dorioliu / MicroScope_Stereo_Calib_based_on_DLT

MicroScope_Stereo_Calib_based_on_DLT

This is an implementation of DLT based stereo camera calibration with Eigen

General,we describe a imaging system of pinhole model as:

$$sp = K[R|t]P (1)$$

s is depth scale coefficient, lower case p represents 2d image point (u,v,1), upper P is a space 3d point (X,Y,Z,1) in world coordinate. usually, in rigid transformation, we descibe a rigid motion from P1(X1,Y1,Z1) to P2(X2,Y2,Z2) with rotation transformation R and the new camera position is C as P2 = R(P1 - C), also sometimes, we hide description of C and introduces a tanslation vecton t = -RC ,in this case, rigid transformation can also be written down as P2 = RP1 + t. So:

$$sp = K[R|t]P = [KR, -KRC]P = HP (2)$$

which C reperents the camera optical center position in world coordinate, H is a preojection matrix with a shape of 3 by 4. The DLT method takes a set of at least 6 pairs of 3d world points and those corrsponding projected 2d iamge points as inputs and outputs the K,R parameters what we want to get by DLT based calibration. please note that all 3d world points must not space on a space planar. So, how to evaluate H and how to do decompsition about H to get K and R ?
In general, we calculate H by build over-determined equation based on the relation between p and P.

$$[u] [ h11, h12, h13, h14 ] [v] = [ h21, h22, h23, h24 ] * [X, Y,Z,1]^T (3) [1] [ h31, h32, h33, h34 ]$$

thus, we get two quations:

$$u = (hllX + h12Y + h13Z + 1)/(h31X + h31Y + h33Z + 1) (4) v = (h2lX + h22Y + h23Z + 1)/(h31X + h31Y + h33Z + 1)$$

now we have I pairs of that point (ui,vi,1) and (Xi,Yi,Zi,1)
each point pair provide two equation.
based on (4) we get a linear algebraic equation:

$$[ -X1 -Y1 -Z1 -1 0 0 0 0 x1X1 x1Y1 x1Z1 x1 ] [ 0 0 0 0 -X1 -Y1 -Z1 -1 y1X1 y1Y1 y1Z1 y1 ] ..... [ -Xi -Yi -Zi -1 0 0 0 0 xiXi xiY1 xiZi xi ] * [h11, h12, h13, h14,h21, h22, h23, h24,h31, h32, h33, h34 ]^T = 0 (5) [ 0 0 0 0 -Xi -Yi -Zi -1 yiXi yiYi yiZi yi ] ..... [ -XI -YI -ZI -1 0 0 0 0 xIXI xIYI xIZI xI ] [ 0 0 0 0 -XI -YI -ZI -1 yIXI yIYI yIZI yI ]$$

this linear equation is a traditional Ax = 0 problem. the sulution of x can be evaluated by SVD decomposition (sigular value decomposition) at that case: A = USV^T, U: 2Ix12, V: 12x12, S: 12x12 is diagonal matrix with descendent value the solution is a sigular vector within V corresponding to the minimal sigular value among S diagonal line coefficent， tipically x = V.col(11) as the repository.
now we have got H, and we use QR decomposition to calculate K , R ,C

$$H = [KR, -KRC] = [Ho, ho] Ho = KR ho = -KRC so C = -Ho^-1 * ho$$

to our konwledge, in QR decomposition, R is a upper triangular matrix, Q is a orthogonal matrix, and is our Ho =KR matrix K is a upper triangular matrix, R is a orthogonal matrix so we can make a samll trick H^-1 = R^-1 * K^-1 = QR than we get rotation matrix R = Q^-1, and get K = R^-1 finnally we devide K by K(2,3) to get a normalized K.