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3D Reconstruction



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3D Reconstruction

After the calibration is done and corresponding features in both images are found, the 3D position of the feature can be computed. With an intensity based approach a dense disparity map can be generated. A feature-based approach only provides disparity information for the features that have been extracted. Interpolation can be used to get more 3D points. The mathematical formulation to get the coordinates of an image point in the camera coordinate system2.4 is given by

begin{equation*}begin{aligned}Z_c = -ffrac{b}{d} X_c = frac{-xz}{f} Y_c = frac{-yz}{f} end{aligned}end{equation*}


where $ f$ is the effective focal length and $ b$ is the baseline, both parameters can be computed using a calibration technique. If $ t_1$ is the translation vector of the left camera and $ t_2$ is the calibration vector of the second camera, the baseline $ b$ can be computed by subtracting the $ x$ coordinates of the translation vectors.


For many applications it is more convenient to use coordinates of the object points given in another coordinate system. To get a rotation matrix and a translation vector, which are both needed to relate two coordinate system to each other, calibration is used. The extrinsic camera parameters define the orientation and translation of the camera coordinate system in reference to a given world coordinate system, which originates at the calibration pattern. To formulate the relation between the two camera coordinate system, let us introduce a new one, the so-called cyclopean view. This coordinate system has its origin in the middle of the baseline $ b$, so that the z-axis bisects the angle between the left and right optical axes, as illustrated in Figure 2.14.

The angle between the optical axes is 2$ phi$. The y-axis of the camera coordinate system has to be parallel. The relation between a point
$ P=(X,Y,Z)$ in the cyclopean coordinate system, and the same point
$ P_l(X_l,Y_l,Z_l),P_r(X_r,Y_r,Z_r)$ can be written as

$displaystyle begin{pmatrix}X_l cr Y_l cr Z_l cr end{pmatrix} = begin{pma... ...d{pmatrix} begin{pmatrix}X + frac{b}{2} - fsinphi cr Y cr Z end{pmatrix}$ (2.37)


and

$displaystyle begin{pmatrix}X_l cr Y_l cr Z_l end{pmatrix} = begin{pmatrix... ...d{pmatrix} begin{pmatrix}X + frac{b}{2} + fsinphi cr Y cr Z end{pmatrix}$ (2.38)


To get the coordinates of point
$ P_w(X_w,Y_w,Z_w)$ in world coordinates, after the coordinates of point
$ P_c(X_C,Y_c,Z_c)$ has been computed, we can use Equation 2.13 to formulate the following equation

$displaystyle begin{pmatrix}X_w cr Y_w cr Z_w end{pmatrix} = left( begin{... ...y}right)^{-1} begin{pmatrix}X_c - T_xcr Y_c - T_ycr Z_c - T_c end{pmatrix}$ (2.39)


Transforming the camera coordinate system into another one is useful when the camera is not oriented in an appropriate manner. Additionally, the coordinate system can be shared amongst different applications with different camera systems.

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Up: Stereo Vision