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• use the same coordinate system as in optical measurements,
• apply minimal offset (translation) to optical frame in order to make all coordinates positive (see left plot below),
• use averaged coordinate af of all tiles (see right plot below),
• use user-defined translation, for example set origin in the evaluated point of beam crossing with the detector plane. 

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Pixel coordinate reconstruction in the detector frame uses

• tile ideal geometry,
• tile center positions,
• tile N·90 rotation and tilt angles:

             (x,y,z)in detector = (Xc,Yc,Zc) + RotationR(N·90+tilt) × (x,y,z)in tile,

where rotation matrix R is a product of three rotations around appropriate axes, for example rotation in X-Y plane around Z is

               / cosT  -sinT    0 \

Rz(T) = | sinT    cosT    0  |

Image Added               \    0        0       1 /

Pixel coordinates in the detector should be accessed by method like

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