When we learn physics in school, it is difficult to understand how to represent a falling ball as a math problem whose solution describes the real world. The first method we learned was Newtonian mechanics which is where everyone draws a force diagram, little arrows representing how a flower wants to evolve.
In college, we may learn something called Lagrangian mechanics which is a little different. For instance, instead of modeling forces, we talk about energy. Energy is often easier to find because it is always mass times velocity squared.
Lagrangian mechanics allows you to solve complex problems easily because you never need to know the forces at work.
But there is a trick to it. It requires identifying something called generalized coordinates, which can sometimes be challenging.
Actually, the invisible science behind Lagrangian mechanics revolves around a method or process known as the Calculus of Variations.
This involves two ideas: 1: that the differential equations of motion governing the dynamics of a classical or mechanical system can be deduced from a cost function and the fact that nature seeks to minimize the integral of this cost function. 2: that the way to identify the equations of motion require that we make small, even infinitesimal, changes in our path through the solution space.
In classical mechanics, the cost function has to do with energy minimization along a solution curve. Specifically, the cost function (Lagrangian) is defined on the tangent bundle (i.e. set of all tangent spaces to the solution curve, which contains velocity vectors). In Euclidean geometry, the “cost” function is actually the Euclidean distance, and is a metric defined on the solution curve or manifold itself (not the tangent bundle). The solution to problem in classical mechanics is a geodesic curve, i.e. one which minimizes the Lagrangian cost function.
In other domains, such as the infrared, it may be possible to construct a cost function based on the least action principle. However we must expand our notion of cost function.
In the example below, the epsilon parameter is a constant between 0 and 1 for each colored curve. Epsilon determines the scale of the effective blackbody curve, which is represented by the value epsilon = 1.
In the absence of any experimental data on the thermal radiator, all values of epsilon are equally likely. One may think of the radiator in as existing (from our perspective) in an undetermined state. By analogy to the Everett interpretation of coherent quantum states, unique versions of the radiator can be said to exist in separated realities. When an experimental observation of the radiator is made, a specific radiation curve is identified. The precise nature of the radiator is then known in “our” reality. In quantum mechanics, this process is called decoherence. A macroscopic object is unlikely to be in a true quantum state as such states require very low temperatures, however the uncertainty in radiant flux can be interpreted using the Everett representation.
The modeling task therefore is to perform a controlled decoherence of a large number of possible co-existant states into a single macroscopic state.
This can be performed using linear algebra, and the idea of a quadratic form as the cost function defined on a solution manifold. The matrix representing the form controls the mixing of individual time streams (realities). [per the Everett interpretation]
Note that the gray body curve is smooth, in contrast to the curve of the selective radiator. An example of a selective radiator would be a metal such as copper which when reduced to powder and burned in an open flame produces a specific set of peaks corresponding to its valence energy levels.
And the possible conclusions are startling!
The following image shows a frequency fingerprint of a radiating object, as seen by an infrared camera or sensor.
The dashed red line is the theoretical fingerprint based on a smooth blackbody radiation curve or model. This assumes that the object has certain properties which may or may not be actually present. These assumptions are the main reason why the actual observed fingerprint (the sold red line) does not match the theoretical model.
When data is taken of an object in space for example, all that we know is the frequency spectrum or fingerprint of the object. We usually do not know what the object is actually made of, it’s shape, etc. We have to make predictions or really educated guesses for that based on the observations.
Our goal at Pinkmoon is to develop a computer algorithm or model which can rapidly identify possible sources of discrepancies between the theory and observations, such as noise in the detector or camera, dynamics such as rotation, or background illumination.
One way to approach this problem is to do a kind of decomposition of the original curve into a set of known components (i.e. transformations or convolutions of the theoretical model). This is similar to a Fourier transform. Because multiple solutions might be possible, a cost function is desired which identifies which level sets or solutions might be more likely than others.
We are using open source tools such as Anaconda, Pyradi, and other technologies including AI to arrive at an algorithm for this problem set.
This article is devoted to PyRadi, which is an infrared modeling toolkit written for Python. Python is a computer language similar to MATLAB which runs on most computer systems and is completely free.
PyRadi is useful because it simplifies the process of designing and evaulating infrared vision systems. Infrared vision systems are able to detect heat signatures of life on Earth, and in other places. Additionally they can also detect many naturally occuring phenomena such as bacteria in the ocean which help our planet support life.
In my experience, infrared systems can be used for remote sensing, which means that they are deployed in space and used to observe the Earth. NASA has a mission similar to this called VIIRS which monitors ocean algae populations and tracks the rate at which they bloom.
PyRadi is completely open ended and can be played with and modified by anyone with internet access. I have been doing this for years.
Here is a summary some of the things you can do with Pyradi.
WHen you want to design a system for seeing infrared radiation, whether on the Earth or moon or both, you need to determine the performance as you generate your design. This is because the performance helps you understand how well your design will work, efficiently.
PyRadi helps with this. If you give it some information about the source of infrared light (just a rough light curve, not too hard to find for most objects), and some other basic information about your system (i.e. telescope, etc) then PyRadi will show you its guess at the performance. This is measured by the amount of signal that your system will be able to receive from it’s target or source.
Here is some output from PyRadi.
The first graph shows several curves which can be thought of as components of a transfer function, as commonly used in control theory. For anyone who doesn’t know this already, a transfer function is represented as a product of several smaller functions. It is part of the solution to a differential equation, which is the basic mathematical model of a system that evolves with respect to time (i.e. dynamical system).
Here is an example:
In this figure, the box labeled Plant is the portion of the system which we are interested in studying or modeling. The “transfer function” is basically all of the interesting aspects of the physical system which we care about. In the case of Pyradi, it includes the following:
- Source basic radiant properties
- Source emisivity or tendancy to illuminate debris
- Intervening atmosphere
- Optics passivity
- Camera sensitivity
In summary, the PyRadi enables you to do what is called a Sensitivity Analysis of your experiment.
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