From OE Reports June 1994

<h2>Recent progress toward a 3D conoscopic camera</h2>

Conoscopic holography is a Fresnel spatially
incoherent-light holographic technique,1,2 from which
it is possible to design a three-dimensional camera,
i.e., a camera that records both the image and the
shape of objects. The basic system consists of a
uniaxial crystal placed between two circular
polarizers, and is called a conoscope. The light
emitted by every object point gives rise to
interferences inside the crystal, which are
mathematically equivalent to those observed in coherent
holography, but whose scale is ten to a hundred times
larger. This feature enables the acquisition of the
holograms on a video camera, their digitization and
their numerical processing. The applications of such a
system, allowing a quick and easy acquisition of an
object as well as its optical and digital
reconstruction, are numerous: they range from 3D
microscopy to 3D cartography from a satellite, and
include 3D television, intelligent robotics, etc.
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The problems to be solved in order to realize such a
camera are essentially: the noninvertibility of the
point spread function (PSF) of the basic system (which
is the familiar Gabor zone lens (GZL)), the
nonlinearity of the shape (i.e., 3D) reconstruction,
and the sensitivity of the 3D reconstruction to noise
and defects in the impulse response.
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The numerical reconstruction of the image of an object
(2D reconstruction) is an important step towards the
reconstruction of its shape; in principle, this 2D
reconstruction is relatively easy since it simulates an
optical reconstruction-as in coherent holography-but it
faces the same problems as the latter, i.e., the
presence of undiffracted light (or DC bias) and of a
conjugate image. By linearly combining different PSFs
of the system, each of them being obtained by
adequately changing the polarization state and the
amplitude of the incident light field (by means of a
light valve and of a rotating amplitude mask, see
figure), we have been able to obtain an invertible
PSF-that is, one that has the same mathematical
expression as the amplitude transmittance of a lens,
and thus yields neither bias nor conjugate image in the
reconstruction process.3
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The hologram of an object is the incoherent
superposition of the GZLs of all the object's points,
but is no longer a convolution when the object is 3D.
By writing the hologram in a differential form instead
of its natural integral form, we have devised an
algorithm to recover the shape of an opaque 3D object
from its conoscopic hologram. Moreover, this algorithm
is not restricted to conoscopic holography and could
also be used on other interferometers with similar PSF.
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Nevertheless, due to the very physics of the
interference phenomenon, the signal-to-noise ratio of
the object's shape decreases in the low spatial
frequencies, which are the ones that determine the
overall shape. This can be understood intuitively in
the following way: the object's shape is coded in the
variations of the PSF (hologram of a point) when the
longitudinal position of the point changes; and it is
essentially the fine fringes (and thus the high spatial
frequencies) that vary. Consequently, the high
frequencies of the shape will be better recorded than
the low frequencies, and the reconstruction, although
validated on simulations, will be unstable for these
latter frequencies. In order to restore them, we have
proposed an iterative algorithm4 that takes advantage
of the knowledge of the object's support (obtained
through the image reconstruction), and makes the shape
reconstruction less sensitive to noise and to defects
in the experimental PSF. The very encouraging
experimental 3D results that have been obtained
validate the potential of conoscopic holography as a
three-dimensional imaging technique.
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<addr>L. M. Mugnier TELECOM Paris, 46 rue Barrault, F-75634
Paris cedex 13, France. 
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E-mail:mugnier@ima.enst.fr</addr>
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D. Charlot, Former Technical Director of Le Conoscope
S.A.; and G. Y. Sirat, Le Conoscope S.A. 12 Avenue des
Prés, F-78180 Montigny-le-Bretonneux, France.
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<b>References</b>
<p>
1.   G. Y. Sirat and D. Psaltis, "Conoscopic
holography," Opt. Lett. 10, No. 1, January 1985.
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2.   D. Charlot, "Holographie conoscopique. Principe et
reconstructions numériques," Ph.D. dissertation,
TELECOM Paris (1987).
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3.   L. M. Mugnier, G. Y. Sirat and D. Charlot,
"Conoscopic holography: two-dimensional
reconstructions," Opt. Lett. 18, No. 1, January 1993.
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4.   L. M. Mugnier, "Vers une inversion des hologrammes
conoscopiques," Ph.D. dissertation, TELECOM Paris
(1992).
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