Andrew Woods ^{1}

Tom Docherty ^{2}

Rolf Koch ^{3}

^{1,2}
Centre for Marine Science & Technology,

Curtin University of Technology, G.P.O. Box U 1987, Perth W.A. 6845, AUSTRALIA.

^{3} School of Mathematical and Physical Sciences,

Murdoch University, South Street, Murdoch W.A. 6150, AUSTRALIA.

**Figure 1**: (a) Stereoscopic camera system and (b) stereoscopic display system

*t*- Camera Separation. The distance between the first nodal points of the two camera lenses.

*C*- Convergence Distance. The distance from the convergence point to the midpoint between the first nodal points of the two camera lenses. *C*=*t/ *(2 tan[ ß + arctan(* h / f *)])

*f*- Lens Focal Length. The focal length of the two camera lenses.

*(alpha)*- Camera Field of View. The horizontal angle of view of the camera.
*(alpha)* = arctan[(*Wc*/2+*h *)/* f *] + arctan[( *Wc*/2- *h*)/*f *]

*V*- Viewing Distance. The distance from the observer's eyes to the display plane.

*e*- Eye Separation. The distance between the observer's eyes. Typically 65mm.

*Ws*- Screen Width. The horizontal size of the display screen.

*P*- Image Parallax. The horizontal distance between homologous points on the screen. *P* = *Xsr - Xsl*

*M*- Frame Magnification. The ratio of screen width (*Ws*) to camera sensor width (*Wc*). *M *= *Ws / Wc*

(*Xo*,*Yo *,*Zo *) - The location of a point in object space (in front of the cameras).

(*Xsl*,*Ysl *),(*Xsr *,*Ysr *) - The location of left and right image points on the screen.

The geometry of a stereoscopic video system can be determined by considering the imaging and display process as three separate coordinate transforms: Firstly from X,Y,Z coordinates in object/camera space to X and Y positions on the two camera imaging sensors (CCDs), secondly from the two sets of CCD coordinates to X and Y positions of the left and right images on the stereoscopic display, and thirdly to a set of X,Y,Z coordinates in image/viewer space.

**Figure 2**: Camera parameters for (a) toed-in camera configuration and (b) parallel camera configuration (Plan View)

This is summarised as follows:

Object Space -> CCD Coordinates -> Screen Coordinates -> Image Space (Xo,Yo,Zo) (Xcl,Ycl),(Xcr,Ycr) (Xsl,Ysl),(Xsr,Ysr) (Xi,Yi,Zi)

The first coordinate transform is shown in equations (1) to (4). The variables and coordinate conventions of this transform are shown in Figure 2 except for the Y axis which for object space is centred at the midpoint between the first nodal points of the camera lenses and positive in the upward direction and for CCD coordinates is positive in the downwards direction from the centre of the CCD.

.......**(1)**

.......**(2)**

.......**(3)**

.......**(4)**

[An error in equations 3 and 4 was corrected 28 October 2004. The error did not occur in the pdf version of this paper.]

The transformation from CCD coordinates to screen coordinates is achieved by multiplying by the screen magnification factor M:

Xsl = M Xcl .......**(5)**

Xsr = M Xcr .......**(6)**

Ysl = M Ycl .......**(7)**

Ysr = M Ycr .......**(8)**

The final transform from screen coordinates to image space coordinates is shown in equations 9 to 11. The variables and coordinate conventions for this transform are shown diagrammatically in Figure 3 except for the Y variables which are positive in the upwards direction from the centre of the screen.

**Figure 3**: Viewing parameters (Plan View)

.......**(9)**

.......**(10)**

.......**(11)**

Special
mention needs to be made about the Y coordinate equation. Two values can be
developed for the image space Y coordinate, one each from the left and right
views, *Ysl* and *Ysr*, however, only one Y position is meaningful.
Therefore a single value of screen Y position must be determined from these two
values. The difference between screen Y coordinates is termed `vertical
parallax' and determines how easily the stereoscopic image can be fused. If
vertical parallax is small we use *Ys *=(*Ysl *+*Ysr *)/2.

The overall coordinate transformation from object space coordinates to image space coordinates is:

.......**(12)**

.......**(13)**

.......**(14)**

These
equations apply to both the parallel camera and the toed-in camera
configurations. Significant simplifications can be made for a parallel camera
configuration. It should also be noted that these equations do not contain any
small angle approximations. It has been found that small angle approximations
can obscure some stereoscopic distortions^{2,3}.

**Figure 4**: Coordinate transformation from Object Space to Image Space
(for *C* = 0.9m, *f *= 6.5mm, *t* = 75mm,
*V* = 0.9m, *e *= 65mm, *Ws *= 300mm).

In order to illustrate the results of the above equations, a computer program was developed to generate plots which display the coordinate transformation from object space to image space. An example of one of these plots is shown in Figure 4. This plot shows the way in which the object space in front of the camera system (in the XZ plane) is transformed to the display system (image space). The grid pattern demonstrates how a rectilinear grid (of 10cm squares) in front of the camera system has been distorted upon display. The two circles represent the viewer's eyes and the bold line is the display. The grid pattern extends to 3m away from the cameras. The curve furthest from the eyes indicates where infinity from the cameras will be displayed on the monitor. The grid pattern is not displayed past 3m to infinity due to its increasing density.

The manipulation of the three camera configuration parameters and the three display configuration parameters are shown diagrammatically in Figures 5 and 6. These figures show how the image display geometry of a predetermined camera and display configuration is affected by changes of configuration parameters.

Click here for Picture (16k, 916 x 668)

**Figure 5**: Variation of camera configuration parameters

Click here for Picture (19k, 884 x 669)

**Figure 6**: Variation of display configuration parameters

Stereoscopic distortions are ways in which a stereoscopic image of a scene differs from actually viewing the scene directly. There are a number of different types of image distortions in stereoscopic video systems. This chapter will discuss various types of image distortions including outlining their origins and their effects on a viewer's perception of a scene.

**Figure 7**: 3D maps of (a) toed-in cameras (b) parallel cameras (c) shear distortion and (d) plot of image distance versus object distance.

The depth plane curvature illustrated here could lead to wrongly perceived relative object distances on the display and also disturbing image motions during panning of the camera system.

**Figure 8**: Vertical parallax caused by keystone distortion

**Figure 9**: Lens radial distortion for 3.5mm lens

**Figure 10**: Experimental results of depth range limit

3. R. Spottiswoode and N. Spottiswoode, The Theory of Stereoscopic Transmission and its Application to the Motion Picture, University of California Press, Berkeley, 1953.

5. C. Smith, "3-D or not 3-D?" New Scientist, Vol.102 #1407, pp. 40-44, April 1984.

9. L. Lipton, Foundations of the Stereoscopic Cinema, Van Nostrand Reinhold Company Inc., New York, 1982.

The program which was used to generate the plots shown in Figures 4, 5, 6 and 7 is now available as shareware. Click here to download "3D-MAP". (Program runs under DOS on a 386 PC or higher) (47k, zip file)

Copyright on this document is retained by Curtin University. This document is not public domain. Permission is hereby given to reprint this paper on a non-profit basis for scholarly purposes provided the document is unaltered and this notice is intact. This paper may not be reprinted for profit or in an anthology without prior written permission. If you wish to reprint this paper on this basis, please contact the primary author at the address shown on the first page of this document.

This paper is also available as a pdf.

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Last modified: 28th October, 2004.

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