal point coordinates alpha c R 1 : Skew coefficient kc R 5 : Distortion coefficients KK R 3 3 : The camera matrix (containing fc, cc, alpha c) Extrinsic parameters for i-th image: omc i R 3 : Rotation angles Tc i R 3 : Translation vectors Rc i R 3 3 : Rotation matrix computed as Rc i = rodrigues(omc i)

7 Initialization 1/2: Intrinsic parameters Input {x i}: Set of detected 2D points. {X i}: Set of known 3D points. Output Initialized intrinsic parameters Initialized extrinsic parameters

8 Initialization 1/2: Intrinsic parameters 1. Roughly compute intrinsic params. 2. Refine intrinsic params. 3. Compute extrinsic params. Step1: fc: Computed based on all the vanishing points. cc: Center of image. alpha c = 0. kc = 0.

9 Initialization 1/2: Intrinsic parameters Step2: Given N sets of corresponding points {x k, X k }, 1. Compute Homography {H k }. 2. Build linear equations Vb = 0 and solve the equations Detail. 3. Compute A from estimated b. v 1,12 (v 1,11 v 1,22 ) v k,12 V (v k,11 v k,22 ) v N,12 (v N,11 v N,22 ) v k,ij = [h k,i1 h k,j1, h k,i1 h k,j2 + h k,i2 h k,j1, h k,i2 h k,j2, h k,i3 h k,j1 + h k,i1 h k,j3, h k,i3 h k,j2 + h k,i2 h k,j3, h k,i3 h k,j3 ]

10 Initialization 2/2: Extrinsic parameters Step3: Compute each extrinsic parameters from estimated A and pre-computed H k as where λ k = r k,1 = λ k A 1 h k,1 r k,2 = λ k A 1 h k,2 r k,3 = r k,1 r k,2 t k = λ k A 1 h k,3 1 A 1 h k,1 = 1 A 1 h k,2

11 Solves the following equations: Non-linear optimization ˆp = arg min p x f (p) 2 2 where p = {fc, cc, alpha c, kc, {omc i}, {Tc i}} and the function f projects the known 3D points X onto 2D image plane with the argument p. Sparse Levenberg-Marquardt algorithm iteratively updates ˆp with an initial estimate ˆp 0 as while( x f (ˆp i 1 ) 2 2 > ɛ) ˆp i = ˆp i 1 + p i ; i = i + 1;

12 Given Image projection x p = (x p, y p, 1) : Pixel coordinate of a point on the image plane. x d = (x d, y d, 1) : Normalized coordinate distorted by lens. x n = (x n, y n, 1) : Normalized pinhole image projection. k R 5 : Lens distortion parameters. KK R 3 3 : Camera calibration matrix. where fc(1) alpha c(1) cc(1) KK = 0 fc(2) cc(2) x p = KKx d = KK(cdistx n + delta x)

13 Given Lens distortion x n = x: Normalized pinhole image projection. k R 5 : Lens distortion parameters. Compute x d = xd: Normalized coordinate distorted by lens. cdist: Radial distortion delta x: Tangential distortion xd = cdist x + delta x [ ] cdist x + delta x(1) = cdist y + delta x(2)

14 Radial distortion where xd1 = [ ] x cdist y cdist cdist = 1 + k(1) r2 + k(2) r4 + k(5) r6 r2 = x 2 i + y 2 i, r4 = r2 2, r6 = r2 3

15 Tangential distortion where delta x = [ ] k(3) a1 + k(4) a2 k(3) a3 + k(4) a1 a1 = 2x i y i a2 = r2 + 2x 2 i a3 = r2 + 2y 2 i

16 Closed form solution for initialization X u s v = A [ r 1 r 2 r 3 t ] Y 0 = A [ r 1 r 2 t ] X Y X = H Y 1 Since r 1 and r 2 are orthonormal, we have following two constraints: h 1 A A 1 h 2 = 0 h 1 A A 1 h 1 = h 2 A A 1 h 2, where A A 1 describes the image of the absolute conic.

17 Closed form solution for initialization Let B 11 B 21 B 31 B = A A 1 B 12 B 22 B 32 B 13 B 23 B 33 = 1 α 2 γ α 2 β γ α 2 β v 0 γ u 0 β α 2 β γ γ(v 0γ u 0 β) v α 2 β 2 β 2 α 2 β 2 0 β 2 v 0 γ u 0 β α 2 β γ(v 0γ u 0 β) α 2 β 2 v 0 β 2 (v 0 γ u 0 β) 2 α 2 β 2 + v 2 0 β Note that B is symmetric, defined by a 6D vector b [ B 11, B 12, B 22, B 13, B 23, B 3 ]

18 Closed form solution for initialization Let the i-th column vector of H be h i = [ h i1, h i2, h i3 ]. Then, we have following linear equation where h i Bh j = v ij b v ij = [h i1 h j1, h i1 h j2 + h i2 h j1, h i2 h j2 h i3 h j1 + h i1 h j3, h i3 h j2 + h i2 h j3, h i3 h j3 ]

19 Closed form solution for initialization The above two constraints can be rewritten as two homogeneous equation w.r.t. unknown b as [ v ] 12 (v 11 v 22 ) b = 0 Given n images, we have Vb = 0, where V R 2n 6.

20 Least Squares Let f ( ) a function projecting 3D points X onto 2D points x with projection parameters p as bmx = f (p) = px. Our task is to find an optimal p such that minimizes x px 2 2. In the case of non-linear function f ( ), we iteratively update the estimate ˆp with an initial estimate ˆp 0 as while( x px 2 2 > ɛ) ˆp i = ˆp i 1 + p i ; i = i + 1;

21 Least Squares: Newton s method Given a previous estimate ˆp i, estimate an update i s.t. i = arg min f (ˆp i + ) X 2 2 = arg min f (ˆp i) + J X 2 2 = arg min ɛ i + J 2 2 where J = f p. Then, solve the normal equations J J = J ɛ i

22 Levenberg-Marquardt (LM) iteration The normal equation is replaced by the augmented normal equations as J J = J ɛ i (J J + λi) = J ɛ i where I denotes the identity matrix. An initial value of λ is 10 3 times the average of the diagonal elements of N = J J.

23 Sparse LM LM is suitable for minimization w.r.t. a small number of parameters. The central step of LM, solving the normal equations, has complexity N 3 in the number of parameters and is repeated many times. The normal equation matrix has a certain sparse block structure.

24 Sparse LM Let p R M be the parameter vector that is able to be partitioned into parameter vectors as p = (a, b ). Given a measurement vector x R N Let x be the covariance matrix for the measurement vector. A general function f : R M R N takes p to the estimated measurement vector ˆx = f (p). ɛ denotes the difference x ˆx between the measured and the estimated vectors.

25 Sparse LM The set of equations Jδ = ɛ solved as the central step in the LM has the form ( ) δa Jδ = [A B] = ɛ. Then, the normal equations J 1 x at each step of LM are of the form [ A 1 x A A ] 1 ( ) ( x B B 1 x A B δa 1 = x B δ b δ b Jδ = J 1 x ɛ to be solved A 1 x B 1 x ) ɛ ɛ

26 Let Sparse LM U = A 1 x A W = A 1 x B V = B 1 x B and denotes augmented matrix by λ. The normal equations are rewritten as [ U ] ( ) ( ) W δa ɛa = W V δ b ɛ B [ U WV 1 W ] ( ) ( 0 δa ɛa WV 1 ) ɛ = B W This results in the elimination of the top right hand block. V δ b ɛ B

27 Sparse LM The top half of this set of equations is (U WV 1 W )δ a = ɛ A WV 1 ɛ B Subsequently, the value of δ a may be found by back-substitution, giving V δ b = ɛ B W δ a

28 Sparse LM p = (a, b ), where a = (fc, cc, alpha c, kc ) and b = ({omc i Tc i }) The Jacobian matrix is where [ ˆx ˆx a = ˆx b = [ fc, ˆx cc, ˆx omc 1, J = ˆx p = ˆx alpha c, ˆx ] kc ˆx Tc 1,, [ ˆx a, ˆx ] = [A, B] b ˆx omc i ˆx Tc i, ˆx omc N ] ˆx Tc N

29 Sparse LM The normal equation is rewritten as ([ ] ) [ ] A [A ] A B + λi p = ɛ x B N A i A i i=1 N B i A i i=1 N i=1 N i=1 B A i B i [ B + λi A ] ɛ p = x B ɛ x i B i

30 Sparse LM J J = N A i A i A 1 B 1 A i B i A N B N i=1 B 1 A 1 B 1 B 1. B i A i. B N A N... B i B i... B N B N

31 Sparse LM J ɛ x = N A i ɛ x i=1 B 1 ɛ x. B i ɛ x. B N ɛ x

32 Sparse LM When each image has different number of corresponding points (M i M j, if i j), each A i and B i have different size as

33 Sparse LM However, the difference does not matter because A A R d int d int A B R d int d ex B A R dex d int B B R dex dex where d int denotes dimension of intrinsic params and d ex denotes dimension of extrinsic params.