How To Calibrate A Sony/Davis Off-Axis Projector

A Little Bit O' Explanation
Above is a little diagram outlining the measurements involved in the calibration of the Sony and Davis projectors used for this project. This equipment was acquired by Professor Levoy during his experiments with color-projection systems on the Stanford home campus, and was later shipped to Florence along with materials required for the Digital Michelangelo Project. After meeting with Prof. Levoy, we decided to place the Davis Projector directly facing the statue, while the Sony Projector was aligned with Lucy's left side. The reason for this decision is that the Davis Projector has a larger field of view, while the Sony's telephoto lens allows an undistorted projection onto the nonplanar mass of the angel's arm, torso, and wing (1). One problem posed by these projectors is that they generate oblique off-axis projections, a function of their having been designed for placement on a classroom ceiling. In order to incorporate the projectors into the project, we needed to quantitatively determine the nature of the off-axis projections they generated. In particular, we calculated the fractional value Y-Offset/E. The appropriate measurements were gathered by projecting an image onto a (supposedly) vertical wall and using a measuring tape to gather the necessary data. It is important to note that, as the projected image was trapezoidal as opposed to rectangular, we also conducted an examination of the pixel aspect ratio of each projector.
A Little Bit O' Trigonometry

A2 = G2 + F2 -2GF * cos(Theta)

cos(Phi) = E / G

(A/2) + Y-Offset = G * sin(Phi)

sin2(Phi) + cos2(Phi) = 1

sin(Phi) = [1-cos2(Phi)]1/2

(A/2) + Y-Offset = G * [1-cos2(Phi)]1/2

Y-Offset = (G *[1-cos2(Phi)]1/2) - (A/2)

Y-Offset = (G *[1-(E2/G2)]1/2) - (A/2)

Y-Offset = (G2-E2)1/2 - (A/2)


Calibration Data
The Davis Projector (800 X 600): Measurements

A = 42 05/16"
B = 57 03/16"
C = 42 10/16"
D = 56 12/16"

E = 83 14/16"
F = 84 05/16"
G = 96 02/16"

Theta = 26.3 Degrees
Y-Offset = 25.65 Inches
Y-Offset/E = 0.306

Note: The pixel aspect ratio of this projector was close enough to "1" that this number was used instead.

The Sony Projector (1024 X 768): Measurements

A = 35 10/16"
B = 47 00/16"
C = 35 08/16"
D = 47 02/16"

E = 119.500"
F = 119.875"
G = 123.875"

Theta = 17.285 Degrees
Y-Offset = 31.936 Inches
Y-Offset/E = 0.11826

Note: E and F were averaged together to produce the value 119.6875", which was then substituted for both E and F where applicable.

After we calculated this information, we were able to utilize the graphical program known as Scanalyze, which was written by members of the Digital Michelangelo Project to facilitate the construction of 3D meshes. By inputting Lucy's mesh into Scanalyze, we could align the program's virtual camera to produce an image that perfectly reflected the physical locations of the projector and of the statue itself. By piping this Scanalyze image into the projector, we mapped an image of the Lucy directly onto her physical geometry.

The Generation of a Scanalyze Image: Necessary Considerations
  • Thanks to the diligence of Szymon Rusinkiewicz, Scanalyze was blessed with several routines which allowed us to specify parameters for an oblique off-axis projection. In particular, we were now able to feed the Y-Offset/E parameter into a Scanalyze command termed "plv_oblique_camera".

  • It was necessary to instruct Scanalyze to modify the resolution of its display to match the resolution of the projector through which the image would be projected. Each projector's width/height ratio is listed along with the various pieces of measured data.

  • Pixel aspect ratio is calculated as follows: A and C are averaged together, as are B and D. For example, let us term P = (A + C)/2 and Q = (B + D)/2. So, if we divide P by the number of pixels along a vertical edge of the screen, we have the length of the vertical edge of a pixel. The length of a horizontal edge of a pixel may be similarly calculated. By dividing the horizontal resolution of a pixel by the vertical resolution, we are left with the pixel aspect ratio for a particular projector. The pixel aspect ratio of each projector was close to "1", which means that the resulting error was negligible.

  • The zoom lens of the Sony Projector was utilized to minimize error resulting from the use of a telephoto lens. Thus, it was necessary to multiply the Sony's Y-Offset/E parameter by 5/3. The number which resulted was inputted into Scanalyze's ply_zoom_angle field.

  • In the above Trigonometry section, (A+C)/2 was substituted for the variable "A" in all calculations.

(1): Addendum (Added 16/03/99) --> It is important to note that we actually merely zoomed the Sony Projector in all the way, zoomed the Davis Projector out all the way, employed the names "telephoto lens" and "wide angle", respectively, and were content.