Bayesian inference of Parameter Values

Scattering calculations can inform us about the hologram produced by a specific scatterer, but they can’t tell us anything about what type of scatterer produced an experimentally measured hologram. For this inverse scattering problem, we turn to a Bayesian inference approach. We calculate the holograms produced by many similar scatterers, and we evaluate which ones are closest to our measured hologram. We can then use known information about the scatterers to determine the parameters of the scatterer (for example, its size and refractive index) that are most likely to have produced the observed hologram.

In the following example, we will infer the size, refractive index, and position of a spherical scatterer. The approach and formalism is described in more detail in [Dimiduk2016]. For more information on Bayesian inference in general, see [Gregory2005].

Here is the full example. We’ll go through it step-by-step afterward:

import holopy as hp
import numpy as np
from holopy.core.io import get_example_data_path, load_average
from holopy.core.process import bg_correct, subimage, normalize
from holopy.scattering import Sphere, calc_holo
from holopy.inference import prior, AlphaModel, tempered_sample

# load an image
imagepath = get_example_data_path('image01.jpg')
raw_holo = hp.load_image(imagepath, spacing = 0.0851, medium_index = 1.33, illum_wavelen = 0.66, illum_polarization = (1,0))
bgpath = get_example_data_path(['bg01.jpg','bg02.jpg','bg03.jpg'])
bg = load_average(bgpath, refimg = raw_holo)
data_holo = bg_correct(raw_holo, bg)

# process the image
data_holo = subimage(data_holo, [250,250], 200)
data_holo = normalize(data_holo)

# Set up the prior
s = Sphere(n=prior.Gaussian(1.5, .1), r=prior.BoundedGaussian(.5, .05, 0, np.inf),
         center=prior.make_center_priors(data_holo))

# Set up the noise model
noise_sd = data_holo.std()
model = AlphaModel(s, noise_sd=noise_sd, alpha=1)

result = tempered_sample(model, data_holo)

result.values()
hp.save('example-sampling.h5', result)

The first few lines import the code needed to compute holograms and do parameter estimation.

import holopy as hp
import numpy as np
from holopy.core.io import get_example_data_path, load_average
from holopy.core.process import bg_correct, subimage, normalize
from holopy.scattering import Sphere, calc_holo
from holopy.inference import prior, AlphaModel, tempered_sample

Preparing Data

Next, we load a hologram from a file using the same steps as those in Loading Data

# load an image
imagepath = get_example_data_path('image01.jpg')
raw_holo = hp.load_image(imagepath, spacing = 0.0851, medium_index = 1.33, illum_wavelen = 0.66, illum_polarization = (1,0))
bgpath = get_example_data_path(['bg01.jpg','bg02.jpg','bg03.jpg'])
bg = load_average(bgpath, refimg = raw_holo)
data_holo = bg_correct(raw_holo, bg)

You will notice that the hologram data is localized to a region near the center of the image. We only want to compare calculated holograms to this region, so we will crop our image with subimage(). We also need to normalize the data so that its mean is 1, since calculations return a normalized result. Since our image is background divided, its mean is already very close to 1, but it is good to get in the habit of normalizing anyway.

# process the image
data_holo = subimage(data_holo, [250,250], 200)
data_holo = normalize(data_holo)

Note

It is often useful to test an unfamiliar technique on data for which you know the expected outcome. Instead of actual data, you could use a hologram calculated from calc_holo(), and modulated by random noise with add_noise().

Defining a Probability Model

Priors

We know that the hologram was produced by a spherical scatterer, so we want to define a Sphere object like we did in the Scattering Calculations tutorial. However, in this case we don’t know what parameters to specify for the sphere (since that is what we’re trying to find out). Instead, we write down a probabilistic statement of our prior information about the sphere. In statistics, we call this a prior. For the case we are investigating here, we would probably have some best guess and uncertainty about the size and index of the particle, obtained from the supplier or from prior work with the particle. We will guess the radius to be 0.5 micrometers (with 50 nm error) and refractive index to be 1.5 (with 0.1 error). We also need to provide a prior for the position of the sphere. We can use a hough() transform to get a pretty good guess of where the particle is in x and y, but it is difficult to determine where it is in z.

Note

One trick to get a better estimate of z position is to numerically propagate the hologram backwards in space with propagate(), and look for where the interference fringes vanish.

Let’s turn our information about priors into code by defining our scatterer:

s = Sphere(n=prior.Gaussian(1.5, .1), r=prior.BoundedGaussian(.5, .05, 0, np.inf),
         center=prior.make_center_priors(data_holo))

We use a Gaussian distribution as the prior because it is the most conservative distribution we can use if all we know is some expected value and some uncertainty on that expected value. For the radius we also know that it must be nonnegative, so we can bound the Gaussian at zero. The make_center_priors() function automates generating priors for a sphere center using center_find() (based on a hough transform). It assigns Gaussian priors for x and y, and picks a large uniform prior for z to represent our ignorance about how far the particle is from the imaging plane. In this case the center prior will be:

[Gaussian(mu=11.4215, sd=0.0851),
Gaussian(mu=9.0945, sd=0.0851),
Uniform(lower_bound=0, upper_bound=170.2)]

Likelihood

Next we need to define a model that tells HoloPy how probable it is that we would see the data we observed given some hypothetical scatterer position, size and index. In the language of statistics, this is referred to as a likelihood. In order to compute a likelihood, you need some estimate of how noisy your data is (so that you can figure out how likely it is that the differences between your model and data could be explained by noise). Here we use the standard deviation of the data, which is an overestimate of the true noise, since it also includes variation due to our signal.

noise_sd = data_holo.std()
model = AlphaModel(s, noise_sd=noise_sd, alpha=1)

Note

alpha is a model parameter that scales the scattered beam intensity relative to the reference beam. It is often less than 1 for reasons that are poorly understood. If you aren’t sure what value it should take in your system, you can allow alpha to vary by giving it a prior, as we did with the sphere parameters.

Sampling the Posterior

Finally, we can sample the posterior probability for this model. Essentially, a set of proposed scatterers are randomly generated according to the priors we specified. Each of these scatterers is then evaluated in terms of how well it matches the experimental hologram data_holo. A Monte Carlo algorithm iteratively produces and tests sets of scatterers to find the scatterer parameters that best reproduce the target hologram. We end up with a distribution of values for each parameter (the posterior) that represents our updated knowledge about the scatterer when accounting for the expected experimental hologram. To do the actual sampling, we use tempered_sample() (ignoring any RuntimeWarnings about invalid values):

result = tempered_sample(model, data_holo)

The above line of code may take a long time to run (it takes 10-15 mins on our 8-core machines). If you just want to quickly see what the results look like, try:

result = tempered_sample(model, data_holo, nwalkers=10, samples=100, max_pixels=100)

This code should run very quickly, but its results cannot be trusted for any actual data. Nevertheless, it can give you an idea of what format the results will take. In our last line of code, we have adjusted three parameters to make the code run faster: nwalkers describes the number of scatterers produced in each generation. samples describes how many generations of scatterers to produce. Together, they define how many scatterering calculations must be performed. For the values chosen in the fast code, a Monte Carlo steady state will not yet have been achieved, so the resulting posterior distribution is not very meaningful. max_pixels describes the maximum number of pixels compared between the experimental holgoram and the test holograms. It turns out that holograms contain a lot of redundant information owing to their symmetry, so a subset of pixels can be analyzed without loss of accuracy. However, 100 pixels is probably too few to capture all of the relevant information in the hologram.

You can get a quick look at our obtained values with:

result.values()

result.values() gives you the maximum a posteriori probability (MAP) value as well as one-sigma credibility intervals (or you can request any other sigma with an argument to the function). You can also look only at central measures:

result.MAP
result.mean
result.median

Since calculation of useful results takes a long time, you will usually want to save them to an hdf5 file:

hp.save('example-sampling.h5', result)

Tying Parameters

You may desire to use tied priors, in which several physical quantities that could be varied independently are constrained to have the same (but non-constant) value. A common example involves multi-particle holograms in which all of the particles are constrained to have the same refractive index, but the index is determined by the Bayesian inference method. This may be done by defining a prior and using it in multiple places :

from holopy.scattering import Spheres

n_prior = prior.Gaussian(1.5, .1)

s1 = Sphere(n=n_prior, r=prior.BoundedGaussian(.5, .05, 0, np.inf),
            center=prior.make_center_priors(data_holo))
s2 = Sphere(n=n_prior, r=prior.BoundedGaussian(.5, .05, 0, np.inf),
            center=prior.make_center_priors(data_holo))

spheres = Spheres([s1, s2])

And then defining the model using :

model = AlphaModel(spheres, noise_sd=noise_sd, alpha=1)

References

[Gregory2005]Gregory, P. (2005) Bayesian Logical Data Analysis. Cambridge University Press