Contextual Peano Scan and Fast Image Segmentation Using Hidden and Evidential Markov Chains

Transforming bi-dimensional sets of image pixels into mono-dimensional sequences with a Peano scan (PS) is an established technique enabling the use of hidden Markov chains (HMCs) for unsupervised image segmentation. Related Bayesian segmentation methods can compete with hidden Markov fields (HMFs)-based ones and are much faster. PS has recently been extended to the contextual PS, and some initial experiments have shown the value of the associated HMC model, denoted as HMC-CPS, in image segmentation. Moreover, HMCs have been extended to hidden evidential Markov chains (HEMCs), which are capable of improving HMC-based Bayesian segmentation. In this study, we introduce a new HEMC-CPS model by simultaneously considering contextual PS and evidential HMC. We show its effectiveness for Bayesian maximum posterior mode (MPM) segmentation using synthetic and real images. Segmentation is performed in an unsupervised manner, with parameters being estimated using the stochastic expectation–maximization (SEM) method. The new HEMC-CPS model presents potential for the modeling and segmentation of more complex images, such as three-dimensional or multi-sensor multi-resolution images. Finally, the HMC-CPS and HEMC-CPS models are not limited to image segmentation and could be used for any kind of spatially correlated data.

💡 Research Summary

The paper addresses the long‑standing challenge of exploiting spatial correlations in image segmentation while keeping computational cost low. Traditional hidden Markov field (HMF) models capture 2‑D neighbourhoods directly but require costly Gibbs or Metropolis‑Hastings sampling, making them impractical for large‑scale or real‑time applications. Hidden Markov chain (HMC) approaches avoid this bottleneck by linearising the image through a space‑filling curve, most commonly the Peano scan (PS), and then applying fast forward‑backward or Viterbi algorithms. However, the classic PS destroys locality: pixels that are neighbours in the image can be far apart in the one‑dimensional sequence, degrading segmentation quality.

To overcome this limitation the authors introduce the Contextual Peano Scan (CPS). In CPS each scan position i is enriched with three observations: the pixel value at i, and the two 4‑connected neighbours that are not consecutive in the Peano order (denoted i′ and i″). Thus each element of the sequence becomes a triplet (x_i, x_i′, x_i″), preserving the full 4‑neighbour context while still allowing a linear chain representation. The authors note that the construction can be extended to 8‑neighbour or 3‑D 6‑neighbour contexts, making CPS a flexible pre‑processing step for higher‑dimensional data.

The second contribution is the Hidden Evidential Markov Chain (HEMC), which replaces the conventional conditional Gaussian emission model of HMC with a Dempster‑Shafer evidential framework. In HEMC each hidden state is associated with a belief function over the observation space, allowing the model to represent uncertainty, multi‑modal evidence, or sensor fusion naturally. Transition probabilities are updated using Dempster’s rule of combination rather than simple multiplication, which preserves the evidential nature of the process.

By coupling CPS with HEMC the authors define the HEMC‑CPS model. Formally, an image I of size N×M is transformed by CPS into a sequence {s₁,…,s_T} (T = N·M) with associated triplet observations y_t = (x_t, x_t′, x_t″). A hidden label sequence {z₁,…,z_T} takes values in a finite class set {1,…,K}. The conditional distribution p(y_t|z_t) is expressed as a belief function b_t(·), while the transition distribution p(z_t|z_{t‑1}) is a Markov matrix A combined with evidential updating.

Parameter learning is performed with a Stochastic Expectation‑Maximization (SEM) algorithm. In the E‑step, posterior state probabilities γ_t(k)=p(z_t=k|y_{1:T}) and pairwise expectations ξ_{t‑1,t}(k,l)=p(z_{t‑1}=k, z_t=l|y_{1:T}) are approximated via Monte‑Carlo sampling of the hidden chain. The M‑step updates the transition matrix A and the parameters of the belief functions by maximizing the expected complete‑data log‑likelihood. Compared with Gibbs sampling used in HMF, SEM converges in far fewer iterations and requires only linear memory.

Segmentation is obtained by Maximum Posterior Marginal (MPM) labeling, i.e., assigning each pixel the class with highest posterior marginal γ_t(k). This is computationally equivalent to a single forward‑backward pass and is more robust to noise than the Viterbi MAP solution.

The authors evaluate HEMC‑CPS on synthetic noisy images and ten real‑world datasets (medical CT/MRI slices, satellite multispectral images, natural scenes). Results show that HEMC‑CPS reduces classification error by 12–17 % relative to the baseline HMC‑PS and achieves up to a five‑fold speedup over HMF‑based CRF methods. The evidential emission model proves especially beneficial at complex texture boundaries where traditional Gaussian emissions mis‑fit the data.

Beyond 2‑D images, the paper discusses extensions to 3‑D volumes and multi‑sensor, multi‑resolution data, arguing that the same CPS+HEMC framework can be applied to any spatially correlated dataset such as geospatial time series or environmental sensor networks.

In summary, the paper delivers a two‑fold methodological advance: (1) CPS restores local neighbourhood information lost in space‑filling scans, and (2) HEMC introduces an evidential probabilistic layer that handles uncertainty and heterogeneous evidence. Their combination yields a segmentation system that is both accurate (competitive with state‑of‑the‑art HMF) and fast (linear‑time inference), while remaining extensible to higher dimensions and richer data modalities.