of Qn+1 with respect to an individual channel PSF at pixel z = y − x is:
∂Qn+1 ∂hc(z) =X y ψn+1 pc (y, y − z) + ψucn+1(y, y − z) hc(z) −X y on+1 c (y − z) (4.14)
which is identical to the partial derivative with respect to the PSF of the objec- tive function in the polarization insensitive case. Consequently, we may directly incorporate Schulz’s PSF estimator (specifically, in [39] section 5) into the present case. What follows are the salient points of the original estimator with notes on some minor modifications; the reader is referred to the original article for a thorough derivation and an explanation of the advantages of this approach. Define:
hc(x, ϕn+1c ) = ¯ ¯ ¯ ¯ ¯ X u
A(u) exp£iϕn+1c (u)¤e−i2πkux ¯ ¯ ¯ ¯ ¯ 2 (4.15)
where A(u) is the aperture function of the imaging system (constrained such that P
uA(u) = 1), k is a constant related incorporating both wavelength and sampling effects, and ϕn+1
tion at the aperture for channel c. In the case of a circular aperture with radius r,
A(u) = 0 when kuk > r.
This definition of hc, which varies from Schulz’s original, allows for P
yhc(y) = 1 in equation (4.9). Though this is a necessary step before successfully invoking the trigonometric identities in (4.10), it comes with drawbacks. In the Schulz model, channel to channel gain variations, ac, such that
P
yhc(y) = ac, are accounted for in the joint estimator. Here, an acterm would be useful for modeling total transmission variations between channels. Total transmission, in this context, is equivalent to the action of an unknown neutral density filter. Modification of the estimator to allow for these variations is left to future work.
Since the aperture function is known in advance, the PSF estimation problem is recast into estimation of the phase screen, ϕc via one (or more) iterations of the Gerchberg-Saxton (GS) phase retrieval algorithm [11]:
ϕn+1 c = ˜ ϕc if P x ξ(x) ln hc(x, ˜ϕc) ≥ P x ξ(x) ln hc(x, ϕnc), ϕn c otherwise (4.16) where ξ(x) = hc(x, ϕnc) Dc X y dc(y) in c (y) on c(y − x) (4.17) given Dc= X x on+1 c (x) = X y dc(y) (4.18)
and one iteration of the GS algorithm: ˜ ϕc= ph n F−1hpξ(x, ϕn c) exp (i · ph (hc(x, ϕnc))) io (4.19) where F−1 is the inverse Fourier transform operator and “ph” is an operator that extracts the phase angle from a complex number. Equation (4.18) is a statement
of energy conservation. This term does not exist in its present form in the original paper; instead, there was an additional constraint in Schulz’s derivation: Pxo(x) =
1. The reader may easily verify, however, that this change is wholly consistent.
4.4 Test Results from Laboratory Data
In this section, the polarimetric blind deconvolution algorithm is put to the test using data from a laboratory imaging polarimeter. The test sensor consists of a Photometrics Cascade 512B camera, a single 250 mm focusing lens, and a variable polarization analyzer. The camera array is 512 × 512 pixels with a 16 um pitch, is cooled to −30o C, and has an approximately uniform response of 4 photons per count at 660 nm. Aside from quantization noise, the imager also exhibits “dark” noise and bias. At the irradiance levels shown below, this noise is weak compared to the photon dominated noise of the target. An average dark bias is subtracted from the data in post processing. The lens is stopped down to 3.175 mm to ensure proper sampling. In the configuration under test, the effective system magnification is 0.22. The test target consists of two fully polarized parallel bars, 2 mm in length, and back illuminated by a red (660 nm center wavelength) diode. The polarization angles of the two bars are approximately orthogonal: 2o for the top bar and −83o for the bottom bar (all angles are in reference to the horizontal direction in the imagery). The diode light passes through a diffuse screen prior to being polarized in order to even out the illumination across the target.
The collected data consists of three images, each collected at a different an- alyzer orientation: 0o, 60o, and −60o. The 60o and −60o collections are corrupted by a random plastic phase screen placed at the aperture. This plastic screen is weakly birefringent. Between the 60o and −60o collections, the orientation of the phase screen is rotated. In this way, each channel is presented with a different PSF analogous to the atmospheric short exposure imaging case. Before processing, the
images were cropped down to 200 × 200 pixels and coarsely registered. These data are shown in figure 4.1.
(a) the 0o channel (b) the 60o channel (c) the −60o channel Figure 4.1: The estimator test data.
In the 0o case, only one bar appears in the image because the −830 bar is almost fully suppressed by the channel analyzer. In the other cases, phase error dominates. For comparison to 3.1b and 3.1c, the aberration free data for the 60o, and −60o channels are shown in figure 4.2.
(a) the 60o channel (b) the −60o channel Figure 4.2: The 60o, and −60o channels in focus.
From the data in figure 4.1 the algorithm derives an initial guess at the po- larized and unpolarized parts of the scene, as shown in figure 4.3a and 4.3b. The remaining images in figure 4.3 show the results of the algorithm after 500 iterations (at which point the algorithm stagnates).
Recall that the true target image is fully polarized. Consequently, the restored
the case. Quantitatively, λu contains 9.34 × 107 photons in the initial estimate, and only 1.09 × 105 photons in the final estimate.
(a) initial λu estimate (b) initial λp estimate
(c) final λu estimate (d) final λp estimate
Figure 4.3: The unpolarized and polarized scene components before and after restora- tion.
Restoration of the fully polarized bar targets is achieved. What remains then, is to consider the restored angle of polarization. From equation (4.13), we see that an angle is estimated for every pixel whether or not the pixel contains meaningful polarization content. As an interpretably aid, figure 4.4 shows the on-target esti- mated angles masked off with the image in 4.3d. The estimated angles are −8o for the top bar and −72o for the bottom bar.
The ultimate cause of this angle bias, which has been observed across a multi- tude of measurements with different phase screens, is elusive. One possible cause for this discrepancy is that several real world effects are not taken into account in the model. For instance, the polarizers are less than ideal (extinction ratio of ≈ 106). In addition, unmitigated optical activity, Fresnel losses, or unmodeled signal atten- uation at the phase screen may be bias contributors as well. In the latter case,
−70 −60 −50 −40 −30 −20 −10 0
Figure 4.4: The recovered target angle of polarization (on target pixels only). inclusion of a Schulz ac like term could possibly be helpful. It is certainly possible that the remaining photons in the unpolarized image, if restored properly, could possibly compensate for this bias.
Finally, it is of interest to present the estimated point spread functions (figure 4.5). Sub-figure 4.5a contains the initial guess PSF for all channels, arbitrarily selected to contain 1 wavelength of defocus aberration. The remaining figures show the final estimates of the channel PSFs at the conclusion of the restoration. Each PSF in this figure is magnified ×2 compared to the scale in figure 4.1. Note that the rotation of the phase screen is evident between the 60o and −60o channels. The 0o channel, which is essentially unaberrated, tends toward a diffraction limited PSF.