Consistency conditions and linear reconstruction methods in diffraction tomography

Because an image can be reconstructed from knowledge of its Radon transform (RT), the task of reconstructing an image is tantamount to that of estimating its RT. Based upon the Fourier diffraction projection (FDP) theorem, from the statistical perspective of unbiased reduction of image variance, we previously proposed an infinite family of estimation methods for obtaining the RT from the scattered data in diffraction tomography (DT). In this work, using the FDP theorem, we define the diffraction Radon transform (DRT), which can be treated as the data function in DT. Subsequently, using strategies similar to those that analyze the consistency conditions on the exponential Radon transform in two-dimensional (2-D) single-photon emission computed tomography with uniform attenuation, we studied the consistency condition on the DRT and we show that there is a hierarchy of estimation methods that actually project the noisy data function onto its consistency space in different ways. In terms of a weighted inner product of the consistency and inconsistency parts of a noisy data function, we further demonstrate that a subset of the family of estimation methods can be interpreted as orthogonal projections onto the consistency space of the DRT. In particular, the statistically suboptimal estimation method in the family corresponds to an orthogonal projection associated with an ordinary inner product of the consistency and inconsistency parts of a noisy data function.