Random heterogeneous, scale-dependent structures can be observed from many image sources, especially from remote sensing and scientific imaging. Examples include slices of porous media data showing pores of various sizes, and a remote sensing image including small and large sea-ice blocks. Meanwhile, rather than the images of phenomena themselves, there are many image processing and analysis problems requiring to deal with \emph{discrete-state} fields according to a labeled underlying property, such as mineral porosity extracted from microscope images, or an ice type map estimated from a sea-ice image. In many cases, if discrete-state problems are associated with heterogeneous, scale-dependent spatial structures, we will have to deal with complex discrete state fields. Although scale-dependent image modeling methods are common for continuous-state problems, models for discrete-state cases have not been well studied in the literature. Therefore, a fundamental difficulty will arise which is how to represent such complex discrete-state fields. Considering the success of hidden field methods in representing heterogenous behaviours and the capability of hierarchical field methods in modeling scale-dependent spatial features, we propose a Hidden Hierarchical Markov Field (HHMF) approach, which combines the idea of hierarchical fields with hidden fields, for dealing with the discrete field modeling challenge. However, to define a general HHMF modeling structure to cover all possible situations is difficult. In this research, we use two image application problems to describe the proposed modeling methods: one for scientific image (porous media image) reconstruction and the other for remote-sensing image synthesis. For modeling discrete-state fields with a spatially separable complex behaviour, such as porous media images with nonoverlapped heterogeneous pores, we propose a Parallel HHMF model, which can decomposes a complex behaviour into a set of separated, simple behaviours over scale, and then represents each of these with a hierarchical field. Alternatively, discrete fields with a highly heterogeneous behaviour, such as a sea-ice image with multiple types of ice at various scales, which are not spatially separable but arranged more as a partition tree, leads to the proposed Tree-Structured HHMF model. According to the proposed approach, a complex, multi-label field can be repeatedly partitioned into a set of binary/ternary fields, each of which can be further handled by a hierarchical field.

Random heterogeneous, scale-dependent structures can be observed from many image sources, especially from remote sensing and scientific imaging. Examples include slices of porous media data showing pores of various sizes, and a remote sensing image including small and large sea-ice blocks. Meanwhile, rather than the images of phenomena themselves, there are many image processing and analysis problems requiring to deal with \emph{discrete-state} fields according to a labeled underlying property, such as mineral porosity extracted from microscope images, or an ice type map estimated from a sea-ice image. In many cases, if discrete-state problems are associated with heterogeneous, scale-dependent spatial structures, we will have to deal with complex discrete state fields. Although scale-dependent image modeling methods are common for continuous-state problems, models for discrete-state cases have not been well studied in the literature. Therefore, a fundamental difficulty will arise which is how to represent such complex discrete-state fields. Considering the success of hidden field methods in representing heterogenous behaviours and the capability of hierarchical field methods in modeling scale-dependent spatial features, we propose a Hidden Hierarchical Markov Field (HHMF) approach, which combines the idea of hierarchical fields with hidden fields, for dealing with the discrete field modeling challenge. However, to define a general HHMF modeling structure to cover all possible situations is difficult. In this research, we use two image application problems to describe the proposed modeling methods: one for scientific image (porous media image) reconstruction and the other for remote-sensing image synthesis. For modeling discrete-state fields with a spatially separable complex behaviour, such as porous media images with nonoverlapped heterogeneous pores, we propose a Parallel HHMF model, which can decomposes a complex behaviour into a set of separated, simple behaviours over scale, and then represents each of these with a hierarchical field. Alternatively, discrete fields with a highly heterogeneous behaviour, such as a sea-ice image with multiple types of ice at various scales, which are not spatially separable but arranged more as a partition tree, leads to the proposed Tree-Structured HHMF model. According to the proposed approach, a complex, multi-label field can be repeatedly partitioned into a set of binary/ternary fields, each of which can be further handled by a hierarchical field.