Kinect intrinsic parameters from field of view

Question

Microsoft state that the field of view angles for the Kinect are 43 degrees vertical and 57 horizontal (stated here) . Given these, can we calculate the intrinsic parameters

Answer 1

If you equate the world origin (0,0,0) with the camera focus (center of projection as you call it) and you assume the camera is pointing along the positive z-axis, then the situation looks like this in the plane x=0:

Here the axes are z (horizontal) and y (vertical). The subscript v is for "viewport" or screen, and w is for world.

If I get your meaning correctly, you know h, the screen height in pixels. Also, zw, yv and xv. You want to know yw and xw. Note this calculation has (0,0) in the center of the viewport. Adjust appropriately for the usual screen coordinate system with (0,0) in the upper left corner. Apply a little trig:

tan(43/2) = (h/2) / f = h / (2f),  so f = h / ( 2 tan(43/2) )

and similar triangles

yw / zw = yv / f            also              xw / zw = xv / f

Solve:

yw = zw * yv / f            and                xw = zw * xv / f

Note this assumes the "focal length" of the camera is equal in the x-direction. It doesn't have to be. For best accuracy in xw, you should recalculate with f = w / 2 tan(57/2) where w is the screen width. This is because f isn't a true focal length. It's just a constant of conversion. If the pixels of the camera are square and optics have no aberrations, these two f calculations will give the same result.

NB: In a deleted (improper) article the OP seemed to say that it isn't zw that's known but the length D of the hypotenuse: origin to (xw,yw,zw). In this case just note zw = D * f / sqrt(xv² + yv² + f²) (assuming camera pixels are square; some scaling is necessary if not). They you can proceed as above.

Answer 2

i cannot add comment since i have a too low reputation here. But I remind that the camera angle of the kinect isn't general the same like in a normal photo camera, due to the video stream format and its sensor chip. Therefore the SDK mentioning 57 degrees and 43 degrees, might refer to different degree resolution for hight and width.

it sends a bitmap of 320x240 pixels and those pixels relate to

Horizontal FOV: 58,5° (as distributed over 320 pixels horizontal)
Vertical FOV: 45,6° (as distributed over 240 pixels vertical).

Z is known your angle is known, so i supose law of sines can get you proper locations then https://en.wikipedia.org/wiki/Law_of_sines