**7. On the parallax of a lens image**

The very common use of photography has always rendered plane images and that is why it is normal to think that a lens image is plane. But we know that the images are threedimensional and that it is the detecting system which makes the result to be in a plane. By closing or opening the aperture of a lens, one always captures the same plane scene but the sharpness of those elements not precisely focused diminishes at large apertures. When opening the diaphragm from his smaller diameter, the out-of-focus situation of the additional rays, which do not fit in the image generated by the center of the lens, happens because of the new perspectives added by the lens area being unobstructed. It is important to notice that they correspond to the viewpoint of each area part being unobstructed on the lens. Continuous parallax is then allowed.

One of the first applications of this property was a three-dimensional photography system employing only one camera, opening to light only one area on each side of the lens, one transmitting through one color filter and the other through another, photographing in only one shot a direct anaglyph.

The horizontal parallax transmitted by a slit and placed on a lens may be employed for generating multiple views and even continuous parallax.

Lunazzi (12) projected a scene directly from objects on a diffractive screen of 15 cm x 30 cm employing an ordinary slide projector objective. The 6x enlarged image gives the precise impression of a holographic one, but has more focusing depth limitations. "Direct holoprojection of objects" was a name given to this technique in which the horizontal extension of a lens is the fundamental property (Figure 4).

Son (13) employed this parallax property to project sequentially a set of views from a multimedia projector. Each slit position on the projector corresponded to a vision zone for the screen and the observer could have a different view within a discrete sequence of lateral positions. The image persistence on the retina gives the illusion of simultaneous viewing, but it is necessary for the system to put all views in the time of one ordinary movie frame (1/24s), so that a set of views may be projected. At 24 views per second, for example, the frame capability of the electronic multimedia system needed is about the square value of that of an ordinary projector. In the present state of the art it seems not possible to achieve this at a high definition resolution. Employing many projectors at close lateral positions is a possibility to reduce the frame rate needs and to obtain a brighter image, but it is only possible if the screen has a low scattering level in order to avoid the simultaneous view of all projections.

diffracting element and as the spectrum is continuous the parallax also is. We can better understand the basic process of a diffractive screen considering it as a primary element, a diffraction grating. If we further approximate the projecting lens to a simple pinhole camera corresponding to its central part, we can see in Figure 5 how the ray tracing based on an

object point explains the resulting image by central symmetry (16).

Fig. 5. Symmetry in double diffraction imaging intermediated by a slit.

DG1 and DG2 are two identical diffraction gratings intermediated by a slit. Each object point has a corresponding image point symmetrically to a central point. Because the observer looks to the image from behind, he sees an inverted depth. In this case of perfect matching between two diffractions, it is interesting to notice a property which resembles a holographic one: the diffraction at a symmetric order, on the second grating for example, generates an image with inverted depth. An image in normal depth is so obtained (17).

Fig. 4. Up: Photograph of an object directly projected on a holographic screen. Above: anaglyphic stereo representation of the same scene, with color channel.
