Analysis of Reconstruction from Noisy Discrete Generalized Radon Data

We consider a wide class of generalized Radon transforms $$\mathcal {R}$$ , which act in $$\mathbb {R}^n$$ for any $$n\ge 2$$ and integrate over submanifolds of any codimension N, $$1\le N\le n-1$$ . Also, we allow for a fairly general reconstruction operator $$\mathcal {A}$$ . The main requirement is that $$\mathcal {A}$$ be a Fourier integral operator whose phase function is linear in the phase variable. We consider image reconstruction from discrete noisy data $$g_{j} = (\mathcal {R} f)_{j} + \eta _{j}$$ , $$j\in \mathbb {Z}^n$$ . Here, f denotes the object to be reconstructed, and $$\{\eta _{j}\}$$ is a family of independent (not necessarily identically distributed) random variables modeling measurement noise. We show that the reconstruction error $$N_\epsilon ^{\textrm{rec}}=\mathcal {A}(\{\eta _{j}\})$$ tends to a limit, i.e., $$N^{\textrm{rec}}(\check{x};x_0)=\lim _{\epsilon \rightarrow 0}N_\epsilon ^{\textrm{rec}}(x_0+\epsilon \check{x})$$ , $$\check{x}\in \bar{D}$$ . Here $$x_0$$ is a fixed point, $$\epsilon $$ is the data sampling scale, $$D\subset \mathbb {R}^n$$ is a bounded domain, $$N^{\textrm{rec}}$$ and $$N_\epsilon ^{\textrm{rec}}$$ are viewed as continuous random functions of the argument $$\check{x}$$ (random fields), and the limit is understood in the sense of probability distributions. This result is proved under some conditions on the first three moments of $$\eta _{j}$$ and some other not very restrictive conditions on $$x_0$$ and $$\mathcal {A}$$ . Using that $$\mathcal {A}$$ is linear, we ignore the reconstruction of the deterministic part of the signal. The existence of the appropriate limit of the latter is shown in an earlier paper by the author. Further, we prove that $$N^{\textrm{rec}}$$ is a zero mean Gaussian random field and explicitly compute its covariance. We also present a numerical experiment with a cone-beam transform in $$\mathbb {R}^3$$ , which shows an excellent match between theoretical predictions and simulated reconstructions.