Band limited data reconstruction in modulated polarimeters

Charles F. LaCasse; Russell A. Chipman; J. Scott Tyo

doi:10.1364/OE.19.014976

1. Introduction

Active and passive polarimeters are powerful tools for remote sensing and these have been developed in all regions of the optical spectrum from UV through the long wave IR [1]. Imaging polarimeters have been used to detect targets in the presence of clutter [2], aid in target identification [3], penetrate scattering media [3–6] and aid in three-dimensional image reconstruction [7], among other tasks. Polarimeters have also been exploited for atmospheric sensing applications, including determination of aerosol properties [8], discrimination of ice/water phase particulates in clouds [9], and observation of plasmas in rocket engine exhaust [10]. As the applications for polarimetry become increasingly diverse, one of the challenges of polarimetry is the understanding and mitigation of polarization artifacts that occur with scenes that vary rapidly with time and space.

The polarization information in a spatially incoherent optical field is typically described using the Stokes parameters [11]

S = {[\begin{array}{l} s_{0} & s_{1} & s_{2} & s_{3} \end{array}]}^{T} = {[\begin{array}{l} I_{x} + I_{y} & I_{x} - I_{y} & I_{45} - I_{135} & I_{L} - I_{R} \end{array}]}^{T} .

In Eq. (1) I_x, I_y, I ₄₅, and I ₁₃₅ are the fluxes observed through ideal linear polarizers at the indicated orientations, and I_L and I_R are fluxes through ideal left- and right-circular polarizers, respectively. Optical detectors cannot measure s ₁, s ₂, and s ₃ directly. Instead, optical polarimeters modify the detected flux of the optical field in a polarization-dependent manner, and then reconstruct the Stokes parameters through an inversion process [1, 11].

Polarimeters are typically grouped into two classes [1]. The first class – Division of Time (DoT) polarimeters – include those polarimeters that make modulated measurements sequentially in time using a single set of temporally varying polarization components and a single detector. The second class – referred to as simultaneous measurement polarimeters (SMP) – analyze the flux in several states simultaneously using strategies that include the use of multiple detectors [12, 13], spatial modulation [14–16], or spectral modulation [17]. DoT polarimeters are widely considered to be more accurate because they are easier to calibrate and control, and tend to be the preferred choice for laboratory measurements where the Stokes parameters under measurement are not a function of time. However, DoT polarimeters are preferably not applied to rapidly time varying polarization signatures because of the false polarization artifacts which arise due to the time-varying signals that are introduced in the reconstruction process. Examples of DoT polarimeters applied to dynamic scenes have focused on employing extremely fast modulation schemes [18, 19].

The traditional categorization into DoT and SMP polarimeters is not necessarily the most useful way to classify polarimeters. An alternative distinction is between wavefront division polarimeters and modulated polarimeters. Wavefront division polarimeters split the light into multiple channels and make the constituent polarization measurements with independent hardware in each channel. Modulated polarimeters introduce a polarimetric modulation in time, space, wavelength, or some combination thereof, and measure the modulated signal using a single detector or detector array. The polarization signal is then determined by demodulating the information carried in the polarization-dependent side bands.

Wavefront division polarimeters have considerable hardware challenges associated with their construction. Multiple optical paths must be equalized, aligned, distortion matched, and cross-calibrated. Differential wavefront and polarimetric aberrations must be controlled from channel to channel. Temporal synchronization must be maintained. Ghost reflection and stray light variations from channel to channel introduce artifacts. When coherent light is used, speckle can present almost insurmountable problems. All of these factors raise issues that lead to increased cost and complexity in wavefront division polarimetry. The wavefront division advantage is that polarimetric imagery is produced at the full resolution of the individual detectors in a single simultaneous measurement cycle.

In contrast, a modulated polarimeter has several advantages over a wavefront division polarimeter working in the same domain. Alignment and synchronization are generally more easily obtained, though DoT polarimeters with moving optics suffer from beam wander artifacts that can degrade image quality [20]. The penalty for using a modulated polarimeter is generally a loss of resolution in the dimension of the modulation, since a single detector is being used to measure multiple quantities. This loss of resolution is associated with polarimetric aliasing, especially when the signal bandwidth is not adequately considered.

In a recent paper, Tyo, et al. [15], considered the sub-class of SMP polarimeters known as Division of Focal Plane (DoFP), or microgrid, polarimeters. These DoFP polarimeters use a focal plane array matched with a pixelated micropolarizer array in order to modulate the flux spatially. DoFP devices have been popular, but can be limited because of the intrinsic spatial mis-registration of their polarization measurements and the resulting edge-enhancement artifacts [21]. This limitation is analogous to the artifacts discussed above that occur when DoT polarimeters are used on time-varying scenes. Tyo, et al.. [15], derived a spatial band limit criterion on the input Stokes parameters that when satisfied guarantees artifact-free reconstruction of the Stokes parameters (in the absence of noise). DoT polarimeters are temporal analogues to microgrid polarimeters, so the spatial analysis of the DoFP polarimeters can be replicated in the time domain for division of time polarimeters. The implication of the band limit criteria is that the assumption that a Stokes signal is constant in time is unnecessarily limiting, just as the assumption for constant Stokes parameters across a 2x2 superpixel in a microgrid is unnecessarily limiting [15]. There is a band limit criteria for DoT polarimeters for artifact free reconstruction (in noise free cases), and both the Stokes parameters and the reconstruction method must satisfy the band limit criteria.

The remainder of this manuscript is organized as follows: section 2 reviews the conventional polarimetric data reduction method. Section 3 describes modulated polarimeters using a systems formalism. Section 4 generalizes the pseudo inverse method of section 2 for an arbitrary window shape. Two sequential in time polarimeter configurations are evaluated in section 5. Section 6 contains the conclusions.

2. Polarimetric Data Reduction Matrix method for DoT Polarimeters

The conventional Polarimetric Data Reduction Matrix (PDRM) method for polarimeter data reduction is well known [11] and is only summarized here. The polarimeter performs a series of flux measurements by using a set of retarders and analyzers that are changed from measurement to measurement. A common example for DoT polarimeters is the rotation of a wave plate in front of a linear analyzer [22], but many other configurations are used [8, 23, 24]. The incident Stokes parameters are analyzed by the polarization optics, and the transmitted flux is measured by a detector. The measured flux as a function of time I(t) is given by the inner product

I (t) = A {(t)}^{T} S (t) = [\begin{array}{l} A_{0} (t) & A_{1} (t) & A_{2} (t) & A_{3} (t) \end{array}] S (t),

where A(t) is the time-varying analyzer Stokes parameters given by the first row of the system Mueller matrix and S is the input Stokes signal. The k^th flux measurement is

I_{k} = A_{k}^{T} S = A_{0 k} s_{0} + A_{1 k} s_{1} + A_{2 k} s_{2} + A_{3 k} s_{3},

where A_mk is the m ^th row of the analyzer vector evaluated at the k ^th time sample. A flux vector of measurements I is related to the incident Stokes parameters and is formed as

I = [\begin{matrix} I_{1} \\ ⋮ \\ I_{K} \end{matrix}] = [\begin{matrix} A_{1}^{T} \\ ⋮ \\ A_{K}^{T} \end{matrix}] S = WS .

where W is the called the polarimetric measurement matrix. The incident Stokes parameters are calculated from the flux vector by matrix inverses as

Ŝ = W^{- 1} I,

where the hat indicates that Ŝ is an estimate of the true value. The matrix W ⁻¹ is generally referred to as the polarimetric data reduction matrix (PDRM) [11].

Among the set of matrix inversions, typically the pseudo inverse

W_{p}^{- 1} = {(W^{T} W)}^{- 1} W^{T}

is preferred since it prevents elements of the null space of W from affecting the measurement.

The above discussion provides a linear algebra perspective to polarimeter operation. A frequency domain approach provides a different perspective, and an example is helpful here. Consider the simple case of the rotating analyzer linear polarimeter; an ideal linear analyzer rotates with a constant frequency f ₀ (in units of rotations per second). The analyzer vector as a function of time is

A (t) = \frac{1}{2} {[\begin{array}{l} 1 & cos (4 π f_{0} t) & sin (4 π f_{0} t) & 0 \end{array}]}^{T},

and Eq. (3) becomes

I (t) = \frac{1}{2} (s_{0} (t) + cos (4 π f_{0} t) s_{1} (t) + sin (4 π f_{0} t) s_{2} (t)) .

Equation (8) indicates that the s ₀ information is contained in the base band, (unmodulated DC) term, the s ₁ information is contained in the in-phase (cosine) side band at 2 f ₀, and the s ₂ information is contained in the quadrature (sine) side band at 2 f ₀. The polarization information could be demodulated using frequency-domain methods, which is often done for systems using photo elastic modulators operating at very high rates (∼10s of kHz) [4]. However, the linear algebra formulation of Eq. (5) is often preferred for imaging polarimeters since it is easier to incorporate calibration, error analysis, and other practical considerations. In principle, only four measurements are required to uniquely invert Eq. (5) for the four Stokes parameters. However, it is common to employ more than four measurements in order to improve the accuracy of the polarimeter and reduce the effects of noise and systematic errors [8, 22, 25, 26].

3. A Linear Systems Formalism for Data Reduction

The polarimetry literature treats S as approximately constant in computing the PDRM with only a few exceptions [8, 27]. However, an analysis of Eq. (3) or Eq. (8) reveals that assuming constant Stokes parameters is overly limiting. Communications systems theory can be employed to allow the Stokes parameters to be varying in time, subject to a bandwidth criterion imposed by the reconstruction process. This result is analogous to results derived for spatially modulated imaging polarimeters [15, 16, 28] and spectrally modulated spectropolarimeters [17].

Assuming the Stokes parameters are functions of time and taking the Fourier transform of the signal I(t) in Eq. (2) yields

\tilde{I} (f) = {\tilde{s}}_{A 0} (f) * {\tilde{s}}_{0} (f) + {\tilde{s}}_{A 1} (f) * {\tilde{s}}_{1} (f) + {\tilde{s}}_{A 2} (f) * {\tilde{s}}_{2} (f) + {\tilde{s}}_{A 3} (f) * {\tilde{s}}_{3} (f),

where the tilde indicates Fourier transform and * is the convolution operator. Equation (9) is a deconvolution problem that can be inverted by careful design of the analyzer Stokes parameters A(t). For the example in Eq. (8) side bands in frequency space carry the polarization information as

\tilde{I} (f) = \frac{1}{2} ({\tilde{s}}_{0} (f) + \frac{1}{2} ({\tilde{s}}_{1} (f - 2 f_{0}) + {\tilde{s}}_{1} (f + 2 f_{0})) + \frac{1}{2 j} ({\tilde{s}}_{2} (f - 2 f_{0}) - {\tilde{s}}_{2} (f + 2 f_{0}))) .

Equation (10) demonstrates that requiring the Stokes parameters to be constant in time is an unnecessary restriction because perfect reconstruction can be achieved assuming that the Stokes parameters are band limited to frequencies below W_B (as seen in Fig. 1). The conditions for no overlap to occur in the frequency domain for the rotating analyzer polarimeter are

W_{B} \leq 2 f_{0},

and

f_{0} \leq \frac{f_{s}}{6},

where f_s is the temporal sampling frequency of the polarimeter. Other types of DoT polarimeters with different specific modulation schemes have their own band limit requirements but a criteria similar to Eq. (11) can be derived for other modulation schemes by applying a no side lobe overlap condition in the Fourier domain.

Fig. 1 A modulated measurement of the Stokes parameters using a rotating analyzer polarimeter, with each parameter band limited such that the signal can be ideally reconstructed. W_B is band width of each parameter, f ₀ is the frequency of analyzer rotation, and f_s is the detector sampling frequency. The dashed blue line indicates that S ₂ is in the quadrature component of the side band. The configuration with maximum allowed bandwidth is shown.

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4. Unifying the PDRM and Linear Systems Methods for General Modulated Polarimeters

This section generalizes the linear systems formalism of section 3 to relate to the traditional PDRM method. The key result is that the PDRM method allows mid frequencies of the Stokes parameters to corrupt signal estimates through improper demodulation. This section demonstrates that many polarization artifacts can be remedied by choosing a different weighting scheme in formulating the PDRM.

Consider a polarized signal that is a function of space, time, and wavelength S(x, y, t, λ). The signal is measured with a polarimeter that contains a single detector with an integration window d(x, y, t, λ), a system impulse response function h(x, y, t, λ), and a polarimetric modulation described by A(x, y, t, λ). The modulated flux is

I (x, y, t, λ) = (A {(x, y, t, λ)}^{T} (h (x, y, t, λ) * S (x, y, t, λ))) * d * (x, y, t, λ),

The functions in Eq. (12) are not necessarily linear, shift-invariant (LSI) functions. When the system is not LSI, the convolution integrals are replaced by overlap integrals of the more general form

f (x) * g (x) = \int_{- \infty}^{\infty} f (α) h (x, α) d α

rather than the more familiar convolution form for LSI systems

f (x) * g (x) = \int_{- \infty}^{\infty} f (α) h (x - α) d α,

where in both cases the input is f (x) and the impulse response is h(x). Furthermore, the impulse response h(x, y, t, λ) is assumed to be scalar in Eq. (12). In reality, the system is described by a polarimetric impulse response matrix that relates how the optical system alters the polarization state between object and image plane before it is ever sampled [29].

Referring back to Eq. (5), the PDRM formalism computes the pseudo inverse $W_{p}^{- 1}$ that operates on the flux vector I. The first part of the pseudo inverse in Eq. (6) is

Z^{- 1} = {(W^{T} W)}^{- 1} = {(\sum_{k = 1}^{N} A_{k} A_{K}^{T})}^{- 1} .

For example, a rotating retarder polarimeter might make N = 16 measurements as the retarder is rotated from 0° to 360°, or a DoFP polarimeter might be decomposed into 4-element (2 × 2) super pixels. In the linear systems formalism, a similar formation of Z is

Z_{i j} = \int A_{i} (t) A_{j} (t) d t,

where A_i, A_j are as defined in Eq. (2). The W ^T W part of the pseudo inverse is the integral of the product of the modulation functions contained in A, so the inversion of this quantity describes how to separate the Stokes parameters. Z and Z ⁻¹ are diagonal matrices for modulation schemes where the modulators A_j are orthogonal over the integral. The quantity Z ⁻¹ will be referred to as the modulator inner product inversion matrix.

If there are an integer number of periods in all of the modulation functions contained in A, Z ⁻¹ will be constant with respect to the initial phase of the modulation. However, in cases where the polarimeter varies in time or in space, as would be the case when there is drift in the absolute angular position of a freely running rotating retarder or in the absolute phase of oscillation of the PEM-based systems [8], then in general Z will not be constant and needs to be computed separately at each reconstruction point. The more general Z in the time domain can be calculated by

Z_{i j} (t) = w (t) * A_{i} (t) A_{j} (t) = \int w (t - t_{0}) A_{i} (t_{0}) A_{j} (t_{0}) {d t}_{0},

where w(t) is the reconstruction window used for the estimation of the Stokes signal. If w is a rectangular window with a length corresponding to integer periods of all modulation frequencies in the system, Eq. (17) reduces to Eq. (16); in general arbitrary window shapes and modulation schemes need the weighted inner product matrix elements calculated for all space time and/or wavelength.

Next, to understand the W ^T term in Eq. (15), examine how it operates on the modulated flux in the standard PDRM formalism,

W^{T} I = W^{T} WS .

The matrix W ^T plays two roles. First, W ^T acts as the homodyne in the demodulation process, since the matrix W includes the modulation strategy of A(x, y, t, λ). The homodyne process remodulates the modulated signal. When the modulation is made up of a superposition of sinusoids, multiplication by a homodyne creates a superposition of signals, with one component of the signal unmodulated (centered at base band) and other components of the superposition are copies of the signal modulated at higher frequencies. These copies can then be filtered out with a low pass filter, leaving only the component of the signal that is unmodulated. Multiplying the input Stokes parameters by W and then by W ^T is equivalent to mixing with a carrier frequency once to move the base band signal up to the side bands and a second time to create a copy at base band along with spurious copies at higher frequencies. In communications theory, the next step is to low pass filters to eliminate the spurious high frequency copies. The low pass filter is implicitly included in the matrix multiplication W ^T W, and this is the second role of W ^T. However, in the matrix multiplication the low pass filter has a rectangular footprint (in time, space, and/or wavelength). This rectangular footprint has a sinc-function frequency response that allows leakage of spurious high frequency signals into the reconstruction as shown below.

Using the linear systems formalism it is preferable to apply band limited low pass filters designed to eliminate the high frequency leakage. This is accomplished by separating the homodyne process from the low pass filter and applying the filter to the inverted signal according to

W_{p}^{- 1} W^{T} I \to w (x, y, t, λ) * Z^{- 1} (x, y, t, λ) A (x, y, t, λ) I (x, y, t, λ) .

In Eq. (19) A is a4 × 1 vector with each element a scalar function of space time and wavelength, not the 4 × N matrix that W conventionally implies. The quantity w(x, y, t, λ) is the windowing function in Eq. (17); this ensures that the reconstruction algorithm properly unfixes the modulators over the particular window that has been chosen for the estimation. Separating w from the inversion process allows for the optimization of the low pass filter operation and for the control over which portions of the frequency domain are included in the estimation. The advantage of this will become clear in the following section. In the special case when the Z matrix is constant over the modulation, the final estimation can be written as

Ŝ (x, y, t, λ) = W^{- 1} {I (x, y, t, λ)} = w (x, y, t, λ) * Z^{- 1} A (x, y, t, λ) I (x, y, t, λ) .

5. Discussion

This section contains examples of polarimeters measuring fluctuating scenes. This section considers polarimetric demodulation and high frequency leakage artifacts in the conventional PDRM and shows how to generalize the PRDM to eliminate them. Two signals with different bandwidths are shown in Fig. 2; the temporally wide signal ( $sinc (\frac{t}{10 t_{0}})$ ) satisfies the band limit requirement imposed by the polarimeter by Eq. (11)(a), while the temporally narrow signal ( $sinc (\frac{t}{5 t_{0}})$ ) has twice the frequency bandwidth as the polarimeter requirement W_B and hence will have errors.

Fig. 2 Two Stokes parameter input signals with different bandwidths; one satisfies the polarimeter band limit criteria (dashed), while the other has twice the allowed bandwidth for ideal reconstruction (solid).

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Section 5.2 analyzes a signal that satisfies the no overlap condition for the rotating retarder polarimeter that is similar to the condition discussed in Eq. (11) for the rotating analyzer polarimeter. Section 5.3 changes the signal to one that has twice the bandwidth of the signal in section 5.2.

5.1. Rotating Retarder Polarimeter

Consider a rotating retarder polarimeter that is free of noise and calibration error. Let the polarimeter be composed of an ideal linear retarder of retardance δ rotating at a frequency f ₀ followed by an fixed ideal linear analyzer with the orientation 0° such that analyzer as function of time is

A (x, y, t, λ) = A (t) = [\begin{matrix} A_{0} (t) \\ A_{1} (t) \\ A_{2} (t) \\ A_{3} (t) \end{matrix}] = \frac{1}{2} [\begin{matrix} 1 \\ \frac{1 + cos δ}{2} + \frac{1 - cos δ}{2} cos 8 π f_{0} t \\ \frac{(1 - cos δ)}{2} sin 8 π f_{0} t \\ - sin δ sin 4 π f_{0} t \end{matrix}] .

Assume instantaneous, periodic sampling such that d_n(t) = δ (t − nt ₀) at a point h(x, y) = δ (x, y). With the periodic sampling there are only samples at the sample number

n = \frac{t}{t_{0}}

. Although the example assumes point sampling for simplicity, a finite integration time is readily included by applying a convolution in the temporal domain by the integration window, or by a multiplication in the frequency domain by the transfer function of the detection process. For the simulation of the polarimeter described by Eq. (21), let

δ = \frac{2 π}{3}

. Sample such that

f_{0} = \frac{f_{s}}{10}

with

f_{s} = \frac{1}{t_{0}}

and acquire 80 measurements centered on the excitation. The modulator inner product matrix for one period is

Z = (\int_{t = 0}^{t = \frac{1}{f_{0}}} A (t) A^{T} (t) d t) = [\begin{matrix} \frac{1}{4} & \frac{cos (δ) + 1}{8} & 0 & 0 \\ \frac{cos (δ) + 1}{8} & \frac{3 {cos}^{2} δ + 2 cos δ + 3}{32} & 0 & 0 \\ 0 & 0 & \frac{{cos}^{2} δ - 2 cos δ + 1}{32} & 0 \\ 0 & 0 & 0 & \frac{{sin}^{2} δ}{8} \end{matrix}] .

The matrix elements are zero when the inner product between two components of the analyzer vector are orthogonal such as the case for A ₂ and A ₃. In Eq. (22) Z ₁₂ = Z ₂₁ ≠ 0 because the modulators A ₀ and A ₁ are not orthogonal; both have DC components.

5.2. Example: Band Limited Polarization Scene

The band limit criteria for the rotating retarder polarimeter of Section 5.1 that corresponds to the condition described by Fig. 1 is W_B ≤ 2 f ₀ and $f_{0} \leq \frac{f_{s}}{10}$ . Assume an excitation with the form ${sinc}^{2} (\frac{n}{10})$ in any one of the four Stokes parameters. The measurement of an arbitrary physical excitation can be decomposed into separate contributions from each Stokes component; Fig. 3 shows the contribution to the total measured flux for each Stokes component.

Fig. 3 The components of the flux I for an excitation of (a) ${sinc}^{2} (\frac{n}{10}) [1 0 0 0]$ , (b) ${sinc}^{2} (\frac{n}{10}) [0 \frac{1}{\sqrt{3}} 0 0]$ , (c) ${sinc}^{2} (\frac{n}{10}) [0 0 \frac{1}{\sqrt{3}} 0]$ , and (d) ${sinc}^{2} (\frac{n}{10}) [0 0 0 \frac{1}{\sqrt{3}}]$ . The vertical red lines indicate an example position of a 10 measurement sequence for comparison to the signal feature size.

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Figure 4 (Media 1) is animated showing how the both measured flux and the flux Fourier transform behave as the bandwidth increases; here Fig. 4(a) contains a single frame of the animation that shows the measured flux for the fully polarized signal

S = {sinc}^{2} (\frac{t}{10 t_{0}}) {[\begin{array}{l} 1 & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]}^{T},

obtained by summing terms in Fig. 3.

Fig. 4 Single frame excerpt from graphic displaying the measured flux as the bandwidth of the input signals increases (Media 1). (a) The measured flux I for $S = {1, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}} {sinc}^{2} (\frac{n}{10})$ in the temporal domain for a rotating retarder polarimeter with a third wave retarder. (b) The Fourier transform of the measured flux, with dashed lines representing imaginary components. The Fourier transform of (a) is given by the sum of all 4 components in (b). The colors in (b) indicate each individual Stokes parameter’s contributions.

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Figure 4(b) (Media 1) shows one frame of the animation of the Fourier transform as a function of bandwidth of the measured flux decomposed into the components arising from the individual modulated Stokes parameters shown in Fig. 3. The Fourier transform of the signal in Fig. 4(a) is obtained by summing all of the components in Fig. 4(b).

The first step in determining the time dependent Stokes parameters is to homodyne the measured flux with the functions given by A.

Figure 5 shows the Fourier transform of the homodyned flux A(t)I(t) using each of the four modulation functions in Eq. (21). The shaded region in Fig. 5 and all other following frequency domain plots indicates the band limit criteria for the modeled polarimeter. The first and second modulators place the s ₀ and s ₁ information in base band. The third modulator places the s ₂ information at base band, and the fourth modulator places the s ₃ information at base band.

Fig. 5 The results of the homodyne processes shown in the Fourier domain. (a) ℱ {A ₀(t)I(t)}, (b) ℱ {A ₁(t)I(t)}, (c) ℱ {A ₂(t)I(t)}, (d) ℱ {A ₃(t)I(t)}. Dashed lines indicate imaginary components. In panels (a) and (b) the base band exhibits coupling between the s ₀ and the s ₁ signals since the modulators A ₀ and A ₁ for these two signals are not orthogonal.

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Multiplication by the modulator inner product inversion matrix Z ⁻¹ separates s ₀ and s ₁ and equalizes the amplitude of all channels, yielding the result in Fig. 6.

Fig. 6 The four components of the estimated Stokes parameters given by Ŝ (f) = ℱ {ZA(t)I(t)} prior to low pass filtering with w(t) according to Eq. (20). Also shown in the dotted line is the rectangular low pass filter that ideally reconstructs the correct individual Stokes parameters. The marking γ is an example of self-error, while the marking ε is an example of cross-error. If the low pass filter w(t) does not reject these frequencies outside of the shaded base band artifacts will arise due to these error terms.

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Figure 6 shows how the time dependent Stokes parameter information has been placed at base band independently for each of the four Stokes parameters. After the signal has been demodulated there are still copies of both the desired signal and all other channels centered at various high frequency multiples of the first harmonic. If the filter choice for w does not reject the contributions from these copies, self-error (error in reconstruction resulting from the desired signal) and cross-error (error caused by channel cross talk) will occur. The artifacts caused by these copies are different from aliasing since there is no overlap in the frequency domain of these channels; aliasing will be discussed in section 5.3

The final step is low-pass filtering to extract the time dependent Stokes parameters. This is the step where most division of time polarimeter reconstruction strategies use the time-limited reconstruction window rather than a band limited strategy advocated here. By applying a band limited low pass filter, only the correct signals at baseband are allowed to pass. For this band limited input the filter perfectly reconstructs the full time-dependent input Stokes signal at base band using Eq. (20). In Fig. 6 it is seen that the ideal band limited filter given in the Fourier domain by $\tilde{w} (f) = rect (\frac{f}{0.2 f_{s}})$ , will reject all frequencies outside of the shaded base band region and provide artifact free reconstruction, with rect(x) = 1 for |x| < 1 and 0 otherwise.

In comparison Fig. 7 shows the effect of using a standard, 10-element rectangular window (as matrix notation in Eq. (5) implies)for $w (t) = rect (\frac{n}{16})$ corresponding to a sliding conventional PDRM with a width of 16 samples $(w (n) = rect (\frac{n}{16}))$ . Each of the four panels shows the Fourier transform of one of the Stokes parameters decomposed into the portions that arise from each of the inputs in Fig. 3. This filter choice leaks error at frequencies that were outside of the original band limit as well as attenuating the contrast of the parts of the spectrum that contained the desired signal.

Fig. 7 The four components of the estimated Stokes parameters in the Fourier domain given by ℱ {w(t) * ZA(t)I(t)} with $w (n) = rect (\frac{n}{16})$ , a 16 sample rect window. (a) ℱ {ŝ ₀}, (b) ℱ {ŝ ₁}, (c) ℱ {ŝ ₂}, and (d) ℱ {ŝ ₃}. Differences from a triangle function in the Fourier domain indicate errors in the data reduction method.

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In Fig. 7, substantial errors in the Fourier domain have been introduced. Using conventional data reconstruction only the DC term has been reconstructed error free. The error at high frequency manifests both as self-error (e.g. base band s ₀ information showing up as high frequency error in s ₀ signal) and cross error (e.g. base band s ₀ information showing up as high frequency error in the s ₁ signal), as well as base band signal attenuation.

Figure 8 shows the reconstructed Stokes parameters in the time domain, where the high frequency reconstruction error is evident. The results from a 16 element rect window are compared to the ideal reconstruction achieved by the band limited filter in Fig. 6. These artifacts have conventionally been described as being due to gradients in the signal [8].

Fig. 8 The time dependent Stokes parameters calculated using a sliding 16 element window compared to the input signal, which is ideally reconstructible a band limited window. (a) s ₀, (b) s ₁, (c) s ₂, and (d) s ₃

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5.3. Example: High Frequency Aliasing

Now consider the same rotating retarder polarimeter with a signal that has twice the band width (band width given by 2W_B) as in Eq. (11) the previous section, $S = [1 \frac{1}{\sqrt{3}} \frac{1}{\sqrt{3}} \frac{1}{\sqrt{3}}] {sinc}^{2} (\frac{n}{5})$ (solid line in Fig. 2). The measured flux (I) is shown in Fig. 9(a), while the Fourier transform split into components are shown in Fig. 9(b). Figure 9 corresponds Fig. 4 (Media 1) from the previous example, and is another singal frame from the animation showing how the measured flux behaves as signal bandwidth increases.

Fig. 9 Single frame excerpt from animation (Fig. 4 (Media 1)) displaying the measured flux for a scene with bandwidth given by 2W_B, with W_B the required bandwidth of the polarimeter for ideal reconstruction. (a) The measured flux I for $S = [1 \frac{1}{\sqrt{3}} \frac{1}{\sqrt{3}} \frac{1}{\sqrt{3}}] {sinc}^{2} (\frac{n}{5})$ in the temporal domain. (b) The Fourier transform of the contributions from individual Stokes parameters to the measured flux, with dashed lines representing imaginary components. The Fourier transform of (a) is given by the summing the 4 components in (b). The colors in (b) indicate contributions of the individual Stokes parameters from Fig. 3. This figure corresponds to Fig. 4 in section 5.2.

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The estimation using a band limited signal recovery algorithm (described in Eq. (20) with $\tilde{w} (f) = rect (\frac{f}{.2 f_{s}})$ ) is shown in Fig. 10. The aliasing artifacts that occur even when processing with a band limited window (as in Fig. 10) demonstrate the resolution penalty incurred by using modulated polarimeters. In the case of the rotating retarder used in this example, the bandwidth requirement is $\frac{1}{10} f_{s}$ compared to the Nyquist sampling requirement of $\frac{1}{2} f_{s}$ for a polarization blind system. Since this signal has twice the required polarimeter bandwidth (the bandwidth is $\frac{2}{10 f_{s}}$ ) there are now unavoidable self aliasing and cross aliasing polarimeter artifacts present, compared to the band limited example which was only degraded by frequency leakage artifacts.

Fig. 10 Shown are the four components of the estimated Stokes parameters reconstructed according to Eq. (20) with $\tilde{w} (f) = rect (\frac{f}{0.2 f_{s}})$ (ideal band limited window), estimating a signal that has a twice the bandwidth required for error free reconstruction. (a) ℱ {ŝ ₀}, (b) ℱ {ŝ ₁}, (c) ℱ {ŝ ₂}, and (d) ℱ {ŝ ₃}. The total signal for the estimate is calculated by summing each of the signals in the corresponding panel, so for example any contribution to the s ₀ estimate in panel (a) from the other three Stokes parameters results in reconstruction artifacts. This figure corespondents to Fig. 7 in section 5.2.

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Figure 11 shows the time domain representation for both the conventional time limited and the band limited reconstruction window advocated here. In both cases, there is unavoidable cross channel aliasing in the base band signal that introduces polarization artifacts. In this case the time limited window might be more visually appealing but an application based metric would need to be used to compare the results for a particular task.

Fig. 11 The time dependent Stokes parameters calculated using the time limited rect window and the band limited sinc window to be compared with the the input signal in Fig. 2. This figure corresponds to Fig. 8 in section 5.2.

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6. Conclusions

The theory of section 4 and the examples in section 5 demonstrate problems associated with traditional PDRM processing of modulated polarimeter data when the Stokes parameters are changing in time. The standard matrix formalism equally weights all of the observations in an N-element window in calculating the inversion of the measurement matrix W. Since the excitation in the conventional PDRM is assumed to be constant, equal weighting only guarantees proper reconstruction for constant signals. The band limited approach presented in this manuscript improves upon the conventional polarimetric data reduction matrix (time limited reconstruction) method for properly band limited time dependent Stokes parameters. When Eq. (11) is satisfied, error-free polarimetric reconstruction is possible for noise-free excitations. Equation (20) shows how to implement a band limited reconstruction algorithm for general modulated polarimeters. This result relaxes the requirement for high modulation frequencies to reduce error in DoT polarimeters [18, 19].

In section 5.3 a signal that was twice the polarimeter bandwidth criteria was considered; this example contained such high frequency fluctuations that polarization artifacts were unavoidable with either reconstruction method. Both algorithms presented in this manuscript produce significant error when applied to signals that contain fluctuations that are of higher frequency than the band width criteria determined by the polarimeter modulation. The choice of filter is much more complicated in the broader bandwidth example; increasing the bandwidth of the filter will allow more of the desired signal through, but will also pass more of the content from other channels. An optimal filter can be designed for a particular application based on the expected signal and polarimeter design.

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Band limited data reconstruction in modulated polarimeters

Abstract

1. Introduction

2. Polarimetric Data Reduction Matrix method for DoT Polarimeters

3. A Linear Systems Formalism for Data Reduction

4. Unifying the PDRM and Linear Systems Methods for General Modulated Polarimeters

5. Discussion

5.1. Rotating Retarder Polarimeter

5.2. Example: Band Limited Polarization Scene

5.3. Example: High Frequency Aliasing

6. Conclusions

References and links

Supplementary Material (1)

Cited By

Figures (11)

Equations (24)

Optics Express