The Fukushima nuclear fallout. On march 11, 2011 after an earthquake and the Tsunami that devastated the northeast coast of Japan, a multiple core meltdown happened at the Fukushima Daiichi Nuclear Power Plant, leading to a substantial release of radioactive material into the environment. Due to a general lack of detailed radiation measurement, Safecast, a citizen science group based in Japan, started building instruments for geo-localized measurements of radioactivity. The idea of the project was to reconstruct a map of radioactivity from these measurements. Let us first consider the following simpler one-dimensional problem. Suppose that you're interested in a continuous time function f of t, that is only known through some discrete set of measurements, the known samples here on the right side. In addition, these measurements might not be uniformly sampled, that is they do not lie on the regular grid. We would like nonetheless, to be able to say something about the value of the function in areas where no data is available. If it is possible to make a few assumptions about the function itself, it is impossible to make an educated guess about the value of the function at locations where the signal is unknown. This process is referred to as interpolation. In our example, we will consider as a particular case of band-limited functions. We can see here, a band-limited interpolation from the known samples. As you can see, is a dotted line or is a continuous-time function goes through all the red samples, so it does a perfect match, adds a known values, and it interpolates in between. Let us consider a continuous time T-periodic and band-limited function f of t. Since it is band-limited, this function can be parameterized by it's fully expansion coefficients up to a finite order M. This means there are two M plus one fully coefficients, that are conveniently represented in vector form by a vector C. So, here we have the expression of the periodic function, which is the expression of a fully series that we study in this class, and the vector is given here by two M plus one coefficients. We assume now that we know capital N pairs; t_i, z_i, where z_i is a value of the function at time t_i. We do not make any assumption about the locations of the t_i's. Given N large enough with respect to the maximum for a coefficient order M, we can make a least squares fit of the data to find the fully expansion coefficient that best approximates the data collected. We need at least N equal two M plus one measurements, but in general, we would like a much larger number because of possible measurement noise. So least square fit can be rewritten in matrix form and solved efficiently. But let us skip the details here. For a full demonstration, C is a corresponding iPad and notebook. Once we have recovered the best guess of the fully expansion coefficients, it is possible to re-sample the function f of t at any location we like. Often, the function is re-sampled on the regular grid. Let us give a first example. The function we chose here satisfies perfectly the assumptions of periodicity and band-limited mass. It is just the sum of two signs of frequency, one hertz and four hertz. The true function is represented by the thick black line. We pick at random 20 samples represented here by red dots. The model order is n is equal to six. The interpolation given by our least squares fit, is a dotted red line. Since all the assumptions are satisfied and because there is no noise, the interpolation recovers perfectly the original signal. You might be wondering what happens if the original signal does not satisfy some of the assumptions, as we might very well be the case in practice. For example, assume the signal is not band-limited. We take here the example of signal continuous everywhere, but not differentiable at zero. Such a signal is not band-limited. However, its three coefficients decay proportionally to the square of the frequency, that is quite fast. So non band-limitedness of the function can be somewhat compensated by picking a larger order for our model and having more samples. We choose here, the order model to be M is equal to 10 and take N is equal to 20 samples, picked again at random. We can observe that our interpolation is still very close to the true function. The approximation is however not so high around a sharp angle at the T is equal to zero. Let us now move on to the real world example of Fukushima. We are interested in measuring radiation levels all over Japan, and beyond. After the Tsunami of 2011 and the nuclear accident. Instruments were loaned to volunteers in a field, and measurements were carried out by fixing the sensors on cars. Sensors would then take a measurement of the radiation every five seconds. Therefore, sampling is generally limited to areas accessible by car. A sub-set of the measurements is shown here on a map of the Fukushima prefecture area. Each color dot is a measurement with yellow being low intensity, and red, high intensity. You can absorb some fairly high measurement levels on the power of the plume starting around a plant situated at the far right of the map and going north east. This is a typical example of where a location of measurements is constrained by available infrastructure. Since the measurements are done with a car, we can only measure on roads as plainly visible on the map. It is just necessary to apply interpolation to get the estimate of the intensity of radiation in areas where no samples are available, such as forests, fields, and soil. The interpolation is done using a model with M is equal to 441 fully coefficients and a number of samples at these close to a million. A function so obtained is enriched sampled on the regular grid and displayed here on the same map as before. A decent estimate of the total radiation field is obtained in this way. A few caveats are to mention. First, radiation is notoriously non band-limited and we will probably need a larger order model for better approximation. Second, large measurement noise is present, and also degrades the quality of the estimation. A few tricks were used in this reconstruction to improve the quality the estimation. The code and detailed explanations are available in the companion iPad and Notebook. Check it out.