Abstract
Image based diagnostics are interpreted in the context of spatial resolution. The same is true for tomographic image reconstruction. Current empirically driven approaches to quantify spatial resolution in chemical species tomography rely on a deterministic formulation based on point-spread functions which neglect the statistical prior information, that is integral to rank-deficient tomography. We propose a statistical spatial resolution measure based on the covariance of the reconstruction (point estimate). By demonstrating the resolution measure on a chemical species tomography test case, we show that the prior information acts as a lower limit for the spatial resolution. Furthermore, the spatial resolution measure can be employed for designing tomographic systems under consideration of spatial inhomogeneity of spatial resolution.
Highlights
Over the last decades absorption spectroscopic linear hard field tomography of gas phase media has been applied to a multitude of engineering problems, including turbines [2, 3, 4, 5], piston engines [6, 7, 8], exhaust gas aftertreatment [9, 10], and coal combustion [11] to reconstruct the spatial distribution of temperatures or concentrations
While discussions of the resolution measures based on point-spread function (PSF), optical transfer function (OTF), or modulation transfer function (MTF) can be found throughout optics textbooks [29, 30], Tsekenis et al [1] provide a thorough discussion of the intricacies of their application in tomography, concluding that a resolution quantity defined on the Modulation Transfer Function (MTF) or OTF amplitude is more robust than resolution measures defined using the PSF
It is only possible to define the resolution of a specific point estimate, in this case the maximum a posteriori estimate
Summary
Over the last decades absorption spectroscopic linear hard field tomography of gas phase media has been applied to a multitude of engineering problems, including turbines [2, 3, 4, 5], piston engines [6, 7, 8], exhaust gas aftertreatment [9, 10], and coal combustion [11] to reconstruct the spatial distribution of temperatures or concentrations. The resolution matrix in limited data tomography often contains null PSFs corresponding to "blind spots" in the tomographic beam arrangement These regions are difficult to interpret in terms of a finite resolution. The underlying assumption hereby is homogeneity and isotropy of the spatial resolution, both being questionable for low beam count measurements This way, prior information introduced by the regularization is indirectly regarded in the resolution measure and the extensive edge phantoms circumvent the problem of blind spots, yielding heuristic empirical spatial resolution estimates. In most real world applications it is common practice to only regard a point estimate, e.g. the maximum a posteriori estimate (MAP), sampled from the posterior distribution This sampling process accounts to a loss in resolution, which we address in this work. A scalar resolution measure based on a thresholding method proposed by Tsekenis et al [1] is presented
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