Published online by Cambridge University Press: 05 September 2014
In this chapter, we apply the theory of sparse stochastic processes to the reconstruction of signals from noisy measurements. The foundation of the approach is the specification of a corresponding (finite-dimensional) Bayesian framework for the resolution of ill-posed inverse problems. Given some noisy measurement vector y ∈ ℝM produced by an imaging or signal acquisition device (e.g., optical or X-ray tomography, magnetic resonance), the problem is to reconstruct the unknown object (or signal) s as a d-dimensional function of the space-domain variable r ∈ ℝd based on the accurate physical modeling of the imaging process (which is assumed to be linear).
The non-standard aspect here is that the reconstruction problem is stated in the continuous domain. A practical numerical scheme is obtained by projecting the solution onto some finite-dimensional reconstruction space. Interestingly, the derived MAP estimators result in optimization problems that are very similar to the variational formulations that are in use today in the field of biomaging, including Tikhonov regularization and l1-norm minimization. The proposed framework provides insights of a statistical nature and also suggests novel computational schemes and solutions.
The chapter is organized as follows. In Section 10.1, we present a general method for the discretization of a linear inverse problem in a shift-invariant basis. The corresponding finite-dimensional statistical characterization of the signal is obtained by suitable “projection” of the innovation model onto the reconstruction space. This information is then used in Section 10.2 to specify the maximum a posteriori (MAP) reconstruction of the signal.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.