PSEUDOCOLORING METHOD FOR 3 CONTOURING

RÉSUMÉ : Les propriétés d’interférence des diagrammes de granularité en lumiére blanche obtenus avec un systéme optique non corrigé de l’aberration chromatique, sont appliquées á la codification en fausses couleurs des contours de niveaux d’égale profondeur d’un objet diffusant tridimensionnel. On enregistre d’abord les différentes images des niveaux de profondeur codifiées en couleur. En une seconde étape, on obtient une image faussement colorée en modifíant la position d’un filtre spatial. On peut obtenir d’ailleurs un contour quantitatif en balayant l’image faussement colorée avec un dispositif spectroscopique. SUMMARY : The interference properties of white light speckle pattern obtained with an optical system not corrected for chromatic aberration are applied to pseudocolor encode the depth level contours of a 3-D diffusing object. At a first step, a recording of the different encoded images of depth levels is done. In a second step, a reconstructed pseudocolored image is obtained, where different color assignements can be tuned by changing the position of a spatial filter slit. Also, a quantitative contouring can be made scanning the pseudocolored image plañe with a spectroscopic device.


INTRODUCTION
Speckle patterns produced by partially coherent white light have been theoretically and experimentally studied in the last few years [1], Namely, the statistical properties of white light speckle have been investigated in the image plañe [2][3][4][5][6] and in the diffraction field [7,8].The dependence of the speckle pattern contrast with surface roughness was established in connection with different parameters that characterize the optical system and the illuminating light : the amplitude point spread function, and the spatial and temporal coherences.K. Nakagawa and T. Asakura [9] have shown experimentally that the average contrast of the white light image speckle depends strongly on the surface roughness : for small valúes of r.m.s.roughness (lower than 0.5 pm) the relevant parameter is the spatial coherence, while for'large valúes of r.m.s.roughness the effect of the point spread function is more important.When the speckle pattern is observed at a defocused plañe, for an object roughness greater than 1 pm, the additional phase-noise distortion pro duced by defocusing does not affect much the speckle pattern contrast.On the other hand, for surface objects having a r.m.s.roughness lower than 0.3 pm, the speckle pattern contrast is affected by the phasenoise distortion due to defocusing [7].
In the present study, the interference properties of white light patterns are applied to pseudocolor encode depth level contours of a 3-D object.For this purpose, an optical imaging system affected by chro matic aberration with an exit pupil consisting in two identical slits, is employed.In this way, the white light speckle pattern results in pairs of tiny spectra.When both spectra of each pair intersect they do it for the same wavelength and interference Young's fringes appear in the intersection región.Therefore, at a fixed distance from the imaging lens, for each object plañe there exists a coded image for a certain wavelength.This spatial codification of color information is decoded, in a further step, to obtain a depth level pseudocoloring of the 3-D object.On the other hand, if the rough surface is a plañe one, a study of the chromatic aberration of the optical system can be done.

GENERAL CONSIDERATIONS
When a rough plañe surface is illuminated by partially coherent quasi-monochromatic light, the speckle intensity distribution in the diffraction and image planes is given by where r(xí, x2) is the spatial coherence function between two different object points xx and x2 that characterizes the quasi-monochromatic illuminating light; K(x) is the amplitude point spread function of the optical system; and O(x) is the complex amplitude of the object.For simplicity, only a one dimensional analysis is done.Taking into account the intensity fluctuations due to random phase changes introduced by the surface roughness, the above expression (1) results (2) where < ••• ) indicates ensemble average.The speckle pattern contrast can be defined as : V = < A /2 ) 1/2/< / ) .Therefore ít should be noted that it depends on the surface roughness, the coherence conditions of the illuminating light, and the employed optical system.
When white light is used, and the optical imaging system is not corrected for chromatic aberration, different image planes appear, each one corresponding to a different color.In this way, taking into account the dispersión constant of the lens, a continuous variation of focused and defocused speckle patterns occurs.At each image plañe, the amplitude point spread function K(x) determines the variation of the speckle pattern contrast with wavelength, in accordance with relation (2).Therefore, using a slit as lens aperture, the resulting polychromatic speckle pattern, at a certain observation plañe, consists of tiny spectra.In this case, each quasi-monochromatic speckle pattern can be treated by the general expression (2).
In this paper the interference conditions of these spectra, when a two-aperture pupil is used, are applied to obtain a mapping of depth level contours.When such a system is employed the image of a 3-D diffusing object consists of pairs of intersecting tiny spectra.Therefore, the intersection región exhibits Young's fringes, the spatial frequency of which is determined by the temporal frequency of the corresponding color.Because this intersection temporal frequency v is fixed by the specific function n = n(2) that describes the chromatic aberration of the lens, as v takes all valúes of the visible spectrum, different planes are characterized for a given fixed image plañe.Thus, a pseudocolor encoding of level contours of a 3-D object can be obtained.

DESCRIPTION OF THE METHOD
In the method we present here, the optical system consists of a white light source S0 limited by a diaphragme D, an achromatic lens L x, and a second lens L2 not corrected for chromatic aberration whose pupil consists of two laterally displaced slits Rx and R2 symmetrically located to the optical axis of the systerm This arrangement is shown in figure 1.For a given 2 characterizing a certain band of the spectra, the amplitude point spread function Kfx', y'), at the image plañe, is the smallest spread diffraction pattern of one slit modulated itself by Young's fringes.Namely, where /¿(x', / ; X) takes into account the slit diffraction pattern at the image plañe corresponding to Xwavelength; p = d¡ Xzu is the spatial frequency of Young's fringes; and < P is a constant phase delay.Because of the chromatic aberration of the employed optical system, the image point spread function for each X lies in a different plañe in accordance with the relationship : n = n(X).
Thus, for_a given object distance z0, the image distance zf(A) for each K¿(x', / ; X) results The white light speckle pattern of a plañe diffusing object is observed at a fixed plañe n\ such that the image point spread functions Kt{x', y '; X), corres ponding to several wavelengths X, lie in planes near te'.In this way, only for a color_represented by a certain wavelength X(l\ K¿(x', y'j A(l)) lies in the 7r'-plane.For all other wavelengths X, the point spread functions Kd(x\ y '; X) are broader than K¡(x', y '; A(I)) in the Tr'-plane.Therefore, an expression of Kd(x\ y' ; X) can be given as  Therefore, when a 3-D diffusing object is considered, every object plañe is imaged in a certain color which is associated to a given wavelength X at the 7i'-plane.In this way, the different point spread functions Kfx', y '; X) that are present in rí determine the existence of Young's fringes whose spatial frequencies belong to a certain band Ap -(dfzi) (1/Ii -1/I2)-These cut-off wavelengths X1 and X2 are determined both by the spectral contení of the white light source, and by the spectral sensitivity response of the recording material to be used.
A recording of the white light « image » speckle pattern is done at the ^'-plane.After processing, the photographic píate H is placed in the optical arrangement shown in figure 2.An achromatic lens L3 images the slit R3 at its focal plañe n0.Taking into account the recorded intensity in H, the amplitude distribution U0(u, v) in the 7i0-plane, is where u = x/f v = y/f and / is the focal length of L3; t(f, rj) is the amplitude transmittance of H, at (f, rj)plane.The integration over X involves all wavelengths of the reconstructing white light source.
Let us consider a certain spatial frequency valué u0 = x0/f that characterizes the location, at 7r0-plane. of one diffracted order of Young's fringes for a certain wavelength X0.A spatial filter slit R4 is placed in 7r0-plane located at u = u0.Taking into account the spatial spectrum of t(f, rj), the expression (5) can be rewritten as follows, In expression (6), T(u/X, v/X) is the spatial spectrum of the amplitude transmittance ?(£, r\) for a given wavelength, regardless the spatial modulation introduced by Young's fringes; rect [fu -u0) / /Av0] is the rectangular function associated with the spatial filter R4.Finally, the d-functions lócate the three diffracted orders 0, ± 1 of Young's fringes.
The slit width Ax0 of R4 determines both the spatial spectral band of the 3-D object image for a certain wavelength that is transmitted to the final image, and the frequency spectral band that encodes each Young's fringes spatial frequency.In this way, R4 acts as a temporal-spatial filter; that is : as the slit width Av0 decreases, the image of each depth level of the 3-D object exhibits a well defined color, but its finer details vanish in the final image.

PSEUDOCOLOR ENCODING OF DEPTH LEVELS
As we stated above, a pseudocolored image of the 3-D object is obtained employing a second lens behind the spatial filter slit R4.Nevertheless, it is possible to carry out a preliminary real time visualization using the optical arrangement shown in figure 1.The color of the Young's fringes associated with the several image planes acts as a primary tool to evalúate the depth level contours of the object, and it can be observed with a microscope focused in the 7i'-plane.In figure 3 is shown the polychromatic speckle pattern where Young's fringes appear only for green color.
When a recording of this speckle pattern is done, the spectral range of the pseudocolored reconstructed image is determined by the spectral band of the image planes that are focused on the photographic píate.The different color encoding planes are related by, where 9 is the angular position of R4, X0 is the obser varon wavelength, and Xr is the recording wavelength.
In this way, changing the valué of 9 it is posible to tune the chromatic interval, so that the colors originally focused on the photographic píate appear in the pseudocolored image.Other false colors, ruled by the expression (7), can be selected in the coded image.Therefore, it is possible to obtain a pseudocolored image in the spectral range where better human eye color discrimination occurs.Besides, if a spectroscopic device is used such that the input slit is placed in the plañe of the reconstructed pseudocolored image, the contouring of the 3-D object, in the input slit direction, is displayed in the output plañe of the analyzer device.In this case, the profile of the 3-D surface can be obtained scanning the pseudocolored image with the slit of the spectroscopic device, and with appropriate calibration a quantitative contouring can be made.In figure 4 and 5, experimental results are shown corresponding to an object composed of two plañe rough surfaces, one of which is normal to the optical axis, and the other one lies in a tilted and axially shifted plañe.The angle of tilt was 3o, and the shift was of 2 mm.Both photographs were taken for two different positions of the spatial filter slit R4.In this way, two different pseudocolored images of the same object are obtained.The optical frequency difference Av, corresponding to the colors associated with each plañe, is, where k is a constant given by : k = c(l/A2 -l/A1); c is the vacuum light velocity; A2 and Ax are the spacings of Young's fringes corresponding to each plañe.Therefore, a greater chromatic difference is obtained by placing the slit R4 in the región of shorter wavelengths.

3-D RECONSTRUCTION CAPABILITY
As it has been pointed out before (Eq.( 7)), the reconstructed image can be obtained with the same colors that were properly focused on H in the registering step.If H is replaced in its original location and the reconstruction is done as it is indicated in figure 6, light rays actually travel the inverse path.In this way, a 3-D image of + 1 magnification of the original object can be obtained in its original position.In this case geometrical aberrations are cancelled.Nevertheless, due to the presence of a narrow slit, depth of field is large and resolution is severely limited.Because of this, the 3-D nature of the images is difficult to observe experimentally.At present, we are intending to overeóme these shortcomings.
The pictures shown in figure 4 and 5 were actually obtained in this way, and a slight missfocusing can be observed in the left side of the images.

CONCLUSIONS
In this paper we have applied the interference properties of white light speckle pattern obtained with an optical system not corrected for chromatic aber ration to pseudocolor encode the depth level contours of a 3-D diffusing object.Due to the special optical arrangement employed the resulting polychromatic speckle pattern consists of tiny spectra with a spatial fringe modulation determined by the optical frequency corresponding to the intersection of spectra.In this way, a spatial encoding of an object-space determined by the amount of axial chromatic aberration of the optical system, is accomplished.In a further step, a pseudocolored image is obtained through a temporalspatial filtering operation.
This method allows the false colors to be tuned; i.e. varying the location of the slit R4 is possible to select different color assignements.In this way, any particular zone of interest of the object can be tuned in order to be shown in the color of maximal human eye discrimination.Also, in the case of a continuous variation of depth, a continuous color encoding is obtained.However, the useful volume is small as it is limited by the amount of axial chromatic aberration.Because of the finite size of the involved slits, image resolution is limited in one direction.Alternatively, if a plañe rough surface is used as object, both the axial chromatic aberration and the lateral chromatic aberration may be studied.In summary, we have presented a simple method for generalizing the speckle pseudocoloring technique to inelude other cases besides gray level pseudocoloring.An advantage of this method consists of employing colors, given by Ai and Á2, that are spectral.When using green and red as false colors, as usual, it is possible with this method to obtain a third color which lies in a straight line in the chromatic diagram, very cióse to the locus of the spectral colors.In this way, false colors of high purity are obtained.

( 3 )
where f d(x', y' ; X) takes into account the slit diffraction pattern at the defocused plañe n'.The attenuation factor y [z;(A)] of Young's fringes is originated by where 1/C is the geometrical bending parameter that characterizes the lens L2.

Fig. 1 .
Fig. 1. -Schematic diagram o f the experimental set-up used in the registering step.Due to chromatic aberrations, L 2 conjugates n-plane and its surroundings (which are shown in detail in the lower parí o f the figure) with n'-plane.

( 4 )
the attenuation factor y [z((A)] may be assumed as and no fringes appear in the diffraction pattern cor responding to the color characterized by X.

Fig. 6 .
Fig. 6. -Schem atic diagram o f the experim ental set-up em ployed f o r 3-D reconstruction.OAl : optical a xis o f the achromatic transform ing lens L x, centered at R2 slit position.OA2 : optical axis o f the not corrected lens L 2. M : mask to block the undesired « 0 order ».
Pseudocoloring of gray level information is a technique for introducing false colors in a black-and-white transparency.The importance of this operation is based on the human eye's ability to distinguish different colors better than gray levels.In the last few years several pseudocoloring optical methods have been proposed.1-10Furthermore, the pseudocoloring technique has been extended to encode image spatial spectral bands11 l:! and holographic interferometric fringe patterns.14In a recent paper15 we presented a further generalization of pseudocoloring for assigning false colors to depth level con tours of 3-D objects.In this Letter we propose a modification of the pseudoco loring speckle method9 for storing two images in a single re cording material in such a way that each recorded image has a different spatial modulation.Therefore, in a decoding step, on illuminating the processed píate with a white light source, there exist two wavelength valúes for which the spatial spectra of both encoded images are centered at the same spatial fre quency valué.Thus, an additive mixture of both images is produced with a pseudocoloring effect which depends on the two mentioned wavelengths.The scheme of the optical arrangement utilized in the en coding step is shown in Fig. 1.The image of a certain scene Ei is recorded by using as a lens pupil a double aperture of separation d This causes a spatial spectra shift Au i given by Aui = d i/XD, where A is the light wavelength, and D is the image distance.Afterward, a record of the image of another scene E2 is done in the same píate but now using as a lens pupil a double aperture of separation d2.Thus, the spatial spectra shift Au2 is given by Au2 = d2/XD.The decoding step is shown in Fig. 2. The developed píate H is illuminated with a collimated white light beam.Then, for a certain valué of the viewing angle 6, there exists a superposition of the £ 1 and E2 images in the colors given by the corresponding wavelengths Xx and X2 such that, d2/d\ = Ái/Á2 In this way, all the common parts of the E¡ and E2 images are pseudocolored by a color mixture which depends on Áj and Á2.So far no relation exists between E\ and E2.However, if Ei and E2 images are complementary (for object transpar en tes, the positive and negative images), the image resulting from the superposition is a pseudocolored one of either E¡ or E2.The corresponding contrast reversed image can be ob tained employing the method of Ref. 10, that is, recording the light scattered from the silver developed grains that form the original image transparency.An alternative approach con sists of employing the optical subtraction technique through Young's fringes modulated speckle.16In this case, a first record of the image of a speckle pattern is made.Then the image of the same speckle pattern is recorded but modulated by the object transparency E1, introducing a 7r-phase delay between both exposures.Thus, a contrast reversed image of E\ is obtained.Another case of interest arises when the image of E2, instead of being the complementary one of E1, is a modified versión of it.In this case, the third pseudocolor encodes the common

Fig. 2 .
Fig. 2. Scheme of the decoding step: 2 is a collimated white light beam; 2 i and 2 2 are the diffracted waves corresponding to Ai andX 2 .
regions of both scenes.Some experiments have been done using as Ei the isochromatic fringe pattern that appears in a loaded photoelastic model.The scene E2 corresponds to the complementary fringe pattern (obtained by rotating the polarization state of the incident light by 90°) but with a dif ferent loading condition.Therefore, in the decoding step, the resulting moire pattern appears in a pseudocolor given by the mixture of Ai and X2.Finally, another point of interest for this pseudocolor en coding consists of studying the spatial changes in one directionof an object transparency.In this case, the E2 image is a shifted versión of the E1 image.The amount of shifting de pends on the size of the finest details to be studied.Thus, the sign of the gray level change is coded by a spectral color which corresponds to Ai or X2.