2D Deconvolution

In this example we show how to quantify the uncertainty of a solution to a 2D deconvolution problem.

First we import the modules needed.

import numpy as np
from cuqi.testproblem import Deconvolution2D
from cuqi.distribution import Gaussian, LMRF
from cuqi.problem import BayesianProblem

Step 1: Deterministic model

Consider the deterministic inverse problem

\[\mathbf{y} = \mathbf{A} \mathbf{x}\]

where \(\mathbf{A}\) is a matrix representing a 2D convolution operation and \(\mathbf{y}\) and \(\mathbf{x}\) are the data and unknown (solution to the inverse problem) respectively.

A linear forward model like \(\mathbf{A}\) is represented by a LinearModel and any data (like some observed data \(\mathbf{y}^\mathrm{obs}\)) as a CUQIarray.

The easiest way to get these two components is to use the built-in testproblems. Let us extract the model and data for a 2D deconvolution.

A, y_obs, info = Deconvolution2D().get_components()

Step 2: Prior model

Now we aim to represent our prior knowledge of the unknown image. In this case, let us assume that the unknown is piecewise constant. This can be modelled by assuming a Laplace difference prior. The Laplace difference prior can be defined as

\[\begin{split}\mathbf{x}_{i,j}-\mathbf{x}_{i',j} \sim \mathrm{Laplace}(0, \delta),\\ \mathbf{x}_{i,j}-\mathbf{x}_{i,j'} \sim \mathrm{Laplace}(0, \delta),\end{split}\]

where \(\delta\) is the scale parameter defining how likely jumps from one pixel value to another are in the horizontal and vertical directions.

This distribution comes pre-defined in CUQIpy as the cuqi.distribution.LMRF. Notice we have to specify the geometry of the unknown.

x = LMRF(location=0, scale=0.1, geometry=A.domain_geometry)

Step 3: Likelihood model

Suppose our data is corrupted by a Gaussian noise so our observational model is

\[\mathbf{y}\mid \mathbf{x} \sim \mathcal{N}(\mathbf{A} \mathbf{x}, \sigma^2),\]

where \(\sigma^2\) is a noise variance that we know.

We can represent \(\mathbf{y}\mid \mathbf{x}\) as a cuqi.distribution.Distribution object. We often call the distribution of \(\mathbf{y}\mid \mathbf{x}\) the data distribution.

y = Gaussian(mean=A@x, cov=0.01)

Step 4: Posterior sampling

In actuality we are interested in conditioning on the observed data \(\mathbf{y}^\mathrm{obs}\), to obtain the posterior distribution

\[p(\mathbf{x}|\mathbf{y}^\mathrm{obs}) \propto p(\mathbf{y}^\mathrm{obs}|\mathbf{x})p(\mathbf{x}),\]

and then sampling from this posterior distribution.

In CUQIpy, we the easiest way to do this is to use the cuqi.problem.BayesianProblem class.

# Create Bayesian problem and set observed data (conditioning)
BP = BayesianProblem(y, x).set_data(y=y_obs)

After setting the data, we can sample from the posterior using the cuqi.problem.BayesianProblem.sample_posterior() method. Notice that a well-suited sampler is automatically chosen based on the model, likelihood and prior chosen.

# Sample posterior
samples = BP.sample_posterior(200)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Automatic sampler selection is a work-in-progress. !!!
!!!       Always validate the computed results.        !!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Using cuqi.sampler UGLA sampler
burn-in: 20%

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Elapsed time: 32.2441041469574

Step 5: Posterior analysis

Finally, after sampling we can analyze the posterior. There are many options here. For example, we can plot the credible intervals for the unknown image and compare it to the true image.

ax = samples.plot_ci(exact=info.exactSolution)

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