README.md

(Tools for a Poisson, Gaussian, and Multiplicative error model)

This is a software package, originally distributed in association with Sipkens et al. (2017), which evaluates a general error model for optical signals, including Poisson-Gaussian noise and changes in the measurement conditions between repeated observations (e.g., between measurement repeats, such as across multiple laser shots). Particular focus is placed on time-resolved laser-induced (TiRe-LII) and the shot-to-shot variations in quantities like the laser energy and particle volume fraction in the probe volume. This code replaces an archived version available on figshare.

Two versions of this code are available:

1. A Matlab version code is available in the upper repository. The Matlab code was developed for use with MATLAB 2016a running on Windows and is demonstrated below. Much of the remainder of this README pertains to this version of the code.

2. A Javascript version of the code is also available in the js/ directory. The Javascript also uses d3.js to build a web app that demonstrates the error model, available at https://tsipkens.github.io/pgm-error/.

Demonstrating the Matlab code

Simulated signals

The simplest demonstration of this program starts with a theoretical, noiseless signal, considered here with respect to TiRe-LII signals. For simplicity, a sample set of data was included with this distribution for reference. It can be loaded using:

data = csvread('data/lii.csv');
t = data(:, 1); % time
J = data(:, 2); % incandescence

This will load three variables into the workspace: (1) lambda contains the wavelength for the loaded signal; (2) t contains a vector of sample times for the signal; and, most importantly, (3) J contains an incandescence trace, evaluated using the Michelsen model from Michelsen et al. (2007). Next, define the relevant error model parameters:

% Define error model parameters
tau = 0.2; % shot-to-shot variation as a dimensionless std. dev.
the = 1; % amplification / scaling factor
gam = sqrt(2); % Gaussian noise level
                 % in percent of max, i.e. 15 = 15%

The parameters are generally described in Sipkens et al. (2017). Briefly, tau controls the amount of variation between observations (e.g., between shots), the controls the level of Poisson noise (indirectly, by accounting for scaling of the signal), and gam controls the cosntant Gaussian component of the background. Now, simulate 500 laser shots:

N_shots = 500; % number of shots to simulate
s_bar = J .* the; % expected mean signal
[s, ~, ~, out] = add_noise(s_bar, tau, the, gam, N_shots);
    % generate observed signals, with error

The second line creates a signal by scaling by the incandescence by the the factor above (necessary as the signal and incandescence are given in different units, requiring a conversion). The third line here is the primary call in this program, taking the generated, noiseless signal and creating 500 noisy signals (including shot-to-shot variations as specified by tau).

Finally, let's visualize the results. First, plot the fifth realization of the noisy signal:

figure(1);
plot(t, s(:,5), 'b.'); % plot the first noisy signal

Now, add the original, average, noiseless signal:

hold on;
plot(t, s_bar, 'k'); % plot overall mean
hold off;

Finally, add the noiseless version of the signal, after accounting for the shot-to-shot variations (e.g. that the volume fraction is much less than average):

hold on;
plot(t, out.s_tilde(:,5), 'b'); % plot noiseless signal after shot-to-shot
hold off;

Experimental signals

One can also analyze experimental signals. We will here demonstrate on noisy signals generated from the above procedure, but a simple substitution for experimental signals has been demonstrated by Sipkens et al. (2017). Analysis proceeds simply by fitting a polynomial to the mean and variance of the data. That is, compute the mean and standard deviation of the signals, s, generated above:

s_ave = mean(s, 2);
s_std = std(s, [], 2);

Alternatively, these quantities are output directly from the add_noise() function, as shown above. Now, fit a quadratic polynomial. It is recommended that one fit to the data in log-space, using get_noisel():

[tau_l, the_l, gam_l] = get_noisel(s);

standard polynomial fitting is available via get_noise(). The degree to which the data prescribes to his simple quadratic structure can be demonstrated by plotting the quadratic fit and the data:

figure(2); % plot average of observed signals verses variance and fits
plot(s_ave, s_std.^2, '.'); % plot observed average verses variance
hold on;
max_plot = the * max(J); % maximum of x-axis in plots

% plot quadratic error model fit to variance
fplot(@(x) ...
    gam_l ^ 2 + ...  % Gaussian
    the_l .* x + ...  % Poisson
    (tau_l ^ 2) .* (x.^2), ...  % multiplicative
    '-k', [0,max_plot]);

hold off;

Sipkens et al. (2017) demonstrated that a variety of TiRe-LII signals will show this quadratic relationship.

A sample script: main.m

Users can execute the main script immediately, provided all attached files are located in the same directory and that directory is included in the MATLAB path. One must then enter main in the MATLAB command line. This code will generate Figure 1 from Sipkens et al. (2017).

The code will display the source error model parameters chosen to generate the data, as well as parameters fit to the signals generated using the given error model. The code will also output two figures: (i) a plot showing a sample signal, expected mean signal, and single-shot expected signal with time (corresponding to Figure 1 in the associated paper) and (ii) a plot show the average verses the variance of the associated signals and the corresponding error model fit to the data.

Error model parameters can be modified by editing the assignment of tau, theta, and gamma in the main.m script.

Data files

Included data files correspond to:

lii.csv - Data for a sample expected mean TiRe-LII signal generated for C-N₂ using the Michelsen model from Michelsen et al. (2007).

gaus.csv - Data corresponding to a Gaussian distribution (used by default in the web app).

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Contact information

The primary author of the code is Timothy A. Sipkens, who can be emailed at tsipkens@uwaterloo.ca. Alternatively, one can contact Kyle J. Daun at kjdaun@uwaterloo.ca who supported code development.

How to cite

Users of this work should cite,

Sipkens, T. A., Hadwin, P. J., Grauer, S. J., & Daun, K. J. (2017). General error model for analysis of laser-induced incandescence signals. Applied optics, 56(30), 8436-8445,

and can consider citing this repository, if this code is used directly.

References

Sipkens, T. A., Hadwin, P. J., Grauer, S. J., & Daun, K. J. (2017). General error model for analysis of laser-induced incandescence signals. Applied optics, 56(30), 8436-8445.

Michelsen, H. A., Liu, F., Kock, B. F., Bladh, H., Boïarciuc, A., Charwath, M., Dreier, T., Hadef, R., Hofmann, M., Reimann, J., & Will, S. (2007). Modeling laser-induced incandescence of soot: a summary and comparison of LII models. Applied physics B, 87(3), 503-521.