Abstract
In emission tomography, iterative image reconstruction from noisy measured data usually results in noisy images, and so regularisation is often used to compensate for noise. However, in practice, an appropriate, automatic and precise specification of the strength of regularisation for image reconstruction from a given noisy measured dataset remains unresolved. Existing approaches are either empirical approximations with no guarantee of generalisation, or else are computationally intensive cross-validation methods requiring full reconstructions for a limited set of preselected regularisation strengths. In contrast, we propose a novel methodology embedded within iterative image reconstruction, using one or more bootstrapped replicates of the measured data for precise optimisation of the regularisation. The approach uses a conventional unregularised iterative update of a current image estimate from the noisy measured data, and then also uses the bootstrap replicate to obtain a bootstrap update of the current image estimate. The method then seeks the regularisation hyperparameters which, when applied to the bootstrap update of the image, lead to a best fit of the regularised bootstrap update to the conventional measured data update. This corresponds to estimating the degree of regularisation needed in order to map the noisy update to a model of the mean of an ensemble of noisy updates. For a given regularised objective function (e.g. penalised likelihood), no hyperparameter selection or tuning is required. The method is demonstrated for positron emission tomography (PET) data at different noise levels, and delivers near-optimal reconstructions (in terms of reconstruction error) without any knowledge of the ground truth, nor any form of training data.
Highlights
I TERATIVE image reconstruction and parameter estimation methods are often compromised by the noise presentManuscript received September 13, 2019; revised November 22, 2019; accepted November 24, 2019
This can be achieved through maximum a posteriori (MAP) methods, which introduce an extra term in the reconstruction objective function to counteract image noise [1], and via reparameterisation methods that use alternative basis functions (e.g. [3]–[5])
This work proposes the use of one or more bootstrap replicates of a noisy measured dataset to find an optimised level of regularisation for the reconstruction of that dataset
Summary
I TERATIVE image reconstruction and parameter estimation methods are often compromised by the noise presentManuscript received September 13, 2019; revised November 22, 2019; accepted November 24, 2019. Regularisation can be included in the objective function in order to compensate for noise in a theoretically-justified manner This can be achieved through maximum a posteriori (MAP) ( known as penalised likelihood) methods, which introduce an extra term in the reconstruction objective function to counteract image noise [1], and via reparameterisation methods that use alternative basis functions The user is still required to select hyperparameters to control the level of regularisation, either in the form of penalty strengths for MAP methods, or through the number of iterations and the basis-function design parameters for the basis function approaches
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