Polarization Modality Handbook

Here we will talk about the S-P coordinate system, and how polarization is relative to some coordinate system. Here we can outline how global polarization needs to be translated into local polarization to interact with a surface or to be captured by a camera. Pillage figures from Schott’s book and Mike’s thesis.

In some cases, the spectral data type is a Stokes Vector, which can be used to store fluxes including radiances, irradiance, radiant intensities, etc. A Stokes Vector is a 1 x 4 vector representation of flux that is capabile of of storing data ranging from unpolarized (randomly polarized) data, to linearly polarized data, to circularly polarized data and all combinations in between.

\begin{equation} \vec{S} = \left [ \begin{matrix} S_0 \\ S_1 \\ S_2 \\ S_3 \end{matrix} \right ] \end{equation}

In other cases, the spectral data type is a Muller Matrix, which can be used to store the optical properties of surfaces including reflectances and emissivities or the optical properties of participating media including scattering coefficients, extinction coefficients, transmissions, etc. A Mueller Matrix is a 4 x 4 matrix that can applied to a Stokes Vector to impart polarization, change polarization and/or remove polarization (depolarize).

\begin{equation} M = \left [ \begin{matrix} m_{00} & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{matrix} \right ] \end{equation}

Stokes-Mueller calculus is a specialization of vector matrix math operations that are employed to perform polarized radiometry calculations. For example, the reflection of an incident Stokes Vector entails multiplying that Stokes Vector by a Mueller Matrix that describes the polarized reflectance characteristics of the surface and yields a reflected Stokes Vector:

\begin{equation} \left [ \begin{matrix} S_0 \\ S_1 \\ S_2 \\ S_3 \end{matrix} \right ] = \left [ \begin{matrix} m_{00} & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{matrix} \right ] \left [ \begin{matrix} S_0 \\ S_1 \\ S_2 \\ S_3 \end{matrix} \right ] \end{equation}

In an effort to optimize both memory usage and computations, the Stokes Vector and Mueller Matrix data types in DIRSIG support either a polarized or unpolarized state. If the radiometric term has any degree of polarization, then the data type is a full Stokes Vector (1 x 4 vector) or Mueller Matrix (4 x 4 matrix). However, if the term is completely unpolarized (randomly polarized) then the term can be represented as a single scalar value.

\begin{equation} \vec{S} = \left [ \begin{matrix} S_0 \\ 0 \\ 0 \\ 0 \end{matrix} \right ] \rightarrow \left [ S_0 \right ] \end{equation}

\begin{equation} M = \left [ \begin{matrix} m_{00} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right ] \rightarrow \left [ m_{00} \right ] \end{equation}

When working with unpolarized terms, the Stokes-Mueller calculations operations can be simplified in an effort to eliminate math operations. For example, the multiplication of two polarized Mueller Matrix terms involves 16 multiplication and 12 addition operations. However, the multiplication of two unpolarized Mueller Matrix terms is single (scalar) multiplication. If any term in the radiometric chain is unpolarized, then the flux will be completely depolarized by that term.

\begin{equation} \left [ \begin{matrix} S_0 \\ 0 \\ 0 \\ 0 \end{matrix} \right ] = \left [ \begin{matrix} m_{00} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right ] \left [ \begin{matrix} S_0 \\ S_1 \\ S_2 \\ S_3 \end{matrix} \right ] \end{equation}

By internally tracking if a Mueller Matrix is unpolarized, a significant number of mathermatical operations can be avoided. In the above example featuring an unpolarized Mueller Matrix, the Stokes-Mueller calculus can simply be simplified to simply a scalar product as:

\begin{equation} \left [ m_{00} \right ] \left [ S_0 \right ] = \left [ \begin{matrix} m_{00} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right ] \left [ \begin{matrix} S_0 \\ S_1 \\ S_2 \\ S_3 \end{matrix} \right ] \end{equation}

This is just one example of how the Stokes-Mueller calculus can be optimized for special cases. Optimizations like these are done at the lowest level of the DIRSIG code and is not something the user needs to worry about. In addition to making polarized DIRSIG simulations maximally efficient, it allows the same exact DIRSIG core to be used for unpolarized simulations.

On of the sources of polarization is that the illumination in the scene is polarized. The primary sources of illumination are the sun, the moon and the sky (which arises from scattered sunlight). The solar, lunar and sky radiation loads in the DIRSIG model are handled by the atmosphere sub-model. In most cases, these sub-models use the SSI atmospheric radiation propagation codes. The primary code used by DIRSIG is the MODTRAN (the MODerate resolution TRANsmission) model.

The DIRSIG model currently supports two internal sources of polarization. One source is an implementation of the polarized version of the classic Torrance-Sparrow BRDF model described by Priest and Germer in their paper "Polarimetric BRDF in the Microfacet Model: Theory and Measurements" This parameterized BRDF model relies on user-supplied a spectral complex index of refraction and surface roughness metric. In practice, the surface roughness is usually treated as a spectral property to acheive better fits to measured data. The second source is an implementation of the polarized background BRDF model described in Jim Shell’s PhD thesis Additional polarized BRDF models are being developed and evaluated by other organizations and more models will be incorporated into DIRSIG at a later time.

The DIRSIG4 radiometry model is a significant redesign of the radiometric modeling used in DIRSIG3. Every object in the DIRSIG simulation has a material assigned to it including internal elements like the sun, moon, sky and the bulk atmosphere. Each material has a radiometry solver assigned to it. A radiometry solver is an object that predicts the flux from an element under a specific set of conditions and using a specific set of material properties. The radiometry solver sub-system is defined by an interface to these computational objects. The most common radiometry solver is the "classic" solver which implements the same surface reflected radiance and emitted radiance algorithm that was used in DIRSIG3. The "classic" solver makes assumptions that each material has a surface emissivity description that can be used to compute the reflectance (assuming that Kirchoff’s Law holds and the reflectance can be computed as 1 minus the emissivity). When the user supplies DIRSIG4 with a DIRSIG3 simulation configuration, scene materials are automatically assigned the "classic" solver. To use other radiometry solvers, the user needs to make changed to the material database, and that process will be discussed in later sections.

The next generation radiometry solver is the "generic" radiometry solver which is a very flexible solver with which the user can assign various surface and bulk optical properties (reflectance, emissivity, scattering and extinction properties) to. These optical properties are defined against defined internal interfaces which allows various optical properties to be abstracted and thus allowing the "generic" radiometry solver to use a very flexible set of surface and bulk optical properties.

In the current version of DIRSIG4, the user has a limited set of optical properties to assign to the "generic" radiometry solver (however, DIRSIG3 had only one). The "classic emissivity" property can be used to model the surface emissivity. This property reproduces the unpolarized emissivity and reflectance support that was modeled in DIRSIG3 (including the classic texture methodology) using the same emissivity files that DIRSIG3 utilized. As an alternative, the user can use the "Priest-Germer" or "ShellBackground" BRDF properties to model a polarizing surface reflectance.

To incorporate polarization signatures into the simulation, the user needs to run a simulation that includes polarized illuminates (provided by a polarized atmosphere model), polarizing surface materials and a way to capture the polarized radiation at the sensor. This section will explain how-to configure these components of the modeling chain.

This section will describe the various options for setting up a polarized receiving instrument in DIRSIG.

To correctly model a polarized scenario you will need to have access to the Air Force Research Laboratory’s (AFRL’s) MODTRAN4-P atmospheric radiation transfer model, which was a polarized varient of the MODTRAN4 code base. This model is currently the only way to provide DIRSIG with model-based atmospheric properties that are polarized. However, the MODTRAN-P model is available only to a select beta-testing community. If you were not already a part of this user community, it is unlikely that you will be able to gain access to this modeling tool.

When the polarization modeling is enabled MODTRAN-P, the radiances produced by the MODTRAN model will be spectral Stokes Vectors (S0, S1, S2 and S3) and a polarization angle. The source of polarization within MODTRAN-P is the scattering phase functions which predict scattered radiances that are polarized. By defintion, the directly transmitted terms (direct solar and lunar) are not polarized since they are not scattered. The model can be used to predict polarized path radiances between the sensor and the scene and to predict a polarized sky.

The DIRSIG4 model supports the MODTRAN-P atmospheric model through a the Classic atmosphere model and the make_adb utility. The configuration and use of this utility for polarization will be discussed later. In DIRSIG4, the atmospheric modeling is handled by an atmosphere sub-system that is defined by a specific internal interface. DIRSIG4 has a growing number of unique atmospheric models that the user can choose from at run-time. At one end of the spectrum is the "simple atmosphere" model which has harded coded solar irradiance, lunar irradiance, and sky irradiance curves. This model is useful for testing purposes because a rigorous atmospheric database does not to be constructed for each simulation. More exotic atmospheric models are under development which support spatially inhomogenous properties but which require more input data and run-time resources.

The polarized atmospheric databases produced by the DIRSIG4 version of make_adb are handled by the "classic atmosphere" model in DIRSIG4 which implements the same treatment of the atmosphere that appeared in DIRSIG3. Polarization support will be added to the new DIRSIG4 atmospheric models as they near completion.

We could include a section that points out how 6S can do polarization but we are not supporting it yet.

At this time, there are only four different optical properties available that can be assigned to a material. They span the three primary properties (reflectance, emissivity and extinction) that can be assigned to a material:

At this time, there are two different radiometry solvers that can be assigned to each material:

To emphasize the flexibility in this radiometry architecture, we should point out that although each material may be using the same radiometry solver, the solver for each material can be configured with a different set of optical properties, input parameters and options.

The Shell-Background BRDF model is a reflectance optical property and it is configured for use by the Generic Radiometry Solver for a specific material by supplying the name "ShellBackground" to the REFLECTANCE_PROP_NAME variable (see below):

The BRDF_FIT_FILE, described below indicates the file containing the fit parameters for the BRDF model. The EMISSIVITY_FILE indicates a DIRSIG emissivity file. The first curve from this file will be converted to reflectance and used by the model.

The fit file is composed of two section types. The variable names used in both cases are identical to those in the thesis. The first section type, SHARED_FIT_PARAMS, holds the wavelength-independent parameters.

The second section type, FIT_PARAMS, holds wavelength-dependent parameters. Multiple instances of this section may be present in a fit file. The model interpolates between available measurements as described by Gartley (dude, write some stuff here!)

At this time, you can configure either a reflectance or emissivity property within one "generic" radiometry solver. You can pair either a reflectance or an emissivity property with the extinction property.

The "Raw" capture method allows the model to produce spectral-polarimetric output that can be processed by external sensor models. When this output is requested, the resulting image data cube is essentially 4-dimensional with 2 spatial dimensions, 1 spectral dimension and 1 Stokes dimension. This image format could be called "Stokes interleaved by Band interleaved by Pixel" which means that the data is organized as such:

Where Yi and Xi are the spatial X,Y pixel locations, Bi is the band/wavelength, and Si is the Stokes Vector index. The image data file elements are double-precision floating-point (native double) that is written in the native endian. An ENVI compatible image header file is automatically generated which should aid in the parsing of this file.

This section will explain how to setup the "Simple" capture method with spectral responses but which output Stokes Vector images directly.

This section will explain how to setup the "Simple" capture method with spectral responses that also include a polarized analyzer.

Basic reflective polarization demo with the chunky bar using the Priest-Germer pBRDF model with aluminum with different surface roughnesses.

A basic emissive polarization demo with the spheres.

An example setup using the Shell Background model with texture.

A four camera "division of aperture" collection setup.

A 2x2 micro-grid array "division of focal plane" collection setup using a data-driven focal plane setup.