Interferometric supermultispectral imagingKazuyoshi Itoh, Takashi Inou - PDF document

Interferometric supermultispectral imagingKazuyoshi Itoh, Takashi Inoue, Tetsuo Yoshida, and Yoshiki IchiokaMultispectral images that have spectral resolution as high as their spatial resolution may be called supermul-tispectral images or simply spectral images. The principle of an interferometric method of supermultispec-tral imaging has been developed by unifying the principles of incoherent holography and Fourier spectrosco-py. The method is expected to inherit the multiplex and throughput advantages from Fourier spectroscopy.We present our first experimental results of the method applied to show that this spectral density function is the desiredspectral image of the light source illuminating thefield.Let us assume that we observe an incoherent object,which may be a light source or a scatterer illuminatedwith spatially incoherent light. It is only required thatthe spherical wavelets emanating from the object sur-face be statistically independent of each other. Theobservation is assumed to be carried out in a sufficient-ly small region (or an appropriate positive lens isplaced between the object and observer) so that thefield within the observation area can be approximatedby a superposition of plane waves with uncorrelatedamplitudes. In this situation, the optical random fieldwithin the observation area can be approximated to behomogeneous.For homogeneous random fields [VA(i')] the correla-tion function or, equivalently, the spatial coherencefunction is given byll1"2JA(r) = (VA(r'+ )V(')), (1)where r' and r are 3-D position vectors, sharp bracketsdenote the statistical or time average, and the asteriskdenotes the complex conjugate. Suffix A implies thatJA(i) and VA(r') are dependent on the location of theobserver. The correlation function of a homogeneousrandom field is expressible in terms of the spectraldensity GA(k) (Refs. 9 and 10):JA(r) = JK GA(k) exp(ik r)dk, (2)where i = 2ET and K is the region of the wave vector inwhich the spectral density is nonzero. The spatialcoherence function, JA(r), is the 3-D Fourier transform(FT) of the spectral density GA(k).It is important to note that, although the spatialcoherence function is idealized here to be shift invari-ant (homogeneous field), it is only true within theobservation area and the coherence function is actuallydependent on the location of the observation area.This implies that, if an observer changes his or herpoint of view (area of observation), the observed coher-ence function or the spectral density of the wave fieldgenerated by a 3-D light source appears to be different.For simplicity we accept that quasihomogeneous fieldthat is dependent on its location of observation area.This concept of local coherence has an analogy withthose of the isoplanatism in the image analysis13andthe quasihomogeneity in the coherence theory.14Let us see the relationship between the object andthe spectral density; then we can infer the structure orthe geometry of the light source from knowledge of thespectral density function of the wave field. Since thehomogeneous random fields are expressible in terms oflinear superposition of plane waves with random am-plitudes, the wave field within the observation area canbe written asVA(r) = aA(k) exp(i * r)di, (3)where aA(k) is the random amplitude of a plane wave0/20/2XFig. 1. Rotational shear volume interferometer. Rotational shearis introduced by tilting the two right angle prisms around the opticalaxis. Longitudinal shear is created by shifting one of the prismsalong the optical axis.associated with the wave vector k. From the defini-tion [Eq. (1)], the spatial coherence function is given byJA(rlr2) = (VA(rl) VA(r2))= JK (aA(kl)aA(ki)) exp[i(kl -k2)r+/2 + ( + k2) r/2]dldk2, (4)where r+ = r + r2, r = r- r2.By comparison of Eqs.(2) and (4) we see that the following equation musthold:(aA(k)a(kS)) = GA(il)(l -k2) (5)From Eq. (5) we see that the spectral density for aparticular wave vector is directly related to the meansquares of the random amplitudes of plane waves asso-ciated with the particular wave vector.Now it is clear what the spectral density means.Since the light source is spatially incoherent in thepresent situation, the plane wave emanating from eachpoint in the light source has its identical angle andconversely the angle of the wave vector specifies onepoint on the object surface. The length of the wavevector specifies the wavelength of radiation. Thus,the density function describes the spatial and spectraldistribution of radiated energy or emanating powerfrom the light source.111. Rotational Shear Volume InterferometerA special instrument called the rotational shear vol-ume interferometer156has been proposed for themeasurement of 3-D spatial coherence. Although itwas already explained elsewhere, we repeat the illus-tration for completeness by referring to Fig. 1. Anincident light beam is split by a beam splitter (BS).The split beams are then reflected by the right angleprisms. Both wavefronts are reversed left-to-right bythe prisms and then superimposed again on the BS.1626 APPLIED OPTICS / Vol. 29, No. 11 / 10April 1990 Since the two prisms are rotated around the opticalaxis, a rotational type lateral shear is created. Thelongitudinal shear is then introduced by moving one ofthe prisms along the optical axis. We take a Cartesiancoordinate system whose z-axis coincides with the op-tical axis and whose x-y plane corresponds to the ob-servation plane. Let e, e, and e denote the unitvectors along the respective axes. Now, let the rota-tion angle of the two prisms be 0/2 and let the longitu-dinal path difference be signified by 2d. Then, theenergy or power generated at the detector elementplaced at a position (xy,O) is readily calculated to beproportional toIA(x,y,d) = JA( ) + JA(io) + (const bias), (6)wherero = 2 sinO(-yex + xiY) + 2dez.Thus, the 3-D intensity pattern, IA(x,y,d), gives thereal part of the 3-D spatial coherence function, JA(ro),along with the bias term. As is clear, we can controlthe lateral scale of the pattern by changing the shearangle 0.17IV. Resolution Limitations and Field of View'5Limitations on the resolution and field of view arise,respectively, from the aperture size and detector spac-ing. Let us consider a specific interferometer whoseentrance aperture is a rectangle of size L1X L, andwhose shear angle is 0. We assume that the interfer-ence pattern is detected by a space-filling array of N XN rectangular detectors of size 11 X 11. The maximumlongitudinal path difference is set at 2L3and parti-tioned into N3intervals of length 213. Thus, JA(r) ismeasured over the volume of 2L, sin0 X 2L, sin0 X 2L3and the unit cell of the sampling lattice is 21, sinG X 21sin0 X 213.The resolution limits are determined by the finiteaperture size and the shear angle:M., = Y = 7r/(L1sinO), (7)AZ = r/L3(8)The finite interval of data sampling limits the size ofobjects that can be reconstructed without aliasing:Ak,, = Aky = 7r/(11 sinO), (9)Ak, = 7r/13.(10)Another expression for the limitations on the objectsize is the field of view. The angular field of view islimited byAOk = X/(211sinO) (11)for small AO.V. Noise Characteristics15The major factors that determine the noise charac-teristics of the present system are the signal quantiza-tion noise and the photon noise. The photon noise isbriefly discussed.Let us assume that a rectangular shape of uniformmonochromatic object is analyzed by a rotationalshear volume interferometer system with a rectangularaperture and a field of view under the following threesituations. We only take account of signal photonsand neglect dark counts and assume a quantum effi-ciency of unity.1. Observation with Very Low Magnification(Point Object)In this case, the object is assumed to be unresolvableand the number of average signal photons per a resolu-tion element, say N8, equals the total number of inci-dent photons, say Nt, and all these photons are regis-tered at one resolution element in the spectral imagespace. Thus, from the previous analysis2we have forthe SNR(12)pi = FT2.2. Observation with Moderate Magnification(Small Object with Dark Background)Since the object is assumed here to be spatiallyresolved, it occupies a certain number of resolutionelements in the spectral image space. Let exactly oneresolution element be occupied by the object when theshear angle is 01 and no more substances are present inthe field of view. Then, the number of occupied ele-ments becomes (sin0/sin0l)2times as large when theshear angle is 0(0 � 0). Thus N becomes smaller forlarge 0 and the SNR is given byP2 = p1(sinO1/sinO)2.(13)3. Observation with Very High Magnification(Large Object)In this case, we might have to place a field stop andlimit the maximum field of view to prevent aliasingand the total number of detected photons becomes lessthan Nt defined previously. Let 02 denote the shearangle when the object exactly covers the maximumfield of view. Then the SNR for this situation is givenbyP3 = pI[sin2O1/(sinO sino2)].(14)VI. Experimental Demonstration6The spatial coherence functions were measured byusing a rotational shear volume interferometerequipped with an image detector (a CCD camera or anIR vidicon camera) connected to an imager processor.A picture of the interferometer in front of an IR vid-icon camera is shown in Fig. 2. The shear angle was1.2°. The camera lens is focused on the apexes of theright angle prisms. In principle, this lens is dispens-able. It allows use of imperfect prisms and removesthe diffraction effects at the apexes. A series of 64 or128 interference patterns (64 X 64 pixels) were takenby successively increasing the path difference of theinterferometer. The path difference was controlled10 April 1990 / Vol. 29, No. 11 / APPLIED OPTICS 1627 Fig. 2. Picture of the rotational shear volume interferometer. Thetwo right angle prisms are rotated by 0.60 around the optical axis.The IR vidicon camera (at the left of the interferometer) is focusedon the apexes of the prisms.Fig. 3. Object is a live flower. The central gray region is red and issurrounded by a white area. Green leaves are visible near the flower.by a piezoelectric translator. We obtained the spec-tral images of size 64 X 64 X 64 or 64 X 64 X 128 fromthese interferometric data via 3-D Fourier transforma-tion. The size of the data is primarily limited by thecomputational capacity of our computer (SUN3/160C-8) and the signal-to-noise ratio of video signals. Thepattern [IA(xy,d)] at each path difference (d) was av-eraged digitally (64-255 frames) to enhance the signal-to-noise ratio of digitized video signals. This processof real time signal averaging enabled us to get digitizedvideo signals that have gray scale resolution of morethan that of the video A-D converter (8 bits) and toreconstruct high resolution spectral images of reason-able quality.The object in the first experiment was a live flowerilluminated by a tungsten lamp. A picture of thisobject is shown in Fig. 3. The flower is composed offive petals whose central part is red (dark area in thepicture). A portion of a green leaf is barely illuminat-ed to be detectable. A part of the series of measuredinterference patterns is displayed in Fig. 4. Thesepatterns are the cross sections of the real part of 3-L)Fig. 4. Series of interference patterns detected by an image sensor(CCD camera). Each pattern has a different longitudinal shear.The image sensor is focused on the detection plane that includes theapexes of the prisms.Fig. 5. Slices of the spectral image of the live flower shown in Fig. 3.Cross sections perpendicular to the k axis are shown. The numberattached to each cross section denotes the wavelength correspondingto the k component.FT of the spectral density associated with the object.The spatial variations of brightness within one of thecross sections are mainly caused by the spatial struc-ture of the object (incoherent hologram), while thevariations from section to section at a particular posi-tion in the sections are mainly due to the spectralstructure of the object (interferogram of Fourier spec-troscopy). The reconstructed spectral image is partlyshown in Fig. 5. Displayed images are the cross sec-tions of the spectral image G(k) perpendicular to the k,axis. Since the image is contracted along the k axisdue to the anisotropy of sampling intervals, the crosssections show approximately the image of constantwavelength. The corresponding wavelengths are dis-played in the pictures in nanometers. The spectralresolution on the k axis is 27 nm at 550 nm. Thecentral bright spot that appears in each section wasidentified with the scintillation noise: it was causedby intensity fluctuations of the service light illuminat-ing the object. Correction is possible in principle.1628 APPLIED OPTICS / Vol. 29, No. 11 / 10 April 1990 titaUCFig. 6. Object is a postage stamp.654.327100[1400112500 17900 23200 28600[800] [560] [430] [350]WAVE NUMBER (Cm-')[WAVELENGTH (nm)]Fig. 8. Spectra extracted from the spectral image of the postagestamp shown in Fig. 7. Each spectrum stands for a particularlocation in the object of a distinct color.Fig. 7. Slices of the spectral image of the postage stamp shown inFig. 6. Cross sections perpendicular to the k axis are shown. Thenumber attached to each cross section denotes the wavelength corre-sponding to the k, component.Figure 6 shows the second object (a postage stamp).It has white rims at all the sides and orange surround-ings and two aspects of the earth composed of the fivecontinents finished in green and the seven seas in blue.The reconstructed spectral image is partly shown inFig. 7. The central pixel that includes the scintillationnoise is replaced by a nearby pixel. The spectral reso-lution is 14 nm at 550 nm. Referring to the wavelengthgiven to each cross section, one can realize the colordistributions described above. Figure 8 shows thespectral at several particular positions in the crosssections (or spectra associated with several wave vec-tors that have particular directions). The widest spec-trum that forms the envelope of the other spectra is thespectrum reconstructed at the white rim. The wavystructure in the envelope seems to be some artifact; theorigin is not yet clear.We applied our method to a thermal object that isself-radiating in the infrared region. The object was acouple of solder iron tips that were supplied with dif-ferent voltages (110 and 95 V). The CCD camera fordetecting the interference signal was replaced by aninfrared vidicon with a spectral response out to 1800nm. A picture of the object is shown in Fig. 9. Thetwo solder tips are in opposite directions. The spec-tral image of this object in the infrared region wasFig. 9. Object under illumination of visible light (solder iron tip).Fig. 10. Slice of an infrared spectral image of the solder iron tipshown in Fig. 9. Cross sections perpendicular to the k axis areshown. The number attached to each cross section denotes thewavelength corresponding to the k, component.measured without lighting. Part of the reconstructedspectral image is shown in Fig. 10. The spectral reso-lution is 43 nm at 1400 nm. Since the spectral sensitiv-ity of the vidicon tube is not sufficient, we cannotdistinguish the spectral differences between the spec-10April 1990 / Vol. 29, No. 11 / APPLIED OPTICS 1629 tral image cross sections of the two tips caused by thesurface temperature difference. However, it is provedthat the present method can be applied to objects thatare self-radiating in the image of 1.1-1.8 Azm. Thesmall horizontal smear at the center of each cross sec-tion is due to electromagnetic noise mixed into thevideo signal.VII. ConclusionsWe have presented a simplified theory of spectralimaging and a brief analysis of the signal-to-noise ra-tios and demonstrated its feasibility of spectral imageacquisition with reasonable signal-to-noise ratios.Since the principle of this method is simple, we expectthat it will have wide use. The method inherits mostof the advantages given to Fourier spectroscopy be-cause it is simply an extension of Fourier spectroscopyto three dimensions. Applications in the infrared re-gions are recommended because of the Fellgett advan-tage. A person who uses a high density detector arraywith a small shear angle will enhance the Jaquinotadvantage of this method.This work was presented in part at ICO-14, Quebec,Canada, 24-28 Aug. 1987, and at the Topical Meetingon Space Optics for Astrophysics and Earth and Plane-tary Remote Sensing, 27-29 Sept. 1988, North Fal-mouth, Cape Cod, MA. This work was supported by agrant-in-aid for Special Project Research onLightwave Sensing, contract 63102003, by the Minis-try of Education, Science, and Culture, Japan.References1. K. Itoh and Y. Ohtsuka, "Interferometric Spectral Imaging," inConference Digest of ICO-13 on Optics in Modern Science andTechnology (Organization Committee of ICO-13, Sapporo, Ja-pan, 1984), pp. 600-601.2. K. Itoh and Y. Ohtsuka, "Fourier-Transform Spectral Imaging:Retrieval of Source Information from 3-D Spatial Coherence,"J. Opt. Soc. Am. A 3, 94-99 (1986).3. G. Vane, Ed., imaging Spectroscopy II, Proc. Soc. Photo-Opt.Instrum. Eng. 834 (1988).4. J. M. Mariotti and S. T. Ridgway, "Double Fourier Spatio-Spectral Interferometry: Combining High Spectral and HighSpatial Resolution in the Near Infrared," Astron. Astrophys.195, 350-363 (1988).5. L. Mertz, Transformations in Optics (Wiley, New York, 1965),Chap. 4.6. 0. Bryngdahl and A. W. Lohmann, "Variable Magnification inIncoherent Holography," Appl. Opt. 9, 231-232 (1970).7. Ref. 5, Chaps. 1 and 2.8. G. A. Vanasse and H. Sakai, "Fourier Spectroscopy," Prog. Opt.6, 261-327 (1967).9. A. M. Yaglom, An Introduction to the Theory of StationaryRandom Functions (Prentice-Hall, Englewood Cliffs, NJ, 1962),Chap. 3.10. S. W. Lang and T. L. Marzetta, "Image Spectral Estimation," inDigital Image Processing, M. P. Ekstrom, Ed. (Academic, Or-lando, FL, 1984), Chap. 6.11. M. Born and E. Wolf, Principles of Optics (Pergamon, London,1970), Chap. 10.12. J. W. Goodman, Statistical Optics (Wiley, New York, 1985),Chap. 5.13. J. W. Goodman, Introduction to Fourier Optics (Wiley, NewYork, 1968), Chap. 2.14. W. H. Carter and E. Wolf, "Coherence and Radiometry withQuasihomogeneous Planar Sources," J. Opt. Soc. Am. 67, 785-796 (1977).15. K. Itoh, T. Inoue, and Y. Ichioka, "Fourier-Transform SpectralImaging with Variable Magnification," in Proceedings, Four-teenth Congress of ICO, H. H. Arsenault, Ed. (Publication Com-mittee of ICO, Quebec, Canada, 1987), pp. 519-520.16. K. Itoh, T. Inoue, and Y. Ichioka, "Interferometric SpectralImaging in the Visible and Near-Infrared Regions," in Techni-cal Digest, Topical Meeting on Space Optics for Astronomyand Earth and Planetary Remote Sensing (Optical Society ofAmerica, Washington, DC, 1988), pp. 76-78.17. C. Roddier and F. Roddier, "Imaging with a Coherence Interfer-ometer in Optical Astronomy," in Image Formation from Co-herence Functions in Astronomy, C. van Schooneveld, Ed. (Rei-del, Dordrecht, The Netherlands, 1979), pp. 175-178.1630 APPLIED OPTICS / Vol. 29, No. 11 / 10 April 1990