Noniterative Tomographic Reconstruction Using Simultaneous Weighted Spline Smoothing and Deconvolution

In [1] we proposed a spatially-variant method for removing noise from tomographic projection data using weighted spline smoothing. The projection model in [1] was based on detectors with the same width as their spacing. A better approximation for the detector response of many tomographs is a rectangular function or Gaussian function whose width is twice the detector spacing. (Such systems are almost adequately sampled in a Nyquist sense.) In this note, we derive a new weighted spline smoothing algorithm that accommodates arbitrary detector models. The implementation described in this note is based on more numerically stable B-splines, rather than the piecewise polynomials of [1]. Simulations demonstrate that the bias/variance tradeoffs using the new algorithm with a more accurate system model are improved relative to the method of [1] and to conventional spatially-invariant smoothing. I. THEORY A. Projection Model Let g(x1, x2) denote the object being imaged, restricted to two dimensions for simplicity. The ideal line-integral projection of this object at an angle φ and radial offset τ is given by lφ(τ) = ∫ g(τ cos φ− t sinφ, τ sinφ+ t cos φ) dt. Assume that the tomographic system has a detector response that is approximately depth independent, and for the remainder drop the dependence on φ. The mean response of the ith detector is approximately: pi = Lil, i = 1, . . . , n, where Lil = ∫ hi(τ)l(τ) dτ (1) and hi(τ) is the line response of the ith detector. Actual detector measurements will fluctuate around the ideal value pi according to a statistical model that depends on the imaging modality. We are only interested in the first two moments of the statistical model, so we assume that the following model is a reasonable approximation: yi ∼ N (pi, σ 2 i ), where σ2 i may have to be estimated from the data and correction factors [1–3]. This work was supported in part by DOE grant DE-FG02-87ER65061 and NCI grant CA60711.