10
White-light Sagnac interferometer for snapshot multispectral imaging Michael W. Kudenov, 1, * Matthew E. L. Jungwirth, 1 Eustace L. Dereniak, 1 and Grant R. Gerhart 2 1 College of Optical Science, The University of Arizona, 1630 E. University Boulevard, Tucson, Arizona 85721, USA 2 U.S. Army TACOM, 6501 E. 11 Mile Road, Warren, Michigan 48397, USA *Corresponding author: [email protected] Received 12 January 2010; revised 3 April 2010; accepted 15 June 2010; posted 22 June 2010 (Doc. ID 122273); published 15 July 2010 The theoretical and experimental demonstration of a multispectral Sagnac interferometer (MSI) is presented. The MSI was created by including two multiple-order blazed diffraction gratings in both arms of a standard polarization Sagnac interferometer (PSI). By introducing these high-order diffractive struc- tures, unique spectral passbands can be amplitude modulated onto coincident carrier frequencies. Extraction of the modulated multispectral images, corresponding to each passband, is accomplished within the Fourier domain. This yields a unique multispectral sensor capable of imaging all the pass- bands in a single snapshot. First, the theoretical operating principles of a PSI are discussed to provide a context for the MSI. This is followed by the theoretical and experimental development of the MSI, which is an extension of a dispersion-compensated PSI. Indoor and outdoor testing and validation of the MSI are performed by observing vegetation, demonstrating the ability of our experimental setup to detect four distinct spectral passbands. © 2010 Optical Society of America OCIS codes: 110.3175, 300.6190, 110.4234. 1. Introduction Multispectral imaging is a valuable technique in re- mote sensing and biomedical imaging [1,2]. Several sensing modalities exist for measuring multispectral data. One method involves time-sequentially ima- ging a scene through multiple filters placed in front of a standard lens and focal plane array (FPA) [3]. This produces a relatively simple and inexpensive in- strument. However, since each spectral passband is measured at a different time, error is introduced when viewing temporally dynamic scenes. Conse- quently, the multispectral passbands must be mea- sured in parallel. For some implementations, these parallel measurements are obtained with multiple cameras, lenses, and filters [4]. Another method, which is likely the most common use and implemen- tation of multispectral imaging, can be considered a division of FPA (DoF) approach. In DoF, different filters are placed onto the pixels of a CCD camera, similar to a standard consumer color camera [5]. For application-specific selection of the passbands, superpixelsare created that typically contain 2 × 2 pixels, where each pixel contains a unique spectral passband. This produces a compact four-band multi- spectral camera [6]. Additional snapshot techniques used to acquire multispectral data include the Image Replicating Imaging Spectrometer (IRIS) [7] and the Coded Aperture Snapshot Spectral Imager (CASSI) [8]. The IRIS enables snapshot multispectral imaging by in- corporating a generalized Lyot filter implemented with Wollaston prisms. These prisms create a mosaic of replicated two-dimensional (2D) images on the FPA, where each image corresponds to a different spectral passband. Consequently, through a trade- off between spatial and spectral resolution, the multispectral images can be measured directly with minimal postprocessing. Conversely, CASSI utilizes 0003-6935/10/214067-10$15.00/0 © 2010 Optical Society of America 20 July 2010 / Vol. 49, No. 21 / APPLIED OPTICS 4067

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Page 1: White-light Sagnac interferometer for snapshot multispectral imaging

White-light Sagnac interferometer for snapshotmultispectral imaging

Michael W. Kudenov,1,* Matthew E. L. Jungwirth,1 Eustace L. Dereniak,1

and Grant R. Gerhart2

1College of Optical Science, The University of Arizona, 1630 E. University Boulevard, Tucson, Arizona 85721, USA2U.S. Army TACOM, 6501 E. 11 Mile Road, Warren, Michigan 48397, USA

*Corresponding author: [email protected]

Received 12 January 2010; revised 3 April 2010; accepted 15 June 2010;posted 22 June 2010 (Doc. ID 122273); published 15 July 2010

The theoretical and experimental demonstration of a multispectral Sagnac interferometer (MSI) ispresented. The MSI was created by including two multiple-order blazed diffraction gratings in both armsof a standard polarization Sagnac interferometer (PSI). By introducing these high-order diffractive struc-tures, unique spectral passbands can be amplitude modulated onto coincident carrier frequencies.Extraction of the modulated multispectral images, corresponding to each passband, is accomplishedwithin the Fourier domain. This yields a unique multispectral sensor capable of imaging all the pass-bands in a single snapshot. First, the theoretical operating principles of a PSI are discussed to provide acontext for the MSI. This is followed by the theoretical and experimental development of the MSI, whichis an extension of a dispersion-compensated PSI. Indoor and outdoor testing and validation of the MSIare performed by observing vegetation, demonstrating the ability of our experimental setup to detect fourdistinct spectral passbands. © 2010 Optical Society of AmericaOCIS codes: 110.3175, 300.6190, 110.4234.

1. Introduction

Multispectral imaging is a valuable technique in re-mote sensing and biomedical imaging [1,2]. Severalsensing modalities exist for measuring multispectraldata. One method involves time-sequentially ima-ging a scene through multiple filters placed in frontof a standard lens and focal plane array (FPA) [3].This produces a relatively simple and inexpensive in-strument. However, since each spectral passband ismeasured at a different time, error is introducedwhen viewing temporally dynamic scenes. Conse-quently, the multispectral passbands must be mea-sured in parallel. For some implementations, theseparallel measurements are obtained with multiplecameras, lenses, and filters [4]. Another method,which is likely the most common use and implemen-tation of multispectral imaging, can be considered a

division of FPA (DoF) approach. In DoF, differentfilters are placed onto the pixels of a CCD camera,similar to a standard consumer color camera [5].For application-specific selection of the passbands,“superpixels” are created that typically contain 2 ×2 pixels, where each pixel contains a unique spectralpassband. This produces a compact four-band multi-spectral camera [6].

Additional snapshot techniques used to acquiremultispectral data include the Image ReplicatingImaging Spectrometer (IRIS) [7] and the CodedAperture Snapshot Spectral Imager (CASSI) [8]. TheIRIS enables snapshot multispectral imaging by in-corporating a generalized Lyot filter implementedwith Wollaston prisms. These prisms create a mosaicof replicated two-dimensional (2D) images on theFPA, where each image corresponds to a differentspectral passband. Consequently, through a trade-off between spatial and spectral resolution, themultispectral images can be measured directly withminimal postprocessing. Conversely, CASSI utilizes

0003-6935/10/214067-10$15.00/0© 2010 Optical Society of America

20 July 2010 / Vol. 49, No. 21 / APPLIED OPTICS 4067

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concepts from compressive sensing in order to pro-duce a snapshot imaging spectrometer that attemptsto minimize the trade-off between spatial and spec-tral resolution. By imaging the scene onto a codedaperture and relaying it through a dispersing prism,an optically compressed measurement of the spectraland spatial data cube ðx; y; λÞ is acquired. Algorithmsare then utilized to estimate the scene’s data cube.Consequently, extraction of a complete measurementrequires significantly more postprocessing time, orspecialized hardware and software, than the IRISsensor.

The multispectral imager described in this paperwasderived fromaSavart-plate snapshot imagingpo-larimeter, initially developed by Oka and Kaneko [9].Their interferometric technique demonstrated thatthe 2D Stokes parameters can be amplitude modu-lated onto spatial carrier frequencies. Reconstructionof the 2D state of polarization is accomplished byFourier filtering each carrier frequency component,and it enables snapshotmeasurement of the complete2D Stokes vector. Furthermore, it was later demon-strated that this technique could be incorporatedwitha Sagnac interferometer [10,11]. However, one limita-tion of this technique is that long-coherence-length(narrowband) illumination is required. For applica-tions requiring passive illumination, limitations areencountered due to the low signal-to-noise ratio ofthe measured data [12]. In an attempt to remedythe narrow-bandwidth limitation, the dispersion-compensated PSI (DCPSI) was developed [13].Through its mathematical development, it was dis-covered that a further extension of the carrierfrequency technique can be realized.Whereas the ori-ginal DCPSI amplitudemodulatesStokes parametersonto carrier frequencies, it is possible, with a minormodification, to modulate unique spectral passbandsonto carrier frequencies. This device creates a newsensor implementation for snapshot multispectralimaging, which we refer to herein as a multispectralSagnac interferometer (MSI).

The MSI is most analogous to the aforementionedIRIS sensor, except, instead of assembling a mosaicof multispectral 2D images in image space (x; y), thesystem creates a mosaic of spatial frequency infor-mation in the Fourier transform domain ðx−1; y−1Þ.This provides a postprocessing advantage with re-spect to CASSI, in that noniterative discrete fastFourier transforms enable extraction of the multi-spectral data. Additionally, like the IRIS or DoF sen-sors, it has a direct trade-off between spatial andspectral resolution; however, the MSI requires morepostprocessing. Rather, the primary advantage ofthe MSI is that spatial image registration betweenthe multispectral bands is inherent, since the dataare amplitude modulated onto coincident carrierfrequencies.

In this paper, we outline the theoretical and ex-perimental development of a an MSI. Section 2 pre-sents an overview of the details regarding the theoryof the DCPSI on which theMSI is based. In Section 3,

the multispectral variant of the DCPSI is introduced,and the key differences between the MSI and theDCPSI are explained. Last, in Section 4, the experi-mental validation and calibration of the MSI is de-tailed, followed by a brief conclusion of the work inSection 5.

2. Dispersion-Compensated PolarizationSagnac Interferometer

The MSI is based on a DCPSI that was originally im-plemented as an imaging polarimeter to detect linearpolarization [13]. The DCPSI contains two blazeddiffraction gratings in each arm of a standard PSI.These gratings enable the generation of white-lightinterference fringes, similar to Ref. [14]. The opticallayout of the DCPSI is depicted in Fig. 1. Light froman object is imaged to infinity by a collimating lens(f col) before entering the DCPSI, where the twoblazed diffraction gratings are denoted by G1 andG2. Incident light initially reflected by the wire-gridbeam splitter (WGBS) is diffracted by grating G2 intothe þ1 order. These dispersed rays propagatethrough the system to grating G1, where they are dif-fracted into the zero order, thus removing the diffrac-tion angle imposed by G2. The rays exit the systemparallel to the optical axis, but offset by a distanceproportional to λx, where G2 is some constant relatedto the DCPSI’s configuration. Conversely, the beamtransmitted by the WGBS is initially incident anddiffracted by G1 into its þ1 order, diffracted by G2into the zero order, and exits the system offset by−λx0. Assuming small angles and that G1 and G2are identical, the shear, SDCPSI, is

SDCPSI ¼ 2mλd

ðaþ bþ cÞ; ð1Þ

where a, b, and c represent the distances between G1and M1, M1 and M2, and M2 and G2, respectively.

Fig. 1. (Color online) The DCPSI contains two blazed diffractiongratings, G1 and G2, at each output of a wire-grid beam splitter(WGBS). These gratings generate a shear (SDCPSI) that is linearlyproportional to the wavelength. The rays from the object are onlydepicted on-axis for clarity.

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Light from the object is reimaged by the objectivelens (f obj) onto the FPA. From Ref. [13], the intensityon the FPA is

IDCPSIðxi;yiÞ¼12

Xd=λmin

m¼0

S00ðmÞ

þ12

Xd=λmin

m¼1

264S

02ðmÞcos

�2πf obj

2md ðaþbþcÞxi

�−S0

3ðmÞsin�

2πf obj

2md ðaþbþcÞxi

�375;

ð2Þwhere the intensity is a summation from orderm ¼ 0to the maximum order m ¼ ðd=λminÞ sinðπ=2Þ, whereλmin is the shortest wavelength observed by the opti-cal system. S0

0ðmÞ, S02ðmÞ, and S0

3ðmÞ are the Stokespolarization parameters, which are weighted by thediffraction efficiency (DE) of both gratings beforebeing integrated over wavelength, such that

S00ðmÞ ¼

Zλmax

λmin

DE2ðλ;mÞS0ðλÞdλ; ð3Þ

S02ðmÞ ¼

Zλmax

λmin

DE2ðλ;mÞS2ðλÞdλ; ð4Þ

S03ðmÞ ¼

Zλmax

λmin

DE2ðλ;mÞS3ðλÞdλ; ð5Þ

where λmin and λmax denote the minimum and max-imumwavelengths passed by the optical system. Thecarrier frequency is

UDCPSI ¼2mdf obj

ðaþ bþ cÞ: ð6Þ

Notable is the absence of any wavelength depen-dence (i.e., dispersion) in the carrier frequency,enabling white-light interference fringes to be gener-ated. This effect was studied in detail per Ref. [13].However, one aspect that was not analyzed was thecarrier frequency’s dependence on the diffraction or-der m. As is demonstrated in Section 3, this depen-dence can be exploited for multispectral imaging,and is accomplished by substituting the single-ordergratings of the DCPSI with multiple-order gratings.

3. Multispectral Variant of the DCPSI

The MSI is nearly identical to the DCPSI depicted inFig. 1, except that the blazed gratings, G1 and G2,now have multiple orders over the sensor’s operatingbandwidth. The difference between the gratings isdepicted in Fig. 2, where themultiple-order structurein Fig. 2(b) has a taller grove profile than its single-order counterpart, per Fig. 2(a). The theoretical DEfor an ideal blazed grating is calculated by

DEðλ;mÞ ¼ sinc2�m −OPD

λ

�; ð7Þ

withOPD ¼ hðn1 − n2Þ; ð8Þ

where h is the groove height, OPD is the optical pathdifference, and n1 and n2 are the indices of refractionfor the incident and blaze medium, respectively [15].

By using Eq. (7), the theoretical diffraction effi-ciency for both single- and multiple-order blazedgratings were calculated. Figure 3(a) demonstratesthe efficiency assuming a single-order grating withh1 ¼ 1:28 μm, n1 ¼ 1:0, and n2 ¼ 1:5. This yields afirst-order blaze wavelength of λ1 ¼ 640 nm, makingit the dominant order for wavelengths spanning425–1000 nm. Similarly, Fig. 3(b) depicts the effi-ciency for amultiple-order gratingwithh2 ¼ 4:07 μm,n1 ¼ 1:0, and n2 ¼ 1:5. This produces a first-orderblaze wavelength of λ1 ¼ 2035 nm, meaning thatthe higher orders (m ¼ 2 through 5) are distributedover 400–1000 nm. The blaze wavelength for thehigher orders, λm, is λ1=m, such that λ2 ¼ 1017 nm,λ3 ¼ 678 nm, λ4 ¼ 508 nm, and λ5 ¼ 407 nm.

As is demonstrated in Fig. 3(b), use of a multiple-order grating will generate several diffraction orderswithin the system, where each order (m) has a un-ique blaze wavelength (λm ¼ λ1=m) and spectral pass-band. Because of the carrier frequency’s dependenceon the order m, each spectral passband is modulatedonto a different spatial carrier frequency. Isolatingeach passband is accomplished by use of Fourier fil-tering, detailed in Subsection 4.A.

Besides including the multiple-order gratings, anadditional modification is needed to the system(Fig. 1) for measurements of the incident spectralcomponents; namely, a linear polarizer (LP) must beinserted in front of the WGBS. If the LP’s transmis-sion axis is oriented at 45° with respect to the x axis,then the Stokes vector incident on the WGBS is

Fig. 2. (a) Single-order blazed grating, where d is the period,n1 and n2 are the indices of refraction for the incident and blazemedium, respectively, and h1 is the depth (OPD∼ 1 wave).(b) Multiple-order blaze grating, where h2 is typically 3–10 timeslarger than h1.

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Page 4: White-light Sagnac interferometer for snapshot multispectral imaging

SWGBS¼12

26641 0 1 00 0 0 01 0 1 00 0 0 0

37752664S0;incS1;incS2;incS3;inc

3775¼

2664S0;incþS2;inc

0S0;incþS2;inc

0

3775; ð9Þ

where S0;inc, S1;inc, S2;inc, and S3;inc are the incidentStokes parameters on the LP and are implicitlydependent upon wavelength (λ). Substituting thevalues from SWGBS for the Stokes parameters presentin Eqs. (3)–(5) yields

S00ðmÞ ¼ S0

2ðmÞ

¼Zλmax

λmin

DE2ðλ;mÞ½S0;incðλÞ þ S2;incðλÞ�dλ; ð10Þ

S03ðmÞ ¼ 0: ð11Þ

Substituting the values for S00ðmÞ, S0

2ðmÞ, and S30ðmÞ

from Eqs. (10) and (11) yields the intensity patternfor the MSI as

IMSIðxi; yiÞ ¼12

XCe½λ1=λmin�

m¼0

½S″

0ðmÞ� þ 12

XCe½λ1=λmin�

m¼1

×�S″

0ðmÞ cos�2πf obj

2md

ðaþ bþ cÞxi��

;

ð12Þ

where

S″

0ðmÞ ¼Zλmax

λmin

DE2ðλ;mÞ½S0;incðλÞ þ S2;incðλÞ�dλ: ð13Þ

It should be noted that the dominant orders experi-mentally observed in the system correspond to theceiling (Ce) of λ1=λmin, where λ1 is the first-order blazewavelength of the diffraction grating. This changesthe maximum limit of the summation from d=λminto Ce½λ1=λmin� observed in Eq. (12). Furthermore, ifS2 is negligible in Eq. (13), then the modulated infor-mation corresponds to the incident S0 component(i.e., the total incident irradiance). Consequently, S2in the scene can cause error in the resulting multi-spectral data. To make the contribution of S2 negli-gible, scenes consisted of diffuse surfaces orientednormal to the optical system. Additionally, the illu-mination source was always oriented to preferen-tially reflect light in the S1 state. Further analysisof the error due to this effect is beyond the scopeof the current study. However, a design that is effec-tively polarization insensitive can be made by repla-cing the WGBS with a standard 50/50 beam splitter.

4. Experimental Verification of the MSI

To verify the operating principles of the MSI, the ex-perimental setup depicted in Fig. 4 was implemen-ted. Spatially and temporally incoherent light isconfigured by aiming the output of a fiber light,sourced by a tungsten–halogen lamp, onto a diffuseobject. A dichroic polymer linear polarizer at 45° isused to linearly polarize the light, such that the S0Stokes parameter can be modulated onto the carrierfrequencies [Eq. (9)]. The WGBS has a clear apertureof 21 mm and consists of antireflection coated alumi-num wires with a 144 nm period. Mirrors M1 and M2are 25 mm diameter 1/10 wave optical flats. A zoomlens, with focal length f c ¼ 70–300 mm, is used toimage the object plane to infinity, while an objectivelens, with focal length f obj ¼ 250 mm, is used for re-imaging. The diffraction gratings, G1 and G2, areblazed for a first-order (m ¼ 1) wavelength of λ1 ¼2035 nm on a BK7 substrate with a period d ¼69:9 μm. A depiction of the ideal theoretical diffrac-tion efficiencies for these grating parameters wasprovided previously in Fig. 3(b). Lastly, an IR block-ing filter is used to reject light with wavelengthslonger than 750 nm.

Another aspect of the setup involves the placementof an iris, set to a 4 mm diameter, between the zoomlens and the WGBS. During initial experimentation,it was observed that the fringe visibility decreased asthe aperture diameter increased. Consequently, a

Fig. 3. Diffraction efficiency (percent) for (a) a single-order blazedgrating in air with h1 ¼ 1:28 μmand n ¼ 1:5 and for (b) a multiple-order blazed grating in air with h2 ¼ 4:07 μm and n ¼ 1:5.

4070 APPLIED OPTICS / Vol. 49, No. 21 / 20 July 2010

Page 5: White-light Sagnac interferometer for snapshot multispectral imaging

test of the optical surface of the WGBS was per-formed, yielding three waves of astigmatism. There-fore, the iris effectively restricts the illuminated areaof the WGBS to reduce the effect of this aberration.

A. MSI Calibration

To calibrate and reconstruct data from the instru-ment, data processing is accomplished in the Fourierdomain [9]. Taking the Fourier transform of the in-tensity pattern Eq. (2) yields

F½IMSIðxi; yiÞ� ¼12

XCe½λ1=λmin�

m¼0

½S″

0ðmÞ�δðξ; ηÞ

þXCe½λ1=λmin�

m¼1

½S″

0ðmÞδðξ −UMSIðmÞ; ηÞ�

þXCe½λ1=λmin�

m¼1

½S″

0ðmÞδðξþUMSIðmÞ; ηÞ�;

ð14Þwith

UMSIðmÞ ¼ 2mdf obj

ðaþ bþ cÞ; ð15Þwhere δ is the Dirac-delta function and ξ, η are theFourier transform variables along xi and yi, respec-tively. Extraction of the Stokes parameters is accom-plished by taking an inverse Fourier transform of thefiltered carrier frequencies, or “channels.” Filtrationand inverse Fourier transformation of the δðξ; ηÞ(channel C0) and δðξþUMSIðmÞ; ηÞ (channel C1ðmÞ)components yields

F−1½C0� ¼12

XCe½λ1=λmin�

m¼0

S″

0ðmÞ; ð16Þ

F−1½C1ðmÞ� ¼ 14S″

0ðmÞ expðj2πUMSIðmÞxiÞ: ð17ÞExtraction of the spectral passbands, S″

0ðmÞ, requiresdemodulation of the exponential phase factor. Since

knowledge of the phase is irrelevant for reconstruc-tion (S″

0ðmÞ is always positive), demodulation isachieved by taking the absolute value of the filteredchannel:

jF−1½C1ðmÞ�j ¼ 14S″

0ðmÞ: ð18ÞA 150 × 150 pixel section, from a raw image of a uni-formly illuminated high-reflectance white diffuser, isdepicted in Fig. 5(a). The Fourier transformation ofthe image is depicted in Fig. 5(b), demonstrating thepresence of a center burst and four carrier frequen-cies, along with their conjugates.

It should be noted that the image and Fourierdomain are rotated by approximately 31°. Whilechanging the shearing direction (i.e., rotating thegratings) will rotate the fringes without rotating theimage, doing so was not feasible in our experimentalsetup due to optomechanical limitations. This rota-tion was induced by rotating the camera to reducereconstruction artifacts caused by aliasing betweenadjacent channels [16]. Specifically, it reduces alias-ing caused by the large sidelobes of the carrierfrequency components along ξ0 and η0, originatingfrom the nature of the 2D sinc function.

This highlights an additional issue regarding theway the channels are multiplexed in the Fourier do-main, namely, how theneighboring, non-band-limitedchannels interact with each other. For instance, if thescene contains spatial frequencies that exceed thehalf-separation of any two neighboring carrier fre-quencies, then one channel’s modulated informationwill interfere with the neighboring channel. This in-terference, or crosstalk, will then appear in the recon-struction as a false signal. Crosstalk of this naturebetween neighboring channels is a concern, and hasbeen addressed in channeled spectropolarimetryper Ref. [16]. While the same technique used inRef. [16] has potential for the MSI, an analysis hasyet to be conducted. Further discussion of error re-lated to this phenomenon is beyond the scope of thecurrent paper.

B. Spatial Resolution Trade Space

Themultiplexing of the spectral passbands in the spa-tial frequency domain results in a spatial resolutiontrade-off. Assuming a band-limited single-lens ima-ging system where the Nyquist sampling frequency(ξny) matches the cutoff frequency of the optics(ξoptics ¼ 1=λðf =#Þ), then the maximum observablespatial frequency in the image is approximately equalto ξny, where ξny ¼ ξoptics. Here, f =# ¼ f =D, where f isthe focal length andD is the diameter of the objectivelens. Introducing carrier frequency channels into theoptical system, as is done with the MSI, limits themaximum spatial resolution by an amount that isproportional to the number of carriers. This is illu-strated in one dimension (1D) in Fig. 6, where it is as-sumed that the carrier frequencies are distributeduniformly between ξ ¼ 0 and ξ ¼ ξny. Consequently,given a total of M channels in 1D, the maximumspatial resolution in each channel is equal to

Fig. 4. (Color online) Experimental setup of the MSI with a, b,and c set to 13.3, 51.4 , and 19:1 mm, respectively. An IR blockingfilter keeps wavelengths >750 nm from entering the system.

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Page 6: White-light Sagnac interferometer for snapshot multispectral imaging

ξc ¼ ξny=ð2M þ 1Þ: ð19Þ

This result canbe readily extended to twodimensions,where ηc ¼ ξc, such that the channels are symmetric.

The MSI’s spatial resolution trade-off is moresevere than observed in other techniques becausethe carrier frequencies are distributed exclusivelyalong 1D of the 2D Fourier transform plane. For in-stance, in the case of the DoF approach discussed inSection 1, new multispectral passbands can be intro-duced in both spatial dimensions, x and y. Assumingthat the multispectral filters are positioned on pixelswithin a square superpixel (e.g., 2 × 2, 3 × 3, 4 × 4 for4, 9, 16, etc. spectral bands), then the maximum spa-tial resolution in each passband follows

ξc;DoF ¼ ξny=ffiffiffiffiffiM

p: ð20Þ

The normalized cutoff frequency versus number ofspectral bands (M) for both of these techniques isdepicted in Fig. 7, and the ratio between the cutofffrequency of the MSI to that of the DoF is

ξcξc;DoF

¼ffiffiffiffiffiM

p

ð2M þ 1Þ : ð21Þ

Hence, the spatial resolution of the MSI is signifi-cantly lower. From this comparison, it is envisionedthat the spatial resolution of the MSI would benefitsignificantly if the carrier frequencies could be posi-tioned in both ξ and η. Future work will focus onmitigating this performance barrier.

C. Spectral Passbands

The operating principles discussed previously indi-cate that the order-dependent carrier frequenciesare visible only when the sensor is illuminated withlight at specific wavelengths. To validate that thepassbands correspond to a given diffraction orderand to measure their relative spectral responsivity,the experimental setup depicted in Fig. 8 was imple-mented. A monochromator, sourced by a tungsten–halogen lamp, is used to generate narrow-bandillumination. The exit slit of the monochromatorwas then imaged by the MSI. Using a focal lengthof 70 mm for the collimating lens ensured that the

edges of the slit were not visible, such that fringeswere present across the entire field of view.

To calculate the relative responsivity, a calibratedspectrometer was used to characterize the output ofthe monochromator. For radiometric calibrationof the spectrometer, a National Institute of Stan-dards and Technology (NIST) -traceable tungsten–halogen lamp was utilized. This enabled the outputof the monochromator to be characterized in radio-metric units, as illustrated in Fig. 9(a). The responseof each channel was then measured by Eq. (18) as themonochromator was scanned from 400 to 800 nm in10 nm increments. An average of approximately100 × 100 pixels from the center of the image wasthen calculated and normalized by the measured re-sponse of the monochromator. The relative responsefor each channel is depicted in Fig. 9(b) alongside thecombined relative response of the FPA and the IRblocking filter; the response of the other optical ele-ments in the system (dichroic LPs, lenses, beam split-ter, etc.) are not included in this curve. Despite this,the relative spectral response of each channel gener-ally follows the trends of the FPA and the filter.

D. Multispectral Reconstruction Validation

To validate the accuracy of the multispectral recon-structions from the MSI, experiments were per-formed that compared MSI results to those of thecalibrated spectrometer (U2S). Specifically, reflec-tance measurements of a healthy and an unhealthytree leaf, taken from an Acacia crassifolia, were mea-sured with the MSI and the U2S. These samples en-abled the characteristic reflectance features ofchlorophyll, which is present in the healthy leaf butlargely absent in the unhealthy leaf, to be used for

Fig. 5. (Color online) (a) Raw image of a uniformly illuminated diffuser demonstrating the fringe pattern in the MSI. (b) Fouriertransformation of the raw image data in (a). Four carrier frequencies are observed, corresponding to m ¼ 2; 3; 4, and 5.

Fig. 6. (Color online) Spatial resolution trade space in 1D. M isthe total number of carrier frequencies in the system and the chan-nels are assumed to be uniformly spaced from ξ ¼ 0 to ξ ¼ ξny.

4072 APPLIED OPTICS / Vol. 49, No. 21 / 20 July 2010

Page 7: White-light Sagnac interferometer for snapshot multispectral imaging

the comparison [17,18]. Reflectance measurementsfrom the U2S were matched to the MSI’s passbandsby

RU2SðmÞ ¼R800400 Ileaf ðλÞTMSIðm; λÞdλR

800400 IdiffuserðλÞTMSIðm; λÞdλ ; ð22Þ

where λ is the wavelength in nanometers, TMSI is therelative response of the passband of order m perFig. 9, and Ileaf and Idiffuser is the measured irradi-ance of the leaf and of a white diffuse reflectancepanel, respectively. The reflectance measurementfrom the MSI is calculated by

RMSIavgðmÞ ¼

Pl

PnIleaf ðl;n;mÞ

Pl

PnIdiffuserðl;n;mÞ ; ð23Þ

where l and n are integers representing pixels in xand y, respectively. Here, the inner 90% of the imageis used in the summation to average over a large area(∼25 mm2) of the sample.

The two experimental configurations for the reflec-tance measurements are depicted in Fig. 10, where

the illumination and observation geometries areidentical for both the U2S and MSI measurements..Because the U2S is nonimaging, defocus was imple-mented in the MSI to obtain reflectance measure-ments that better averaged the spatial details overthe observed area of the sample. The measuredreflectance from the U2S and the MSI yields theresults portrayed in Fig. 11.

The root mean square (RMS) error between theMSI and U2S data is calculated by

ε ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14

X5m¼2

ðRU2SðmÞ − RMSIðmÞÞ2vuut : ð24Þ

The RMS error for the healthy and unhealthy leafsamples yields 0.0115 and 0.0117, respectively.

Fig. 7. Normalized cutoff frequency versus the number of pass-bands present in the system. The DoF approach maintains higherspatial resolution since it utilizes both spatial dimensions x and y.

Fig. 8. (Color online) monochromator configuration for sendinglight into the MSI for verification of the fringe visibility. The band-width of the light exiting the monochromator was approximately10:5 nm using a 3 mm exit slit.

Fig. 9. (a) Measured monochromator output in W=m2. (b) Mea-sured relative spectral response of the different passbands withinthe MSI. Also included (solid light-gray line) is the relative re-sponse of the FPA multiplied by the measured transmission ofthe IR blocking filter.

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Consequently, the error in the spectral reconstruc-tion is of the order of 1:2% between the U2S andthe MSI.

E. Multispectral Spatial Reconstructions

To demonstrate the imaging capabilities of the MSI,both the healthy and unhealthy leaf samples wereimaged simultaneously with the sensor. A relativereflectance calculation was performed using

RMSIðl;n;mÞ ¼ Ileaf ðl;n;mÞIdiffuserðl;n;mÞ : ð25Þ

The band-integrated image, taken by demodulatingthe first component in Eq. (14), is depicted inFig. 12(a), and an image from each band (or order)is depicted in Figs. 12(b)–12(e). An additional calcu-lation, demonstrating the applicability of the results,is the normalized difference vegetation index (NDVI)provided in Fig. 13. The NDVI is calculated by

NDVIðl;nÞ ¼ Iðl;n; 2Þ − Iðl;n; 3ÞIðl;n; 2Þ þ Iðl;n; 3Þ : ð26Þ

If the leaf has a high chlorophyll content, order 3 willcontain less reflected energy than order 2, yielding ahigh NDVI. Conversely, if the leaf has little to nochlorophyll, order 3 will have more energy comparedto the previous case, yielding a lower NDVI. This is

observed in the NDVI image, where the upper leftportion of the scene (quadrant 2) contains the un-healthy leaf, while the lower right region (quadrant4) contains the healthy leaf.

Fig. 10. (a) System setup for U2S reflectance measurements. Theincidence angle of the source (tungsten–halogen lamp) was ap-proximately 37° with respect to the surface normal of the sample.(b) System setup for MSI reflectance measurements. The sample isdefocused to average the spatial details over the observed area.

Fig. 11. Relative reflectance of the healthy and unhealthy leaves,normalized to the irradiance in order 2, as measured with the MSIand U2S. The healthy vegetation experiences more absorption inorder 3 due to the presence of chlorophyll.

Fig. 12. Relative reflectance images of a healthy and unhealthyleaf. (a) Band-integrated image (m ¼ 0). (b)–(e) Images from ordersm ¼ 2 through m ¼ 5.

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Since the near-infrared (NIR) band (order 2)spans wavelengths from 685 to 755 nm full widthat half-maximum, order 2 integrates across the vege-tative red edge. Consequently, the NDVI calculationperformed here is more analogous to the calculationperformed in Ref. [18]. Gitelson demonstrates thatan NDVI calculation performed between two bands,located at 525–555 nm and 695–705 nm, can be lin-early related to the vegetation’s chlorophyll concen-tration. This is in opposition to an NDVI calculationperformed with Landsat 7’s band 4 (780–900 nm)with ETM+ [19]. The MSI and Landsat 7 NIR bandsare overlaid onto the measured relative reflectancefor Acacia crassifolia per Fig. 14. Ultimately, the sys-tem can be designed to incorporate the appropriatebands to perform an NDVI measurement analogousto Landsat 7, or other passbands, given the require-ments of a specific application.

Application of the MSI’s multispectral imagingcapabilities is further illustrated in outdoor test re-

sults. A photo of an outdoor scene, taken with a colordigital camera, is depicted in Fig. 15. The recon-structed spectrally band-integrated image from theMSI is depicted in Fig. 16(a), while an image fromeach band is depicted in Figs. 16(b)–16(e). The NDVIimage is portrayed in Fig. 17. This demonstrates thesensor’s ability to operate in a remote sensingcapacity.

Fig. 13. NDVI image of the leaf, calculated using Eq. (26). A high-er signal is associated with the presence of chlorophyll, as illu-strated with the healthy leaf in the lower right (quadrant 4) ofthe image.

Fig. 14. Relative reflectance of a healthy leaf from Acacia crassi-folia measured with the U2S (solid black curve). The “red edge”begins at 700 nm and peaks at approximately 765 nm. The Land-sat 7 (band 4) spans 775–900 nm (dark dashed curve) while theMSI spans 685–755 nm (gray dotted curve).

Fig. 15. (Color online) Photo taken with a standard color digitalcamera of an outdoor scene. Healthy vegetation is present, in ad-dition to brick, concrete, and a relatively clear sky.

Fig. 16. Relative reflectance images of an outdoor scene. (a)Band-integrated image (m ¼ 0). (b)–(e) Images from orders m ¼2 through m ¼ 5.

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5. Conclusion

The development, calibration, validation, and experi-mental results of a different kind of multispectral im-ager have been presented. The MSI was created bymodifying the DCPSI originally used for snapshotpolarimetric imaging. By replacing the single-orderblazed diffraction gratings with multiple-order grat-ings, each diffraction order’s distinctive spectralresponse is encoded onto unique spatial carrier fre-quencies. The multispectral reconstruction’s RMSerror, as calculated between theMSI and a calibratedspectrometer, was measured to be approximately1:2% for both a healthy and unhealthy leaf fromAcacia crassifolia. The imaging potential of the MSIwas demonstrated through NDVI measurements ofthe same leaf samples, in addition to outdoor mea-surements of scenes containing vegetation. Ulti-mately, the sensor compromises spatial resolutionto capture the spectral components. However, it is en-visioned that, with further optimization, the spatialand spectral resolution can be further maximized byutilizing more of the spatial frequency domain.

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