## Realization of quantitative-grade fieldable snapshot imaging spectropolarimeter

Optics Express, Vol. 12, Issue 26, pp. 6559-6573 (2004)

http://dx.doi.org/10.1364/OPEX.12.006559

Acrobat PDF (462 KB)

### Abstract

We discuss achievement of a long-standing technology goal: the first practical realization of a quantitative-grade, field-worthy snapshot imaging spectropolarimeter. The instrument employs *Polarimetric Spectral Intensity Modulation* (PSIM), a technique that enables full Stokes instantaneous “snapshot” spectropolarimetry with perfect channel registration. This is achieved with conventional single beam optics and a single focal plane array (FPA). Simultaneity and perfect registration are obtained by encoding the polarimetry onto the spectrum via a novel optical arrangement which enables sensing from moving platforms against dynamic scenes. PSIM is feasible across the electro-optical sensing range (UV-LWIR). We present measurement results from a prototype sensor that operates in the visible and near infrared regime (450–900 nm). We discuss in some detail the calibration and Stokes spectrum inversion algorithms that are presently achieving 0.5% polarimetric accuracy.

© 2004 Optical Society of America

## 1. Introduction

20. J. Q. Peterson, G. L. Jensen, and J. A. Kristl, “Imaging polarimetry capabilities and measurement uncertainties in remote sensing applications,” Proc. of SPIE **4133**, 221–228 (2000). [CrossRef]

21. J. Q. Peterson, G. L. Jensen, J. A. Kristl, and J. A. Shaw, “Polarimetric imaging using a continuously spinning polarizer element,” Proc. of SPIE **4133**, 292–300 (2000). [CrossRef]

17. J. A. Shaw and M. R. Descour, “Instrument effects in polarized infrared images,” Opt. Eng. **34**, 1396–1399 (1995). [CrossRef]

*Polarimetric Spectral Intensity Modulation*(PSIM)[4][5] technique and present measurement results from a prototype sensor operating in the visible and near infrared regime (450–900 nm). We discuss in some detail the calibration and Stokes spectrum inversion algorithms that are presently achieving 0.5% polarimetric accuracy. Oka and Kato [6

6. K. Oka and T. Kato, “Spectroscopic Polarimetry with a Channeled Spectrum,” Opt. Lett. **24**, 1475–1477 (1999). [CrossRef]

## 2. PSIM sensor principle of operation

### 2.1. Overview

*stationary*birefringent crystals, followed by a

*stationary*polarizer, packaged in a

*polarization module*. When placed optically upstream from the spectrometer dispersing element (in our design, outside the spectrometer before the slit), this polarization module produces polarization-dependent interference fringes of the measured intensity spectrum (cf. Fig. 1). There are no moving parts. The full Stokes vector spectrum can subsequently be retrieved from the modulated intensity spectrum, based on the fringe patterns. Both 1D (line-imaging) and 2D imaging spectrometers can be employed. In a single “snapshot” (integration period), the focal plane in the slit-based line-imaging spectrometer records spatial information along one dimension (columns) and modulated spectra along the other dimension (rows).

*µ*m) wavelengths. An example of the PSIM spectrum from a laboratory demonstration is also shown within Fig. 1. The incident light was essentially 100% polarized. Note the 0 and 90 degree cases are of opposite phase. The modulation effect is sinusoidal in

*wavenumber*, hence the chirped appearance in wavelength. The 45 and 135 degree cases contain other frequency components, as can be seen from the shallower nulls at some wavelengths. In short, this modulation is produced by a wavelength dependent interference phenomenon of the fast and slow waves in two birefringent crystals. We show later how the spectrum can be demodulated to retrieve the full Stokes spectrum.

## 2.2. Mueller matrix derivation

**M**

_{sys}is relevant for describing the detected spectral intensity

*𝓘*(

*ν*). If the sum and difference retardances of the two crystals are denoted ∑

*ϕ*and Δ

*ϕ*, respectively, then the following expression for

*𝓘*(

*ν*) can be written:

*ν*=Optical frequency (Hz)

*c*=Speed of Light

*𝓘*(

*ν*)=Detected spectral intensity

**m**

_{sys,1}(

*ν*)=First row of system Mueller matrix,

**M**

_{sys}

*ν*)=Incident Stokes vector, [

*I*

*Q*

*U*

*V*]

^{T}

*ϕ*=

*ϕ*

_{1}-

*ϕ*

_{2}

*ϕ*=

*ϕ*

_{1}+

*ϕ*

_{2}

*ϕ*

_{1}=2

*πν*(

*n*

_{e}-

*n*

_{o})

*ℓ*

_{1}/

*c*

*ϕ*

_{2}=2

*πν*(

*n*

_{e}-

*n*

_{o})

*ℓ*

_{2}/

*c*

*n*

_{o},

*n*

_{e}=Ordinary and extraordinary indices of refraction

*ℓ*

_{1},

*ℓ*

_{2}=Crystal lengths.

## 2.3. Practical system model

*λ*

_{∘}is the center wavelength of the pixel,

*𝓘*

_{raw}(

*λ*

_{∘}) is the raw uncorrected digitized pixel reading,

*R*(

*λ*) is the pixel responsivity in A/D counts per radiance unit,

*C*(

*λ*

_{∘}) is the offset of the pixel

*λ*

_{∘}in A/D counts,

*K*(

*λ*) is the spectral blur function, and Δ

*n*(

*λ*) is the birefringence and the integration is over the spectral blur subtended by a single pixel.

*L*(

*λ*

_{∘}) in radiance units is:

## 2.3.1. Spectral blur kernel model

*exclusive*of spectral averaging effect due to the finite width of the pixel. One is led to the following model for

*L*(

*x*

_{o},

*y*

_{o}), the amount of signal measured by an

*entire*pixel centered at (

*x*

_{o},

*y*

_{o}) with spectral integer index

*x*

_{o}, from gaussian system blurring of dispersed light source

*s*(

*λ*):

*K*(

*x*) the spectral blur kernel function for varying spectral index

*x*, taken as a continuous variable. The function

*λ*(

*x*) describes the spatial position to wavelength mapping. If a diffraction grating is used as the dispersive element, then the mapping is nominally a linear one (i.e.

*λ*(

*x*)=

*mx*+

*λ*

_{b}). However, to obtain well calibrated results, one must make use of slightly more elaborate models which are described in the following sections. The normalized blur parameter σ expresses the gaussian system blur width in pixel integer index units. Therefore,

*K*(

*x*), and not the gaussian kernel, is the appropriate blur function

*K*to use in the high-fidelity system model (Eq. 4).

*K*(

*x*) is mirror-symmetric and normalized to unit area. The spectral blur kernel

*K*(

*x*) is approximately gaussian for moderate normalized blur parameter values, and approaches the ideal tophat averaging response for negligible system blur.

## 3. Calibration for Stokes inversion

### 3.1. Calibration sequence for estimating system model

## 3.1.1. Global spectral calibration

*λ*(

*x*

_{o},

*y*

_{o}) is parameterized by a quadratic polynomial in

*x*

_{o}and linear term in

*y*

_{o}(accounting for any slant due to slight rotational misalignment of the grating in its mount), where

*x*

_{o}is the spectral pixel integer index on the FPA, i.e. along the spectral dispersion dimension, and

*y*

_{o}is the spatial along-slit dimension integer index. In addition, the calibration determines the blur-width parameter σ of the spectral blur kernel

*K*(

*λ*). To motivate the importance of the spectral blur estimation, a σ value that is 50% too large will result in about 3–4% typical error. For a 10% error in σ, the typical relative prediction error is <1%, although it occasionally will peak to about 5% near the fringe nulls.

## 3.1.2. Spectroradiometric non-uniformity calibration (NUC)

*C*(

*λ*

_{∘}) and

*R*(

*λ*

_{∘}) values for each pixel.

## 3.1.3. Floating retardance and spectral spline calibration

**m**

_{sys}

_{,1}row term to achieve a fully-determined system model (Eq. 4). The principal parameters of

**m**

_{sys,1}, by means of Eq. 2, are the PSIM polarization module waveplate lengths

*ℓ*

_{1}and

*ℓ*

_{2}as well as their spectral birefringence (

*n*

_{e}(

*λ*)-

*n*

_{o}(λ)). However, there are factors that complicate such a simple reckoning of

**m**

_{sys,1}. Furthermore, Eq. 2 for

**m**

_{sys,1}only applies to ideal axial propagation through the PSIM polarization module, although by design, deviations from this have been rendered a negligible factor. Remaining factors include stray birefringence and waveplate retardance temperature sensitivity. The significance of these are gauged against the

*noise-equivalent retardance*Δ

_{ne}, defined as the change in optical path difference (OPD), (

*n*

_{e}-

*n*

_{o})

*ℓ*, that causes the relative fringe intensity registered by a pixel to shift by 1/

*SNR*, where SNR is signal-to-noise ratio. For our prototype sensor, at a 0.5% level of significance or SNR of 200:1, Δ

_{ne}is conservatively 1 nm.

*ℓ*

_{1},

*ℓ*

_{2}and [

*n*

_{e}(

*λ*)-

*n*

_{o}(

*λ*)]. For example, a 1°

*C*change shifts the OPD of a 1 mm length quartz crystal by about 1 nm, comparable to Δ

_{ne}.

*ℓ*

_{1}Δ

*n*(

*λ*

_{∘}) and

*ℓ*

_{2}Δ

*n*(

*λ*

_{∘}) from their nominally-known values. This is performed by viewing the scene through a linear polarizer. This permits estimation of the

**m**

_{sys,1}(

*λ*;

*ℓ*

_{1},

*ℓ*

_{2},Δ

*n*(

*λ*

_{∘})) in Eq. 4, expressed in the functional form of Eq. 2. The floating-retardance calibration is more fully described in §4.3.

*δ*(

*λ*).

## 4. Stokes inversion

### 4.1. Linear spectrum signal model

**d**, within a spectral analysis window centered at pixel

*x*

_{o}is:

**d**is a data vector of length 2

*N*+1 samples, in radiance units ensuing from the flat-fielded measurements (Eq. 4):

**M**

_{1}(

*δ*;

*x*

_{o},

*y*

_{o}) is the forward system matrix for the spectral samples centered about pixel (

*x*

_{o},

*y*

_{o}), and s

_{1}(

*x*

_{o},

*y*

_{o}) is the incident Stokes vector for wavelength

*λ*(

*x*

_{o},

*y*

_{o}) at the

*center*of the pixel. The linear-spectrum model assumes that s

_{1}(

*x*

_{o},

*y*

_{o}) exhibits linear variation with wavelength

*within*a 2

*N*+1 sample-length analysis window. An inversion can be performed at each pixel, yielding a smoothing effect similar to a boxcar moving average filter with an impulse response of the same extent as the PSIM analysis window. Thus, although an inversion is computed at each pixel, the effective resolution is on the order of the window length.

*x*-

*x*

_{o}) is presumed to exhibit linear dependence on wavelength, expressed by a spectral constant (pedestal) and a spectral linear ramp signal for each Stokes component:

*x*is the continuous wavelength variable normalized in pixel units. For modeling purposes which will be made clear, we stack the coefficients of the above Stokes linear spectrum model into an

*augmented*Stokes vector:

*x*

_{o}=FPA detector index in dispersion dimension (indexing wavelength),

*i*=Relative pixel index within analysis window:

*i*∊[-

*N*,

*N*]

*y*

_{o}=Pixel index along the spatial (slit) dimension,

*δ*=Deviation from the nominal spectral retardance of the waveplates.

*ℓ*

_{1},

*ℓ*

_{2}=Waveplate thicknesses

*n*(

*x*

_{o},

*y*

_{o})=Pixel-indexed spectral birefringence.

*K*(

*x*), is given by Eq. 6. The values of Δ

*n*(

*x*

_{o},

*y*

_{o}),

*ℓ*

_{1}, and

*ℓ*

_{2}are taken to be their nominal values, based on the published values for the spectral birefringence of the waveplate material and the specified or measured thicknesses of the waveplates. The signed multiplicative deviation of Δ

*n*(

*x*

_{o},

*y*

_{o})

*ℓ*

_{x}from the nominal value is termed the floating retardance perturbation and denoted by

*δ*(

*λ*). It is spectrally dependent, thus in turn dependent on (

*x*

_{o},

*y*

_{o}). We will have more to say about it later (§4.3). All spectral integrations extend over an interval in which the spectral blur kernel takes on significant values. In practice, ±(1/2+3σ) seems to be sufficient. It is conceivable that, in some practical systems, σ, and hence

*K*(

*x*), may require characterization as functions of

*x*and

*y*. For instance, it is conceivable that the point spread function of the collection optics could vary significantly over the operating band.

*x*+

*i*)

*within*(inside) the spectral integral. It does not suffice to employ a staircase approximation, where for example the

*mQ*

_{1}column within Eq. 11 would be formed by multiplying the

*mQ*

_{0}column by a zero-mean linear ramp vector. The linear-spectrum model is motivated by the failure of the

*constant-spectrum*model which, when employed against sources exhibiting even modest spectral variation such as a linear trend, induces objectionable ripple artifacts in the inverted Stokes spectrum.

## 4.2. Linear spectrum Stokes inversion

*N*≥4, the linear-spectrum forward model of Eq. 7 and Eq. 11 forms an overdetermined system that we invert in a least squares sense using the pseudoinverse

*δ*;

*x*

_{o},

*y*

_{o}), which results from the singular value decomposition of

**M**

_{1}[9]. The system matrix for a properly designed instrument is always well conditioned so all singular values are retained in the inversion.

*δ*is computed as follows:

*Î*

_{1},

*Q̂*

_{1}, etc. (

*cf*. Eq. 10) represent the slopes of the linear ramp terms (

*x*-

*x*

_{o}). These slope values are presently discarded. Since the linear ramp atoms are zero-mean over the analysis window,

*[Î*

_{0}

*Q̂*

_{0}

*Û*

_{0}

*V̂*

_{0}] thus represents the mean value of the Stokes vector within the analysis window.

*chirped*as shown in Fig. 1. This corresponds to the fringe density decreasing by a factor of two between the ends of an octave spectral range. Consequently, our inversion technique similarly increases the window length of the data vector

**d**by a factor of two while traversing from the short to the long wavelength end of the range, in essence yielding a

*constant Q*smoothing effect.

## 4.3. Floating-retardance calibration for δ

*δ*in the formulation is the temperature sensitivity of the retardance (

*cf*. §3.1.3). Ideally from the signal processing standpoint, the polarization module would be temperature stabilized to eliminate the need to track

*δ*in the analysis. Currently, the value of

*δ*is estimated using a highly

*linearly*polarized input to the sensor of sufficiently high SNR. To find the optimum value of

*δ*, the following problem is solved:

**X**

^{+}denotes the pseudoinverse of

**X**,

**M**

_{1}is the system matrix, Eq. 11, for the linear-spectrum model, and the diagonal matrix C is included to incorporate the knowledge that the

*V*Stokes component is zero:

*δ*need only account for temperature drift of retardance and is, in fact, able to encompass a large range of temperatures. The Stokes inversion algorithm using this method to find the actual retardance has been named

*floating retardance inversion*.

## 4.4. Enhanced Stokes inversion strategies

*N*+1 to encompass the entire sensor spectral range. This is not inconceivable and such

*superresolution*performance is routine in the mature field of autoregressive (AR) spectral estimation.

## 4.4.1. Over-complete spectral dictionary model

**z**

_{1}(

*x*) is comprised of pedestal and linear ramp spectral functions, which we term signal elements

*z*

_{{I,Q,U,V}p}(

*x*):

*P*spectral signal elements per Stokes component,

**z**

_{P}(

*x*). Consequently, the augmented Stokes vector s

_{P}now contains 4

*P*terms, the system matrix,

**M**

_{P}, in turn expands its number of columns, with elements expressed more generally as:

*x*. To handle increasingly complex spectral structure, one might intuitively expand the number of signal elements in the Stokes signal dictionary

**z**

_{P}(

*x*), for example by adding Taylor series terms. We are motivated to include signal elements to represent spectral resonance shapes directly because their representation using higher-order Taylor series terms may not converge quickly.

*P*>(2

*N*+1), the system becomes under-determined. Moreover, depending on the signal elements, even if (2

*N*+1)≥4

*P*,

**M**

_{P}may yet remain under-determined due to rank deficiency. Rather than increasing the analysis window size 2

*N*+1 to compensate, we can purposely seek the under-determined Stokes inversion s

_{P}which fits the signal

**d**against an

*over-complete*spectral dictionary

**M**

_{P}.

## 4.4.2. Over-complete Stokes inversion via basis pursuit

*over-determined*system, the pseudoinverse

_{P}which minimizes

*L*

_{2}norm error, i.e., the sum of squared errors

*ε*

^{T}

*ε*between the predicted and actual data, where:

*under-determined*system, such as the overcomplete spectral dictionary,

**M**

_{P}. In such case, there are arbitrarily many possible solution vectors ŝ

_{P}=

**d**, and the pseudoinverse returns the

*minimal-length vector*ŝ

_{P}that satisfies

**M**

_{P}ŝ

_{P}=

**d**. The terminology for such an under-determined problem is

*best-basis*matching [10

10. S. Chen, D. Donoho, and M. Saunders, “Atomic Decomposition by Basis Pursuit,” SIAM Review **43(1)**, 129–59 (2001). [CrossRef]

*L*

_{1}norm (rather than

*L*

_{2}) ŝ

_{P}satisfying

**M**

_{P}ŝ

_{P}=

**d**, casting the problem as a linear programming (LP) exercise [10

10. S. Chen, D. Donoho, and M. Saunders, “Atomic Decomposition by Basis Pursuit,” SIAM Review **43(1)**, 129–59 (2001). [CrossRef]

_{P}assigns non-zero weights to the fewest possible signal elements in the dictionary, rather than attributes or diffuses the energy in

**d**across many elements. Remarkably, if a signal possesses a sufficiently sparse representation, the

*L*

_{1}optimization of basis pursuit necessarily yields this unique solution [11].

## 4.4.3. Populating the spectral dictionary with signal elements

**z**

_{P}(

*x*) with a reasonable number of signal elements that nevertheless span the space of potential Stokes spectra. The notion of populating

**z**

_{P}(

*x*) with replicas of resonance signal templates for all potential

*λ*translations of spectral location and dilations of spectral width is daunting. Fortunately, by appealing to design of steerable

*framelets*, such a brute-force approach appears unnecessary.

*z*(

*x*), a steerable frame is that set of vectors or functions whose linear combinations represent the possible shifts and dilations of

*z*(

*x*). As a simple example, the signal function cos (

*ωt*+

*β*), for

*any*shift

*β*, can be succinctly represented by a linear combination of two functions, cos (

*ωt*) and sin (

*ωt*). For a given class of signal elements, the existence of a steerable frame therefore enables a compact signal dictionary to implicitly represent all possible signal shifts and dilations.

*λ*shift and dilation invariance, but are, however, quite simple compared to signal elements that would represent spectral resonance features, e.g., those ensuing from the Drude-Lorentz model for refractive index, manifested through Fresnel reflectance [14]. At issue is whether we can find a steerable framelet representation for such physically-motivated signal elements of our choosing. Recent work by Daubechies

*et. al.*indicates that this may indeed be achievable [15

15. I. Daubechies, B. Han, A. Ron, and Z. Shen, “Framelets: MRA-Based Constructions ofWavelet Frames,” Applied and Computational Harmonic Analysis **14(1)**, 1–46 (2003). [CrossRef]

## 5. Measurement results

19. J. K. Boger et al., “An error evaluation template for use with imaging spectro-polarimeters,” Proc. of SPIE **5158**, 113–124 (2003). [CrossRef]

*n*≈1.5), maximum DoPs are ≈20–30% at an angle of incidence of 65°. Higher DoPs can be generated by stacking multiple plates.

*n*=1.478 at 500 nm). There error is less than 1% over much of the range. These results were obtained using the HyperSpectral Polarimeter for Aerosol Retrievals (HySPAR) which is shown in Fig. 3. This instrument has a 120° field of view and operates over 450–900 nm. The instrument was developed to perform multi-angle spectropolarimetric measurements for retrieval of aerosol microphysical parameters. It has been shown that the addition of polarimetric information can improve accuracy over intensity-only methods[3

3. M. Mishchenko and L. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” Journal of Geophysical Research **102(D14)**, 16989–17013, 1997. [CrossRef]

2. J. Chowdhary, B. Cairns, M. Mishchenko, and L. Travis, “Retrieval of aerosol properties over the ocean using multispectral and multiangle photopolarimetric measurements from the Research Scanning Polarimeter,” Geophys. Research Lett. **28**, 243–246 (2001). [CrossRef]

1. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Reviews **16**, 527–610 (1974). [CrossRef]

## 6. Conclusion

## Acknowledgments

## References and links

1. | J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Reviews |

2. | J. Chowdhary, B. Cairns, M. Mishchenko, and L. Travis, “Retrieval of aerosol properties over the ocean using multispectral and multiangle photopolarimetric measurements from the Research Scanning Polarimeter,” Geophys. Research Lett. |

3. | M. Mishchenko and L. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” Journal of Geophysical Research |

4. | P. Kebabian, “Polarimetric spectral intensity modulation spectropolarimeter,” US Patent 6,490,043 (2002). |

5. | F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. L. Kebabian, “Polarimetric Spectral Intensity Modulation (P-SIM): Enabling Simultaneous Hyperspectral and Polarimetric Imaging,” in |

6. | K. Oka and T. Kato, “Spectroscopic Polarimetry with a Channeled Spectrum,” Opt. Lett. |

7. | M. Born and E. Wolf, |

8. | E. Collett, |

9. | G. H. Golub and C. F. V. Loan, |

10. | S. Chen, D. Donoho, and M. Saunders, “Atomic Decomposition by Basis Pursuit,” SIAM Review |

11. | D. L. Donoho and M. Elad, “Optimally Sparse Representation in General (Non-Orthogonal) Dictionaries Via L1 Minimization,” Tech. rep., Stanford University, Department of Statistics (2002). |

12. | W. T. Freeman and E. H. Adelson, “The Design and Use of Steerable Filters,” IEEE Trans. Pattern Analysis and Machine Intelligence |

13. | R. Manduchi, P. Perona, and D. Shy, “Efficient Deformable Filter Banks,” IEEE Trans. on Signal Processing |

14. | F. Iannarilli, “Spectro-Polarimetric Remote Surface Orientation Measurement,” US Patent 6,678,632 (2004). |

15. | I. Daubechies, B. Han, A. Ron, and Z. Shen, “Framelets: MRA-Based Constructions ofWavelet Frames,” Applied and Computational Harmonic Analysis |

16. | J. D. Howe, “Two-color infrared full-Stokes imaging polarimeter development,” IEEE Aerospace Conference (1999). |

17. | J. A. Shaw and M. R. Descour, “Instrument effects in polarized infrared images,” Opt. Eng. |

18. | G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, “Micropolarizer array for infrared imaging polarimetry,” J. Opt. Soc. Am. A. |

19. | J. K. Boger et al., “An error evaluation template for use with imaging spectro-polarimeters,” Proc. of SPIE |

20. | J. Q. Peterson, G. L. Jensen, and J. A. Kristl, “Imaging polarimetry capabilities and measurement uncertainties in remote sensing applications,” Proc. of SPIE |

21. | J. Q. Peterson, G. L. Jensen, J. A. Kristl, and J. A. Shaw, “Polarimetric imaging using a continuously spinning polarizer element,” Proc. of SPIE |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

(280.1100) Remote sensing and sensors : Aerosol detection

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 12, 2004

Revised Manuscript: December 14, 2004

Published: December 27, 2004

**Citation**

Stephen Jones, Frank Iannarilli, and Paul Kebabian, "Realization of quantitative-grade fieldable snapshot imaging spectropolarimeter," Opt. Express **12**, 6559-6573 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-26-6559

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### References

- J. E. Hansen and L. D. Travis, �??Light scattering in planetary atmospheres,�?? Space Science Reviews 16, 527-610 (1974). [CrossRef]
- J. Chowdhary, B. Cairns, M. Mishchenko and L. Travis, �??Retrieval of aerosol properties over the ocean using multispectral and multiangle photopolarimetric measurements from the Research Scanning Polarimeter,�?? Geophys. Research Lett. 28, 243-246 (2001). [CrossRef]
- M. Mishchenko and L. Travis, �??Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,�?? Journal of Geophysical Research 102(D14), 16989-17013, 1997. [CrossRef]
- P. Kebabian, �??Polarimetric spectral intensity modulation spectropolarimeter,�?? US Patent 6,490,043 (2002).
- F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. L. Kebabian, �??Polarimetric Spectral Intensity Modulation (P-SIM): Enabling Simultaneous Hyperspectral and Polarimetric Imaging,�?? in Infrared Technology and Applications XXV, B. F. Andresen and M. Strojnik, eds., Proc. SPIE 3698, 474�??481 (1999).
- K. Oka and T. Kato, �??Spectroscopic Polarimetry with a Channeled Spectrum,�?? Opt. Lett. 24, 1475�??1477 (1999). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1965).
- E. Collett, Polarized Light (Marcel Dekker, 1993).
- G. H. Golub and C. F. V. Loan, Matrix Computations (The Johns Hopkins University Press, Baltimore, Maryland, 1983).
- S. Chen, D. Donoho, and M. Saunders, �??Atomic Decomposition by Basis Pursuit,�?? SIAM Review 43(1), 129�??59 (2001). [CrossRef]
- D. L. Donoho and M. Elad, �??Optimally Sparse Representation in General (Non-Orthogonal) Dictionaries Via L1 Minimization,�?? Tech. rep., Stanford University, Department of Statistics (2002).
- W. T. Freeman and E. H. Adelson, �??The Design and Use of Steerable Filters,�?? IEEE Trans. Pattern Analysis and Machine Intelligence 13, 891�??906 (1991). [CrossRef]
- R. Manduchi, P. Perona, and D. Shy, �??Efficient Deformable Filter Banks,�?? IEEE Trans. on Signal Processing 46, 1168�??1173 (1998). [CrossRef]
- F. Iannarilli, �??Spectro-Polarimetric Remote Surface Orientation Measurement,�?? US Patent 6,678,632 (2004).
- I. Daubechies, B. Han, A. Ron, and Z. Shen, �??Framelets: MRA-Based Constructions ofWavelet Frames,�?? Applied and Computational Harmonic Analysis 14(1), 1�??46 (2003). [CrossRef]
- J. D. Howe, �??Two-color infrared full-Stokes imaging polarimeter development,�?? IEEE Aerospace Conference (1999).
- J. A. Shaw and M. R. Descour, �??Instrument effects in polarized infrared images,�?? Opt. Eng. 34, 1396-1399 (1995). [CrossRef]
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