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Fizeau Fourier transform imaging spectroscopy yields both spatial and spectral information about an object. Spectral information, however, is not obtained for a finite area of low spatial frequencies. A nonlinear reconstruction algorithm based on a gray-world approximation is presented. Reconstruction results from simulated data agree well with ideal Michelson interferometer-based spectral imagery. This result implies that segmented-aperture telescopes and multiple telescope arrays designed for conventional imaging can be used to gather useful spectral data through Fizeau FTIS without the need for additional hardware.
©2008 Optical Society of America
1. Introduction
Imaging spectroscopy is the process of collecting of both spatial and spectral information for a scene. Several grating- or prism-based imaging spectrometers have been deployed for remote sensing applications. An alternative to dispersive spectroscopy is the Fourier transform technique [1]. The conventional design for such a system is based on a Michelson imaging interferometer [2-4]. Panchromatic intensity measurements are made with various optical path differences (OPDs) between the arms of the interferometer, and spectral information is recovered through post-processing. Alternatively, Fourier transform imaging spectroscopy (FTIS) can be performed with a Fizeau imaging interferometer by introducing OPDs between subapertures of a multiple telescope array (MTA) or a segmented-aperture system [5-7]. Kendrick et al. [6] demonstrated the Fizeau FTIS concept with a two-telescope array system. An advantage of Fizeau FTIS is that it can be implemented with a MTA or segmented-aperture system, using existing optical delay lines or subaperture piston actuators that are normally used to phase up such systems for conventional imaging. Thus, such systems do not require any additional hardware or modifications to implement Fizeau FTIS, whereas the implementation of Michelson FTIS would typically require the addition of a Michelson interferometer science instrument to an existing imaging system.
In a previous paper [7], we analyzed the imaging properties of Fizeau FTIS. A unique aspect of Fizeau FTIS is that spectral information is not obtained for a finite bandwidth of low spatial frequencies. Section 2 reviews the relevant equations and discusses this problem in more detail. Nonlinear reconstruction algorithms are required for reconstructing the missing information to yield spectral imagery that is more easily interpretable. In Sec. 3, we describe an algorithm that uses a gray-world model to reconstruct the missing data. The need for nonlinear reconstruction algorithms suggests that the Fizeau configuration is suboptimum for performing FTIS. Simulation results presented in Sec. 4, however, indicate that spectral imagery of moderate quality can be obtained with Fizeau FTIS. This result demonstrates that a MTA or segmented-aperture system designed for conventional panchromatic imaging has the added value of being able to fulfill some spectral imaging tasks without additional instrumentation. Section 5 is a discussion and summary.
2. Imaging model
The imaging properties of Fizeau FTIS are discussed in detail in Ref. [7]. Here we only present equations necessary for formulating the image reconstruction algorithm. Furthermore, we only consider the case of introducing OPDs between two groups of subapertures for performing Fizeau FTIS, i.e., one group of subapertures has a fixed optical path length while the optical path length through the other group is varied. The generalized pupil function of the system can be written as
where T 1(ξ,η,ν) and T 2(ξ,η,ν) are the respective generalized pupil functions for the two subaperture groups, (ξ,η) are pupil plane coordinates, ν is the optical frequency, τ=OPD/c is the time-delay introduced between the subaperture groups, and c is the speed of light. The coherent impulse response [8] corresponding to each Tq(ξ,η,ν) is
where q=1 and 2 is an index for the subaperture groups, λ=c/ν is the optical wavelength, f is the system focal length, and (x,y) are focal plane coordinates. It is convenient to define the spectral point spread functions (SPSFs), h p,q(x,y,ν), as
for p and q=1 and 2, such that the incoherent point spread function (PSF) for the entire system can be written as a function of the time delay, τ, between the subaperture groups, as
The image intensity as a function of τ is given by
where κ=λ 2/π, S o(x,y,ν) is the spectral density of a spatially incoherent object, and the system is assumed to have unit magnification for notational convenience.
In performing Fizeau FTIS, a number of images are recorded by a focal plane array with different time delays between the subapertures. Following the normal Fourier transform spectroscopy processing steps, the spectral image, S i(x,y,ν), is then computed as the Fourier transform of the measured intensity modulation at each image point, i.e.,
where
is the average image intensity at each image point. Reference [7] discusses the linear relationship between S i(x,y,ν) and S o(x,y,ν) involving the SPSFs, which enables the real part of the complex-valued spectral image S i(x,y,ν) to be written as
for ν>0. The problem of the missing low spatial frequencies can be understood by taking the spatial Fourier transform of S (Re) i (x, y,ν) to yield [9]
where A pup is the combined area of all of the subapertures, G o(fx,fy,ν) is the spatial Fourier transform (FFT) of S o(x,y,ν), and the terms Hp,q(fx,fy,ν) are spectral optical transfer functions (SOTFs) defined as the normalized Fourier transform of the corresponding SPSFs, i.e.,
Note that the SOTFs are given by cross correlations between subaperture group pupil functions. Thus, the terms H 1,2(fx,fy,ν) and H 2,1(fx,fy,ν) vanish necessarily in some finite region around the dc spatial frequency (fx,fy)=(0,0) for non-overlapping subaperture groups, as is the case for a MTA or segmented aperture system. Low spatial-frequency data in this region are missing from Fizeau FTIS spectral imagery.
The measured image intensity I(x,y,τ) does, however, contain panchromatic information about the missing low spatial frequencies that can be used to facilitate the reconstruction of this missing data. The spatial Fourier transform of I avg(x,y) is given by
Note that the SOTF terms H 1,1(fx,fy,ν) and H 2,2(fx,fy,ν) are given by the autocorrelation of the pupil functions T 1(ξ,η,ν) and T 2(ξ,η,ν), respectively, and are non-zero in the region around the dc spatial frequency. Thus, G avg(fx,fy) contains spectrally-integrated information about the missing data.
3. Reconstruction algorithm
We wish to reconstruct an object spectral density estimate Ŝ o(x,y,ν) from the data and any available a priori knowledge about the object. The algorithm we describe for doing this uses a Wiener filter [10] to reconstruct spatial frequencies inside the support of the SOTFs H 1,2(fx,fy,ν) and H 2,1(fx,fy,ν) and reconstructs the remaining spatial-frequency data inside the support of the conventional optical transfer function (OTF) from the panchromatic intensity measurements using a gray-world approximation. The gray-world approximation assumes that to first order S o(x,y,ν) can be approximated as
where f(x,y) is the total flux at each image point (x,y) and ψ(ν) is a gray-world spectrum. In practice, the gray-world approximation can be fairly good for natural scenery within certain spectral intervals. For example, the gray-world approximation yields a root mean square (RMSE) error of 0.099 [W·m−2·sr−1·µm−1] and a normalized RMSE of 6.5% for the non-gray spectral imagery used for the simulation in Section 4.
In practice, the FTIS data is discretely sampled, whereas the equations in Section 2 are in terms of continuous variables. For convenience, we use the same notation for both the continuous and discretely sampled quantities. For example, while I(x,y,τ) is a function of the continuous variables x, y, and τ in Eq. (5), we also use I(x,y,τ) to represent the image intensity measured at discrete image points and time delays, such that x ∊{−Δx Nx/2, −Δx(Nx−2)/2, …, Δx(Nx−2)/2}, y ∊{−Δy Ny/2, −Δy(Ny−2)/2, …, Δy(Ny−2)/2}, and τ ∊{−Δτ Nτ/2, −Δτ(Nτ−2)/2, …, Δτ(Nτ−2)/2}, where Δx and Δy are the sample spacings in the focal plane, i.e., the detector pixel pitch, Δτ is the time delay sample spacing, and Nx, Ny, and Nτ are the number of samples for each variable. Additionally, the unprocessed spectral image S i(x,y,ν) is calculated using a fast Fourier transform (FFT), and is discretely sampled along the spectral dimension with a sample spacing of Δν=1/(NτΔτ). Likewise, the sample spacings for spatial-frequency domain quantities calculated using FFTs are Δfx=1/(Nx Δx) and Δfy=1/(NyΔy).
A gray-world spectrum ψ(ν) for the reconstruction can be estimated from the unprocessed spectral imagery by minimizing the following cost function
where the summation is over all spatial- and temporal-frequency samples, F 1(fx,fy), given by
which minimizes the value of Eψ, analogous to Eq. (17) in Ref. [11], for any particular ψ(ν), and ε is a small number included in the denominator of Eq. (14) to prevent infinite values when numerically computing F 1(fx,fy).
Additionally, the object spatial power spectrum Φo(fx,fy,ν), which is needed for constructing the Wiener filter, can be estimated from the unprocessed spectral imagery. Using the statistics of natural scenery [12-15] and the gray-world approximation, Φo(fx,fy,ν) is modeled as
where A 0, A, and α are parameters of the model. The parameters A and α, as well as the noise power spectrum Φn (assumed to be white) are obtained by minimizing the cost function
where the dc spatial frequency samples, (fx,fy)=(0,0), are not included in the summation. In practice, we find it necessary to blur |Gi (Re)(fx,fy,ν)| slightly (by convolution with a 5-8 pixel wide kernel) for use in Eq. (16) to prevent underestimating Φo(fx,fy,ν) caused by the logarithmic weighting of very small numbers. The parameter A 0 is estimated as
The object and noise power spectra are then used to form a Wiener filter [10], given by
where SNR 1(fx,fy,ν) is the estimated power signal-to-noise ratio (SNR) for Gi (Re)(fx,fy,ν), given by
The Wiener-filtered spectral image S 1(x,y,ν) is calculated as the inverse spatial 2-D FFT of
Additionally, a gray-world object spectral density estimate S 2(x,y,ν) is generated from G avg(fx,fy) using the gray-world approximation. In the gray-world approximation, we can write
where F 2(fx,fy) is the 2D FFT of f(x,y) in Eq. (12). The power SNR of G avg(fx,fy) can be estimated as
By inverting Eq. (21) and including a noise regularization term identical to that for the Wiener filter, F 2(fx,fy) can be calculated as
and the gray-world reconstruction is given by
Finally, the reconstructions G 1(fx,fy,ν) and G 2(fx,fy,ν) are combined to yield a final spectral image G 3(fx,fy,ν), given by
where c>1 is a weighting parameter used to emphasize the Wiener filtered spectral data G 1(fx,fy,ν) over the gray-world reconstruction G 2(fx,fy,ν). If G 1(fx,fy,ν) and G 2(fx,fy,ν) were independent Wiener filter reconstructions of G o(fx,fy,ν), then a value of c=1 would yield a G 3(fx,fy,ν) with minimum RMS error compared to G o(fx,fy,ν) [16]. Since, however, for Fizeau FTIS G 2(fx,fy,ν) is a gray-world reconstruction, we use a large value of c (=103 in Section 4) to weight G 1(fx,fy,ν) more heavily than G 2(fx,fy,ν) in Eq. (25). With this weight, G 3(fx,fy,ν) is essentially equal to G 1(fx,fy,ν) throughout the support of the SOTF terms H 1,2(fx,fy,ν) and H 2,1(fx,fy,ν), except where SNR 1(fx,fy,ν)≪SNR 2(fx,fy,ν) and equal to G 2(fx,fy,ν) at the remaining spatial frequencies passed by the conventional OTF H(fx,fy,ν).
4. Simulation results
Simulated FTIS data is used to illustrate the reconstruction algorithm. The optical system is an array of six telescopes in a ring configuration as shown in Fig. 1. Various optical system and detector parameters used in the simulation are given in Tables 1 and 2. The simulation is restricted to short wavelength infrared (SWIR) wavelengths λ=1.95–2.50 µm where the gray-world approximation is often quite good. To perform Fizeau FTIS, the subaperture telescopes are divided into two groups (q=1 or 2) as indicated by the shading in Fig. 1. During data collection the OPD between these groups of subapertures is modulated. Figure 2(a) shows an AVIRIS data set [17] used for the object So(x,y,ν) in the simulation, while Figure 2(b) shows a narrowband reference image of the object S ref(x,y,ν) used for comparison with reconstruction results later. The reference image was obtained by Wiener filtering the spectral image obtained from a noiseless Michelson FTIS simulation with the same optical system. Thus, S ref(x,y,ν) is bandlimited to those spatial frequencies passed by the system OTF H(fx,fy,ν) and has the same spectral sampling and spectral resolution as the Fizeau
Fig. 1. Pupil of six-telescope array optical system, where the telescopes in group q=1 are white and those in group q=2 are gray.
Table 1. Optical System Parameters.
Table 2. Detector Parameters.
Fig. 2. (a). Panchromatic view of object spectral density S o(x,y,ν) in units of W/m2/sr, and (b) narrowband view of reference imagery S ref(x,y,ν) in units of W/m2/sr/µm for ν=140 THz (λ=2.13 µm).
Fig. 3. Plot of the average radiance for each spectral band of S o(x,y,ν). The errorbars represent +/− one standard deviation of the radiance for each spectral band.
FTIS imagery. Figure 3 gives an indication of the validity of the gray-world approximation for the AVIRIS data. As a side note, the AVIRIS data used for S o(x,y,ν) has a ground sample distance (GSD) of 20m. The required object distance to match this GSD to the detector pixel pitch Δx is 35556 km, which is approximately equal to a geosynchronous orbital altitude of 35,786 km for a remote sensing system.
Table 3. FTIS parameters.
Fig. 4. Movie (668 KB) of Fizeau FTIS image intensity I(x,y,τ) [the still frame shows I(x,y,0)]. The visibility of the intensity modulation is very small for most of the movie, but is greatest near τ=0, which occurs halfway through the movie. [Media 1]
Table 3 lists the parameter values related to performing Fizeau FTIS. Since the spectral bandwidth of the simulation is limited to the SWIR wavelengths, I(x,y,τ) can be coarsely sampled with respect to τ [18]. This reduces the size of the data set, and does not have any SNR penalty in the resulting spectral image for a fixed total data collection time in the shot noise limit [19]. Thus, note that the OPD sample spacing cΔτ=2.87 µm is almost a factor or 3 larger than the Nyquist sample spacing of min(λ)/2=0.975 µm. The total data collection time for making the FTIS measurements is given by t exp Nτ=6.9 sec (neglecting time required for detector readout and OPD adjustments). The spectral resolution of the AVIRIS data used for S o(x,y,ν) is Δλ=10 nm, and the FTIS parameters were chosen to yield a somewhat coarser spectral resolution than this to prevent artifacts due to the discrete nature of the simulation. Figure 4 shows a movie of simulated intensity measurements, I(x,y,τ), as a function of τ.
After computing the complex-valued spectral image S i(x,y,ν), the first step in the reconstruction algorithm is to estimate the gray-world spectrum ψ(ν) from G i(fx,fy,ν) by minimizing the objective function Eψ given by Eq. (13). Figure 5(a) shows the resulting ψ(ν), as well as the average spectrum of the reference spectral image ψ avg(ν), given by
Intuitively, one might expect that the best reconstruction results would be obtained when these spectra are equal, i.e., that the ideal gray world spectrum equals the average spectrum of the scene ψ(ν)=ψ avg(ν). This, however, is not universally true. A normalized root mean square error (RMSE) metric can be used to evaluate the accuracy of the gray-world approximation. We define the normalized RMSE at each spatial frequency between the reference data G ref(fx,fy,ν) [the 2D spatial Fourier transform of S ref(x,y,ν)] and a gray-world approximation using ψ(ν) as
Fig. 5. (a). Gray-world spectrum ψ(ν) (solid line) estimated from the spectral image data G i(fx,fy,ν) and the average spectrum of the reference spectral image ψ avg(ν) (dotted line), and (b) azimuthal average of the spatial-frequency domain normalized RMSE E(fx,fy) corresponding to each gray world spectrum.
where the coefficients β(fx,fy) are chosen to minimize E(fx,fy) [11] and are given by
Figure 5(b) shows the azimuthal average of the normalized RMSE E(fx,fy) using the gray-world spectra shown in Fig. 5(a). While ψ avg(ν) yields E(0,0)=0, i.e., ψ avg(ν) matches the true object spectrum at the dc spatial frequency, the normalized RMSE of ψ avg(ν) at the remaining spatial frequencies is ~15% worse than that obtained with ψ(ν), the gray-world spectrum estimated from the data. Note that the error shown in Fig. 5(b) increases dramatically for spatial frequencies beyond 0.3 cycles/pixel, which are outside the support of the system OTF.
The next step in the reconstruction algorithm is to estimate the object power spectrum model parameters A and α and the noise power spectrum Φn by minimizing E Φ. Figure 6 shows comparisons of |Gi (Re)(fx,fy,ν)|2 and |G avg(fx,fy)|2 with the corresponding power spectrum model using the estimated model parameters. Notice that the smooth power spectrum model fits the spatial frequency data well on average. The power spectrum model is used only to estimate the Fourier-domain SNR quantities SNR1(fx,fy,ν) and SNR2(fx,fy), and an exact match between the power spectrum model and the Fourier data is not necessary. As a side note, the estimated value of α=1.07 is within the expected two-sigma range of α=1.2 ±0.3 for natural scenery [14].
Next, the SNR estimates SNR 1(fx,fy,ν) and SNR 2(fx,fy) are used to compute the Wiener and gray-world reconstructions S 1(x,y,ν) and S 2(x,y,ν), respectively. These two estimates are then combined using Eq. (25) to form a final reconstruction S 3(x,y,ν). Figure 7 shows each of these reconstructions in comparison to the unprocessed spectral image S i (Re)(x,y,ν) at a particular spectral band. For comparison, the corresponding reference image S ref(x,y,ν) is shown in Fig. 2(b). The missing low spatial-frequencies cause the amplitude of S i (Re)(x,y,ν) to be appreciable only at edges within the scene, as is evident in Fig. 7(a). The Wiener filter sharpens the spectral image considerably, but cannot reconstruct the missing low spatial frequencies. Thus, S 1(x,y,ν) is still zero mean and contains low spatial-frequency artifacts as seen in Fig. 7(b). The gray-world reconstruction S 2(x,y,ν), shown in Fig. 7(c), has the general appearance of a conventional image [the small negative values in S 2(x,y,ν) are Gibb’s ringing artifacts], but does not have much spectral utility since the spectrum at each pixel is identical. The reconstruction S 3(x,y,ν) contains desirable properties from both S 1(x,y,ν) and S 2(x,y,ν), namely the features of conventional imagery and spectral utility.
Fig. 6. Power spectrum fitting results for: (a) |G i (Re)(0,fy,ν)|2 data (solid line) and the power spectrum fit (dotted line) for ν=140 THz and (b) |G avg(fx,0)|2 (solid line) and the corresponding gray-world power spectrum model.
Fig. 7. Spectral image data: (a) S i (Re)(x,y,ν), (b) S 1(x,y,ν), (c) S 2(x,y,ν), and (d) S 3(x,y,ν) for ν=140THz. The corresponding reference image S ref(x,y,ν) is shown in Fig. 2(b).
Fig. 8. Normalized RMSE [calculated with respect to S ref(x,y,ν)] for each image point for: (a) S i (Re)(x,y,ν), (b) S 1(x,y,ν), (c) S 2(x,y,ν), and (d) S 3(x,y,ν).
Figures. 8, 9, and 10(a) shows the normalized RMSE, between S ref(x,y,ν) and each of S i (Re)(x,y,ν), S 1(x,y,ν), S 2(x,y,n) and S 3(x,y,ν) evaluated at each image point (x,y), at each spatial frequency (fx,fy), and for each spectral band ν, respectively. In each case, the normalized RMSE η is defined in a manner similar to Eq. (27). Additionally, Fig. 10(b) shows the standard RMSE for each spectral band of the various reconstructions and Table 4 lists the total reconstruction errors for each reconstruction. The quality of both S i (Re)(x,y,ν) and the Wiener reconstruction S 1(x,y,ν) appears to be rather poor in terms of η evaluated for each image point and each spectral band. Figure 9, however, indicates that the spectra at spatial frequencies passed by the SOTF terms H 1,2(fx,fy,ν) and H 2,1(fx,fy,ν) is fairly good. The quality of the gray-world reconstruction S 2(x,y,ν) is fairly good in terms of η, but does not offer much spectral utility. It is interesting to note that better η values were obtained in the spatial frequency domain for G 2(fx,fy,ν) than for G 1(fx,fy,ν). We believe this results from the combination of three effects: (i) the error associated with the gray-world approximation is
Fig. 9. Normalized RMSE [calculated with respect to G ref(fx,fy,ν)] for each spatial frequency for: (a) G i (Re)(fx,fy,ν), (b) G 1(fx,fy,ν), (c) G 2(fx,fy,ν), and (d) G 3(fx,fy,ν).
Fig. 10. (a). Normalized RMSE and (b) standard RMSE [calculated with respect to S ref(x,y,ν)] for each spectral band for: S i (Re)(x,y,ν), S 1(x,y,ν), S 2(x,y,ν), and S 3(x,y,ν).
Table 4. Overall Error.
Fig. 11. Comparison of Fizeau and Michelson FTIS reconstruction results: (a) RMSE and (b) fractional RMSE [calculated with respect to S ref(x,y,ν)] for each spectral band.
smaller than the errors due to noise, (ii) when estimating the gray-world spectrum ψ(ν) via Eq. (13), the noise is reduced by effectively averaging over many spatial frequencies, and (iii) when using I avg(x,y) to compute S 2(x,y,ν), noise is also reduced by averaging over many time delays. Comparing the error metric values, the quality of the final reconstruction S 3(x,y,ν) is better than that of S 2(x,y,ν).
Finally, a Michelson FTIS simulation was performed for the same optical system and viewing conditions. The Michelson FTIS data was reconstructed using just the Wiener filter portion of the algorithm described in Section 3 since it does not suffer from missing low spatial frequencies. Figure 11 compares the RMSE for the final reconstructions obtained from the Fizeau and Michelson FTIS simulations. Note that the RMSEs are comparable at some wavelengths and the Michelson is up to two times better than the Fizeau at other wavelengths. For comparison, the Hyperion instrument, which is part of NASA’s Earth Observing-1 Mission [20], has a spectrally-averaged SNR of approximately 140 for the visible and 60 for the SWIR spectral regions [21]. This implies that the fractional RMSE for Hyperion is approximately 2% in the SWIR region, compared with the fractional RMSEs of ~10% obtained from the Fizeau and Michelson FTIS simulations for the particular parameters we chose.
5. Discussion and summary
Fizeau FTIS poses a challenging image reconstruction problem because of missing low spatial frequencies. Unlike the optical superresolution problem [22], in which one attempts to reconstruct high spatial-frequency information beyond the diffraction limit through extrapolation (which is very sensitive to noise), we wish to interpolate and reconstruct missing low spatial frequencies in the Fizeau FTIS problem. Furthermore, panchromatic information about the low spatial frequencies is available from the raw Fizeau FTIS measurements. The algorithm proposed here uses a gray-world approximation, which is typically applicable to natural scenery as long as a significant contrast reversal of the scene does not occur within the spectral bandwidth of the imagery. For example, while the gray-world approximation might be valid for imagery spanning the visible or near infrared wavelengths, the gray-world approximation is usually not valid for imagery containing vegetation that spans both the visible and near infrared wavelengths due to large contrast reversals associated with the chlorophyll reflectance spectrum. To overcome this problem, Fizeau FTIS could be performed twice for the same scene, using color filters to limit the spectral bandwidth of the imagery first to visible wavelengths and then to near infrared wavelengths.
Results with numerically simulated data indicate that moderate quality spectral imagery can be obtained through Fizeau FTIS using the proposed algorithm. The Fizeau FTIS results were found to be within approximately a factor of two of Michelson FTIS results (in terms of the estimated spectral SNR. The algorithm does assume knowledge of the system PSF, which may not be available in all cases, and the simulations did not explore the impact of incomplete knowledge or errors in the OPDs introduced for performing FTIS. While the performance of Fizeau FTIS is slightly less than Michelson FTIS, the ability of Fizeau FTIS to gather spectral information without the need for additional hardware makes it attractive for segmented-aperture or multiple telescope array systems.
Acknowledgement
This work was supported by Lockheed Martin Corporation.
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