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We present the analysis and design of spatio–temporal channeled Stokes polarimeters. We extend our recent work on optimal pixelated polarizer arrays by utilizing temporal carrier generation, resulting in polarimeters that achieve super-resolution via the tradeoff between spatial bandwidth and temporal bandwidth. Utilizing the channel space description, we present a linear-Stokes design and two full-Stokes imaging polarimeter designs that have the potential to operate at the full frame rate of the imaging sensor of the system by using hybrid spatio–temporal carriers. If the objects are not spatially bandlimited, the achievable temporal bandwidth is more difficult to analyze; however, a spatio–temporal tradeoff still exists.

See the corresponding SPIE talk on youtube here.

Micropolarizer arrays are occasionally used in partial Stokes, full Stokes, and Mueller matrix polarimeters. When treating modulated polarimeters as linear systems, specific assumptions are made about the Dirac delta functional forms generated in the channel space by micropolarizer arrays. These assumptions are 1) infinitely fine sampling both spatially and temporally and 2) infinite array sizes. When these assumptions are lifted and the physical channel shapes are computed, channel shapes become dependent on both the physical pixel area and shape, as well as the array size. We show that under certain circumstances the Dirac delta function approximation is not valid, and give some bounding terms to compute when the approximation is valid, i.e., which array and pixel sizes must be used for the Dirac delta function approximation to hold. Additionally, we show how the physical channel shape changes as a function of array and pixel size, for a conventional 0∘, 45∘, −45∘, 90∘ superpixel micropolarizer array configuration.

Currently, the best performing micropolarizer array is the 2 × 4 pattern introduced by LeMaster and Hirakawa. In this Letter, we extend the available set of patterns with the aim of improving reconstruction quality by leveraging the Fourier domain and designing information carriers that yield optimal bandwidth. First, the family of 2 × L patterns widens the optimization space of the 2×4 pattern by facilitating variable allocation of bandwidth for channels surrounding polarization and intensity carriers. Second, the 2 × 2 × N patterns present an intriguing option for use within a hybrid spatiotemporal modulation scheme, where the multiple temporal measurements enable maximum theoretical spatial resolution of reconstructed Stokes parameters.

Designing polarimetric systems directly in the channel space has provided insight into how to design new types of polarimetric systems, including systems which use carriers in hybrid domains of space, time, or spectrum. Utilizing linear systems theory, we present a full Stokes imaging polarimeter design which has the potential to operate at half the frame rate of the imaging sensor of the system by utilizing a hybrid spatio-temporal carrier design. The design places channels on the faces and the edges of the Nyquist cube resulting in the potential for half the Nyquist limit to be achieved, provided that the spatial frequency of the objects being imaged are bandlimited to less than 0.25 cycles per pixel. If the objects are not spatially bandlimited, then the achievable temporal bandwidth is more difficult to analyze. However, a spatio-temporal tradeoff still exists allowing for increased temporal bandwidth. We present the design of a “Fast Stokes’’ polarimeter and some simulated images using this design.

Little publicly available data exists for polarimetric measurements. When designing task specific polarimetric systems, the statistical properties of the task specific data becomes important. Until better polarimetric datasets are available to deduce statistics from, the statistics must be simulated to test instrument performance. Most imaged scenes have been shown to follow a power law power spectral density distribution, for both natural and city scenes. Furthermore, imaged data appears to follow a power law power spectral distribution temporally. We are interested in generating image sets which change over time, and at the same time are correlated between different components (spectral or polarimetric). In this brief communication, we present a framework and provide code to generate such data.

Matlab software to manipulate the unit cell parameters, or derive unit cell parameters by choosing channels for any type of irradiance filter array on a rectangular lattice. See the Optics Express article “Focal plane filter array engineering I : rectangular lattices.”

Capabilities also include generating image reconstructions via linear filtering in the channel domain, and the MSE between the truth images and the reconstructed images.

If this software is used for any purpose, attribution must be made. If the software is used for a publication, the above paper and this code must be cited.

A tutorial on usage is at https://www.youtube.com/watch?v=_hR0-QCUm_M

Focal planes arrays (FPA) measure values proportional to an integrated irradiance with little sensitivity to wavelength or polarization in the optical wavelength range. The measurement of spectral properties is often achieved via a spatially varying color filter array. Recently spatially varying polarization filter arrays have been used to extract polarization information. Although measurement of color and polarization utilize separate physical methods, the underlying design and engineering methodology is linked. In this communication we derive a formalism which can be used to design any type of periodic filter array on a rectangular lattice. A complete system description can be obtained from the number of unit cells, the pixel shape, and the unit cell geometry. This formalism can be used to engineer the channel structure for any type of periodic tiling of a rectangular lattice for any type of optical filter array yielding irradiance measurements.

Imaging polarimeters have been largely used for remote sensing tasks, and most imaging polarimeters are division of time or division of space Stokes polarimeters. Imaging Mueller matrix polarimeters have just begun to be constructed which can take data quickly enough to be useful. We have constructed a Mueller matrix (active) polarimeter utilizing a hybrid modulation approach (modulated in both time and space) based on a micropolarizer array camera and rotating retarders. The hybrid approach allows for an increase in temporal bandwidth (instrument speed) at the expense of spatial bandwidth (sensor resolution). We present the hybrid approach and associated reconstruction schemes here. Additionally, we introduce the instrument design and some preliminary results and data from the instrument.

Appeared in *Proc. SPIE* 9613, Polarization Science and Remote Sensing VII, 961312 (September 1, 2015); doi:10.1117/12.2188675

Recently we designed and built a portable imaging polarimeter for remote sensing applications. Polarimetric imaging operators are a class of linear systems operators in the Mueller matrix reconstruction space, resulting in a set of measurement channels. The nature of remote sensing requires channel crosstalk to be minimized for either general Mueller matrix reconstruction or task specific polarimetric remote sensing. We illustrate crosstalk issues for a spatio-temporally modulated Mueller matrix reconstruction operator, and show how to minimize channel crosstalk by maximizing bandwidth between channels. Specifically channel cancellation allows increases in channel bandwidth. We also address the impact that systematic deviations from the ideal operators and i.i.d. noise have on the system channel structure.

This paper appeared in *Proc. SPIE* 9613, Polarization Science and Remote Sensing VII, 961305 (September 1, 2015); doi:10.1117/12.2188653

Goudail and Beniere claim that in certain circumstances, specifically in a polarimeter with systemic independent additive Gaussian white noise (AGWN), fewer measurements are better. This claim is counter to what is derived in most statistical estimation or imaging science books. In this paper, I analyze conditions in which fewer measurements (or statistical samples) result in better estimation of an

object through an imaging system for independent additive Gaussian white noise when using the maximum likelihood estimate and reconcile the differences. I also seek to derive a better estimator using the full vector electric field propagation through a polarimetric instrument to derive a more robust imaging operator.

NOTE : This document was removed and deleted due to a cease and desist letter from Advanced Optical Technologies, Inc. in Albuquerque, NM. If you are interested in the document, please contact them. However, a general literature search of the terms *per pixel classification, segmentation, object classification, *and *per pixel vs object OR segmentation classification *will yield a great deal of open and public domain results on the subject.

Two commonly used performance metrics for two class classification algorithms are the receiver operating characteristic (ROC) curve and Empirical Risk. These two performance metrics can be related in order to directly compare scores once a point on an ROC curve is known. ROC is a more general description overall, but with certain assumptions we can compute one in terms of the other.

Support vector machines are conceptually simple. Suppose we have data points in some *n* dimensional space, each one assigned to a class in the set *{+1, -1}*. Now suppose that we want to construct an *n-1* dimensional surface (called a manifold) that separates the data in the two classes as best as possible. The first surface to try is just an *n-1* dimensional plane. The plane is flat, so will often be a poor separating surface (if for example a spherical surface better separates the classes).

Waveplate retardance can be corrected via measuring known optical elements.

NOTE : There is a better algorithm , *the eigenvalue calibration method*, in the public domain here.

I received an award for 2nd place for best poster.

The extra-tropical transitions (ET) of tropical cyclones are significant contributors to weather related disasters globally. One way to reduce the societal impact of these disasters is to provide early warning of these events, which can potentially be accomplished via full numerical simulation, but using full numerical modeling has proven to be difficult due to the apparent chaotic nature of the underlying system dynamics. Early warning can also be accomplished via machine learning techniques and classification of ET events. Our previously published work used support vector machines (SVM) to attain a probability of detection (PD) of ~76% with a corresponding false alarm rate (FAR) of ~27% on a subset (a single pressure surface of potential temperature) of back-fitted full-physics numerical prediction model data. In this study we extend this subset to the the full volume of the Western Pacific for both potential temperature and equivalent potential temperature for input into the SVM, attaining ~80% PD with a ~26% FAR. We also apply SVM to the full model data for the Western Pacific, a data set of ~600 million points, which is difficult computationally. SVM is a supervised learning technique, and does not elucidate the natural manifold of physically meaningful data embedded in the model data, for this reason we apply Riemannian Manifold Learning (RML) to the model data for all variables and the volume of the Western Pacific region. The implementation of RML is being accomplished utilizing the CUDA framework, which uses consumer grade graphics cards (NVidia GTX 465s) for fast parallel computing. RML will potentially reduce the dimension of the relevant data, and give us a meaningful manifold. The resulting manifold can then be submitted to a supervised learning algorithm for binary classification.

S. R. Felker, J. S. Tyo, E. A. Ritchie, and I. J. Vaughn, “Support vector machine techniques to predict tropical cyclone re-intensification following extratropical transition,” in AMS Conference on Hurricanes and Tropical Meteorology (2010)

I. J. Vaughn, “The imaging equation for a microgrid linear Stokes polarimeter,” Proc. SPIE 8160, 816008 (2011)

Copyright SPIE, 2011

I. J. Vaughn, B. G. Hoover, J. S. Tyo, “Classification using active polarimetry,” Proc. SPIE 8364, 83640S (2012),

Copyright SPIE, 2012.

T. Wakayama, K. Komaki, I. J. Vaughn, J. S. Tyo, Y. Otani and T. Yoshizawa, “Evaluation of Mueller matrix of achromatic axially symmetric wave plate,” Proc. SPIE 8873, 88730P (2013)

While several analyses of polarimeter noise-reduction have been published, little data has been presented to support the analytical results, particularly for a laser polarimeter based on measurements taken at discrete, independent rotation angles of two birefringent waveplates. This paper derives and experimentally demonstrates the reduction of both system and speckle noise in this type of laser polarimeter, achieved by optimizing the rotation angles of the waveplates by minimizing the condition numbers of the appropriate matrix equation. Results are demonstrated experimentally in signal-to-noise ratio (SNR) variations for a range of materials and spatial bandwidths. Use of optimal waveplate angles is found to improve the average SNR of the normalized Mueller matrix over speckle by a factor of up to 8 for a non-depolarizing material, but to provide little improvement for a depolarizing material. In the limit of zero spatial bandwidth, the average SNR of the normalized Mueller matrix over speckle is found to be greater than one for a non-depolarizing material and less than one for a depolarizing material.