Extracting 3D coordinates given 2D image points, depth map and camera calibration matrices

Nicolas Burrus has created a great tutorial for Depth Sensors like Kinect.

http://nicolas.burrus.name/index.php/Research/KinectCalibration

I'll copy & paste the most important parts:

Mapping depth pixels with color pixels

The first step is to undistort rgb and depth images using the estimated distortion coefficients. Then, using the depth camera intrinsics, each pixel (x_d,y_d) of the depth camera can be projected to metric 3D space using the following formula:

P3D.x = (x_d - cx_d) * depth(x_d,y_d) / fx_d
P3D.y = (y_d - cy_d) * depth(x_d,y_d) / fy_d
P3D.z = depth(x_d,y_d)

with fx_d, fy_d, cx_d and cy_d the intrinsics of the depth camera.

If you are further interested in stereo mapping (values for kinect):

We can then reproject each 3D point on the color image and get its color:

P3D' = R.P3D + T 
P2D_rgb.x = (P3D'.x * fx_rgb / P3D'.z) + cx_rgb
P2D_rgb.y = (P3D'.y * fy_rgb / P3D'.z) + cy_rgb

with R and T the rotation and translation parameters estimated during the stereo calibration.

The parameters I could estimate for my Kinect are:

Color

fx_rgb 5.2921508098293293e+02 

fy_rgb 5.2556393630057437e+02 
cx_rgb 3.2894272028759258e+02 
cy_rgb 2.6748068171871557e+02 
k1_rgb 2.6451622333009589e-01 
k2_rgb -8.3990749424620825e-01 
p1_rgb -1.9922302173693159e-03 
p2_rgb 1.4371995932897616e-03 
k3_rgb 9.1192465078713847e-01

Depth

fx_d 5.9421434211923247e+02 

fy_d 5.9104053696870778e+02 
cx_d 3.3930780975300314e+02 
cy_d 2.4273913761751615e+02 
k1_d -2.6386489753128833e-01 
k2_d 9.9966832163729757e-01 
p1_d -7.6275862143610667e-04 
p2_d 5.0350940090814270e-03 
k3_d -1.3053628089976321e+00

Relative transform between the sensors (in meters)

R [ 9.9984628826577793e-01, 1.2635359098409581e-03, -1.7487233004436643e-02, 

-1.4779096108364480e-03, 9.9992385683542895e-01, -1.2251380107679535e-02,

1.7470421412464927e-02, 1.2275341476520762e-02, 9.9977202419716948e-01 ]

T [ 1.9985242312092553e-02, -7.4423738761617583e-04, -1.0916736334336222e-02 ]