x l (X¡ Z) ¢ 2. <strong>Principles</strong> <strong>of</strong> <strong>Stereoscopic</strong> <strong>Depth</strong> <strong>Perception</strong> <strong>and</strong> Reproduction 2.4.2 <strong>Stereoscopic</strong> video geometry The mathematical basis for stereoscopic image formation was first described by Rule (1941). In the 1950s, the seminal paper by Spottiswoode, Spottiswoode & Smith (1952) extended this work considerably. Various accounts <strong>of</strong> the geometry <strong>of</strong> stereoscopic camera <strong>and</strong> display systems have been published since, including, Gonzalez & Woods (1992), Woods, Docherty & Koch (1993), <strong>and</strong> Franich (1996). They describe the geometric formulae for both the parallel camera configuration, where the optical axes <strong>of</strong> the left <strong>and</strong> right monoscopic cameras run parallel, <strong>and</strong> the converging camera configuration, where the optical axes <strong>of</strong> the cameras intersect at a convergence point. The equations by which the horizontal disparities can be calculated for both types <strong>of</strong> camera configurations are given below, based on the calculations by Gonzalez & Woods (1992) <strong>and</strong> Franich (1996). Parallel cameras Figure 2.11 illustrates the parallel camera set-up, where the optical axes <strong>of</strong> both cameras run parallel, equivalent to viewing a point at infinity. Included in the figure are the video capture variables that have an impact on the horizontal (left-right) image disparity captured at the camera’s imaging sensors. The magnitude <strong>of</strong> disparity is the distance (in metres) between X-coordinates <strong>of</strong> homologous points in the left <strong>and</strong> right image, if the two camera planes were superimposed on each other. As can be seen from the figure, a real-world point w, with coordinates X, Y, <strong>and</strong> Z, will be projected onto the left <strong>and</strong> right camera imaging sensors, <strong>and</strong> the horizontal disparity will be scaled by the following factors: the camera base separation, or baseline, B, the focal length <strong>of</strong> the lenses <strong>of</strong> the cameras, , <strong>and</strong> the distance from the cameras to the real world point, Z w . The image capture process for each <strong>of</strong> the two cameras consists <strong>of</strong> a translation along the X-axis followed by a perspective transformation (Franich 1996, Gonzalez & Woods 1992). The camera projection coordinates (left camera: x l¡ y l ; right camera: x r¡ y r ) <strong>of</strong> real world point w(X,Y,Z) are given by: X £ B 2 Z (2.1) 74 ¥¤
y l (Y¡ Z) ¢ x r (X¡ Z) ¢ y r (Y¡ Z) ¢ ¤ ¤ 2.4. <strong>Stereoscopic</strong> video production Figure 2.11: Parallel camera geometry Y Z (2.2) X ¤ B 2 ¤ Z (2.3) Y Z (2.4) As one can determine by comparing equations (2.2) <strong>and</strong> (2.4), the real world point (X,Y,Z) is projected onto the same Y coordinate for both left 75
Change language