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Parallel Lens Axes Computing a stereoscopic image involves drawing two monocular viewpoints from different positions. In the past, people have created computer-generated stereoscopic pairs by rotating an object through a few degrees, but we advise against this procedure. The use of rotation to generate stereoscopic pairs will result in vertical misalignment of corresponding left and right image points. Such a vertical misalignment causes the eye muscles to work in an extraordinary way which most people experience as discomfort. If the eyes try to fuse vertical parallax, which contains no depth information, the result can be painful. |
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The rectangle ABCD, on the left, has its vertical axis indicated by the dotted line. When rotated about the axis, as shown, corner points A', B', C' , and D ' are now higher or lower than corresponding points A, B, C , and D . Despite the fact that this example involves a simple figure, it's typical of what happens when rotation is employed for producing stereoscopic pairs. Instead of rotation, we recommend producing the two views as a horizontal perspective shift along the x axis, with a horizontal shift of the resulting images (HIT - horizontal image translation) to establish the ZPS (zero parallax setting). We will now describe a mathematical model of a stereo camera system, using the method of horizontal perspective shifts without rotation to produce the proper stereoscopic images. |
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The Camera Model |
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The geometry of the basic algorithm for producing computer-generated electro-stereoscopic images is illustrated here, showing left and right cameras located in data space. Their interaxial separation is given by tc. The camera lens axes are parallel to each other in the z direction. The distance from the cameras to the object of interest is do . |
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We have chosen a camera model because of the diverse approaches and nomenclatures that people use for computer graphics. It's our belief that everybody understands something about photography these days because of the wide acceptance of 35-millimeter and video cameras. Imagine that the cameras shown are two still or video cameras whose lens axes are parallel, mounted on a flat base. t c is the interaxial separation or distance between the cameras or, more accurately, the centers of the lenses. The cameras are mounted so that the lenses' axes - lines passing through the centers of the lenses and perpendicular to the imaging planes - always remain parallel. Imagine that we're looking at an object in this data space. The two cameras will produce two different perspective viewpoints, because they are horizontally shifted the distance tc. The two cameras are using lenses whose focal lengths are identical. Short-focal-length lenses produce a wide-angle view, and long-focal-length lenses produce a narrow angle of view. For example, wide-angle lenses for 35-millimeter photography are generally anything under 50 millimeters, and long-focal-length or telephoto lenses are generally anything above 50 millimeters. Let's assume we have two video cameras with lenses looking straight ahead, each seeing its perspective viewpoint of an object in space. Let's switch the video signal between the cameras at field rate. 1 If we were to take a look at the images on a television monitor, we would see two images which are not exactly superimposed, because the cameras are tc apart. Since the cameras are pointing straight ahead with parallel lens axes, they will see different parts of the object. Let's horizontally shift these two images relative to each other, so that one portion of the image is superimposed. Wherever the image is superimposed the parallax is zero, and that portion of the object (or scene) appears at the plane of the screen. This has been called "convergence," but that term is easily confused with human physiology and the rotation of the eyes needed for fusion, as discussed earlier. Therefore, we use the term HIT (horizontal image translation), which is used to achieve ZPS (zero parallax setting). If the images are horizontally shifted so that some part of the image is perfectly superimposed, that part of the image is at ZPS. Remember that our object exists not only in the x and y axes, but also in the z axis. Since it is a three-dimensional object, we can achieve ZPS for only one point (or set of points located in a plane) in the object. Let's suppose we use ZPS on the middle of the object. In this case, those portions of the object which are behind the ZPS will have positive parallax, and those which are in front of the ZPS will have negative parallax. his parallel lens axes algorithm for stereoscopic computer-generated images uses two camera viewpoints, with parallel lens axes, set some distance tc apart. Both viewpoints, or camera lenses, have the same angle of view. The degree to which we horizontally shift or translate (HIT) the images is more than a matter of taste. Earlier we explained the advantages of creating images with low parallax. We must think of using ZPS to produce the best compromise of parallax for the entire image. This approach involves no rotation and leads to geometric-distortion-producing vertical parallax, but we will have achieved the two perspective viewpoints needed for a stereoscopic image. If HIT is used as described, with other conditions (angle of view of lenses, distance of object from the lenses, tc ) kept constant, there is no change in depth content. It will take a moment for your eyes to reconverge for the different values of parallax, but the total depth content of the image remains the same. After your eyes readjust, the image will appear to be just as deep as it was before. However, you may have produced, through HIT, images which appear to be entirely in front of or behind the screen. As discussed earlier, it's best to place the ZPS at or near the center of the object to reduce the breakdown of accommodation and convergence and to reduce the ghosting artifact. The Parallax Factor We suggest the use of small tc values and a wide angle of view, at least 40 degrees horizontal. To help understand why, let us take a look at the relationship: Pm =Mfc tc (1/do -1/dm ) This is the depth range equation used by stereographers, describing how the maximum value of parallax (Pm) changes as we change the camera setup. Imagine that by applying HIT, the parallax value of an object at distance d o becomes zero. We say we have achieved ZPS at the distance do . An object at some maximum distance dm will now have a parallax value of Pm . The aim is to produce the strongest stereoscopic effect without exceeding a maximum of 1.5 degrees of parallax. The form of this equation is helpful in understanding this. The value of magnification ( M ) will change the value of Pm. For example, the image seen on a big screen will have greater parallax than the image seen on a small screen. A 24-inch-wide monitor will have twice the parallax of a 12-inch monitor, all things being equal. Reducing tc will reduce the value of screen parallax. We also note that reducing lens focal length fc (using a wide-angle lens) also reduces Pm. The most important factor which controls the stereoscopic effect is the distance tc between the two camera lenses. The greater tc , the greater the parallax values and the greater the stereoscopic depth cue. The converse is also true. If we are looking at objects that are very close to the camera - say coins, insects, or small objects - tc may be low and still produce a strong stereoscopic effect. On the other hand, if we're looking at distant hills tc may have to be hundreds of meters in order to produce any sort of stereoscopic effect. So changing tc is a way to control the depth of a stereoscopic image. Small values of tc can be employed when using a wide angle of view (low fc ). From perspective considerations, we see that the relative juxtaposition of the close portion of the object and the distant portion of the object, for the wide-angle point of view, will be exaggerated. It is this perspective exaggeration that scales the stereoscopic depth effect. When using wide-angle views, tc can be reduced. 2 The reader ought to verify the suggestions we're making here. Producing computer-generated stereoscopic images is an art, and practice makes perfect. It is up to you to create stereoscopic images by playing with these parameters. As long as you stick with the basic algorithm we've described (to eliminate geometric distortion), you have a free hand in experimenting with tc , do , and angle of view ( fc ). This parallel lens axes approach is particularly important when using wide-angle views for objects which are close. In this case, if we use rotation, the geometric distortion would exacerbate the generation of spurious vertical parallax. Remember that the strength of the stereoscopic effect is controlled by the interaxial distance tc, which controls the parallax values of the object or the scene. Once the interaxial setting is established, HIT is used to control the ZPS of the images, and is often achieved in software by moving two portions of the buffer relative to each other, as discussed in the next chapter. The horizontal movement of the left and right images in the upper and lower portions of the buffer produces HIT, which is used to control ZPS. In terms of producing user-friendly software, consideration should be given to the fact that when an image is booted up, it should have the proper ZPS. For a single-object image, a good place for ZPS is the middle of the object. If this suggestion is not followed, the user will have a difficult time because of the breakdown of accommodation and convergence, and because of the concomitant crosstalk. It's important to implement the ZPS condition in software as a default condition. In addition, when changing the size of the image it will also be desirable to maintain the ZPS, and this can also be implemented in software. Given our parallel lens axes camera model, when tc is changed the ZPS will also change; and it is desirable to implement, in software, a constancy of ZPS even when tc is varied. This effort will help the user produce a better-quality, easier-to-look-at stereoscopic image. Windows and Cursors One important concept for windows is that alphanumerics must appear in both the left and right fields, and it's often appropriate for this material to be at ZPS. Alphanumerics (and the vertical and horizontal lines that make up the window) should appear in both the left and right fields in the same x,y locations. There may be occasions when it is desirable to have windows which are in CRT or viewer space. This may be appropriate when multiple windows are used. Cursors have the same benefits as they have in the planar display mode but, in addition, for a stereoscopic display they can also provide z -axis-locating with a great degree of accuracy. People who have been active in photogrammetry use stereoscopic cursors, typically in the form of a cross, in order to measure height or depth. A cursor appearing in the two subfields (see the next chapter) in exactly the same location will have zero parallax, but it's also important to be able to move the cursor in z -space. By moving one or both of the cursors in the left and right subfields in the horizontal direction (the x direction), the cursor will appear to be moving in z -space. By counting x pixels, the exact location of the cursor can be determined. If additional information about the capture or photography of the image is provided, one will then be able to translate the counting of x pixels into accurate z measurements. The depth sense of binocular stereopsis is exquisitely sensitive, and a skilled operator will be able to place a cursor to an accuracy of plus-or-minus one pixel. References 1. Lipton, Lenny, and Meyer, Lhary. A time-multiplexed two-times vertical frequency stereoscopic video system. International Symposium Digest , SID 1984, Vol.XV. 2. MacAdam, D.L. Stereoscopic perceptions of size, shape, distance and direction. SMPTE Journal , 1954, 62:271-93. |
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All materials © Copyright 1996-97, StereoGraphics Corporation |
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