Bayesian Image Processing in Two Dimensions

Abstract

A Bayesian image processing (BIP) formalism which incorporates a priori amplitude and spatial probability density information was applied to two-dimensional source fields. For valid, moderately restrictive a priori information, strikingly improved results for ideal and experimental radioisotope phantom imaging data, compared to a standard non-Bayesian formalism (maximum likelihood, ML), were obtained. The applicability of a fast Fourier transform technique for "convolution" calculations, a reduced-region restriction for the initial "deconvolution" calculations, and a relaxation parameter for accelerating convergence are considered.

Range Condition and ML-EM Checkerboard Artifacts

“…In order to mitigate the artifacts, Bayesian criterion was introduced in [12]. By regularization techniques such as an a priori constraint or a penalty function, the ML criterion (6) may be rewritten as the maximum a posteriori (MAP) criterion: (8) where U (·) stands for a priori information and is usually expressed as a potential function on the neighborhood system, i.e., index (m′,n′) specifies the neighbors around (m,n) and α(f) may reflect any known information about the radiotracer concentration distribution [12]. By the use of the EM algorithm, an iterative MAP-EM scheme to reach the maximum point of (8) can also be derived [12].…”

“…More detailed description of the range condition can be found in [3,20,21], and its application to the exponential Radon transform is reported in [22][23][24] and to the Radon transform is reported in [25][26][27]. In this paper, we adapt the general expression of the range condition from [3], which states that the attenuated Radon transform p(s,θ) must satisfy the following integral equation: (12) Due to the presence of noise, the measured projection data p(s,θ) may not satisfy (12) in general, i.e., there might not exist any object function f(x, y) such that p(s,θ) is the aRT of f(x, y). In the following, we apply the range condition to separate the noise n(s,θ) into two parts: one part satisfies the condition (12) and the other part does not.…”

“…The component n N (s,θ) belongs to the null space of the operator χ B Nμ and does not meet condition (12). More detailed description on the null space of the Radon transform operator was given in [20].…”

<title>Image processing and image reconstruction with the use of a-priori information</title>

“…p(XIy) dX . (2) Equation (2) states that, given a current estimate of X, a sample from n can be obtained from the conditional predictive density p(nIX,y), which redistributes y to different pixels according to the multinomial probabilities relating image pixels to detector bins. Iterative algorithms for image processing or image reconstruction can be designed according to Equations (2) and (3).…”

Statistically based image reconstruction for emission tomography

“…Clearly, the specification of P(a) will have an effect on the reconstruction results, and it is necessary, therefore, to be very careful in using any prior information in an application t o medical imaging. Without attempting to be complete, the set of assumptions that workers have used as priors can be listed as smoothness of the image, except at boundaries, [ll-131, limited deviation from a plausible solution [14], image entropy [15,161, and known experimental information [17]. Regularization by the method of sieves [18,19] or by stopping the maximization of the likelihood at some specific point [20] are forms of constraining the likelihood solution, but do not lend themselves to being placed in the Bayesian formulation.…”