Markov random fields (Mrfs) have been used widely to model image texture as the basis for both maximum likelihood (ML) and maximum a posterior (MAP) estimation. However, stationary Gaussian random fields are not well suited to many image reconstruction situations since they tend to oversmooth discontinuities. Some authors have suggested alternatives to the quadratic potential in the Gaussian random field model to reduce this oversmoothing. This work investigates the use of inhomogeneous random fields in image reconstruction.
A preliminary investigation used a Gaussian Markov random field is used to describe the spatial variation of the interaction parameters of a Gaussian Markov random field describing the image intensities. Simulation examples indicated that the reconstruction based on the inhomogeneous model are far better than those using a single homogeneous Gaussian random field model.
In a second study inhomogeneous Gaussian random fields are proposed as a general model for most image processing applications, replacing the application specific edge-preserving homogeneous models. The simplicity of the Gaussian model allows rapid calculation and the flexibility of the spatially varying prior parameter allows varying degrees of spatial smoothing. This approach is in the spirit of adaptive kernel density methods where only the choice of the variable window width is important. This proposal was tested out on simulated and real SPECT data. The results indicate that the inhomogeneous Gaussian model allows greater flexibility than the edge-preserving log-cosh model of Green; small features are not masked by the smoothing and constant regions obtain sufficient smoothing to remove the effects of noise.
Current work is looking at the influence of different hyper-priors on the fully Bayesian parameter estimation, and at applying the methods to a wider range of data.
Aykroyd, R. G. and Zimeras, S. (1999). Inhomogeneous priors for image reconstruction. JASA, 94, no 447, 934-946.
Aykroyd, R. G. (1998). Bayesian estimation for homogeneous and inhomogeneous Gaussian random fields. IEEE-PAMI, 20(5), 533-539.
Aykroyd, R. G. and Zimeras, S. (1997). Automatic reconstruction with inhomogeneous models. Research Report No. STAT-97/07 (117K) .
Zimeras, S. (1997). Statistical models in medical image processing. PhD Thesis. Department of Statistics, University of Leeds.