[Paper Review] Accuracy of MAP segmentation with hidden Potts and Markov mesh prior models via Path Constrained Viterbi Training, Iterated Conditional Modes and Graph Cut based algorithms

Ana Georgina Flesia, Josef Baumgartner+2 more|arXiv (Cornell University)|Jul 11, 2013

Medical Image Segmentation Techniques17 references3 citations

TL;DR

This paper evaluates three algorithms—Path Constrained Viterbi Training (PCVT), Iterated Conditional Modes (ICM), and Graph Cut—for Markov random field-based image segmentation using hidden Potts and Markov mesh priors. It demonstrates that PCVT achieves superior noise suppression and segmentation accuracy, especially in complex, multimodal images, while maintaining robustness to non-distinct histogram modes.

ABSTRACT

In this paper, we study statistical classification accuracy of two different Markov field environments for pixelwise image segmentation, considering the labels of the image as hidden states and solving the estimation of such labels as a solution of the MAP equation. The emission distribution is assumed the same in all models, and the difference lays in the Markovian prior hypothesis made over the labeling random field. The a priori labeling knowledge will be modeled with a) a second order anisotropic Markov Mesh and b) a classical isotropic Potts model. Under such models, we will consider three different segmentation procedures, 2D Path Constrained Viterbi training for the Hidden Markov Mesh, a Graph Cut based segmentation for the first order isotropic Potts model, and ICM (Iterated Conditional Modes) for the second order isotropic Potts model. We provide a unified view of all three methods, and investigate goodness of fit for classification, studying the influence of parameter estimation, computational gain, and extent of automation in the statistical measures Overall Accuracy, Relative Improvement and Kappa coefficient, allowing robust and accurate statistical analysis on synthetic and real-life experimental data coming from the field of Dental Diagnostic Radiography. All algorithms, using the learned parameters, generate good segmentations with little interaction when the images have a clear multimodal histogram. Suboptimal learning proves to be frail in the case of non-distinctive modes, which limits the complexity of usable models, and hence the achievable error rate as well. All Matlab code written is provided in a toolbox available for download from our website, following the Reproducible Research Paradigm.

Motivation & Objective

• To compare the performance of three segmentation algorithms—PCVT, ICM, and Graph Cut—under different Markov random field priors.
• To evaluate the impact of parameter estimation, computational efficiency, and automation level on segmentation accuracy.
• To investigate the influence of image histogram structure (e.g., distinct vs. non-distinct modes) on algorithm robustness.
• To provide a reproducible, open-source MATLAB toolbox for PCVT and ICM, and a wrapper for Graph Cut, supporting benchmarking of Markovian segmentation methods.
• To offer a unified theoretical and empirical framework for comparing hidden Potts and hidden Markov mesh models in image segmentation.

Proposed method

• Uses Maximum A Posteriori (MAP) estimation to infer hidden pixel labels: $ s^* = \arg\max_s p(s|I,\theta) $, where $ I $ is the observed image and $ \theta $ the model parameters.
• Applies two prior models: a second-order anisotropic Markov Mesh (for PCVT) and an isotropic Potts model (for ICM and Graph Cut).
• Employs Path Constrained Viterbi Training (PCVT) for hidden Markov mesh models, optimizing over a constrained set of 20 most probable label sequences per diagonal.
• Uses Iterated Conditional Modes (ICM) with pseudo-likelihood approximation and smoothness parameter re-estimation for the Potts model.
• Implements Graph Cut via a wrapper to Boykov and Kolmogorov’s C code, using $\alpha$-expansion moves for multi-label optimization.
• Employs EM-based Gaussian mixture modeling for emission likelihoods, with all models sharing the same emission distribution.

Experimental results

Research questions

• RQ1 How do PCVT, ICM, and Graph Cut compare in terms of Overall Accuracy, Kappa coefficient, and Relative Improvement across synthetic and real dental X-ray images?
• RQ2 How does the presence of non-distinctive histogram modes affect the robustness and performance of each segmentation algorithm?
• RQ3 What is the impact of parameter estimation quality—particularly smoothness parameter estimation—on segmentation accuracy and noise suppression?
• RQ4 How does the choice of prior model (Potts vs. Markov mesh) influence the final segmentation outcome and computational complexity?
• RQ5 Can PCVT outperform standard methods like ML and ICM by escaping local optima and avoiding smoothed versions of the maximum likelihood solution?

Key findings

• In experiment (d), PCVT achieved the highest Kappa coefficient (0.8165 ± 0.0005), significantly outperforming ICM (0.7834 ± 0.0005), Graph Cut (0.8089 ± 0.0005), and ML (0.7573 ± 0.0005), with all differences being statistically significant.
• PCVT reduced background noise more effectively than other methods, with errors concentrated in the object region, indicating superior structural preservation.
• Graph Cut showed convergence issues in high-complexity scenarios (e.g., four-class problems), performing best in binary segmentation tasks.
• ICM was the most robust to the number of classes, consistently delivering stable results across all experiments without convergence failure.
• PCVT produced segmentations that were not simple smoothed versions of the ML solution, indicating its ability to escape saddle points in the solution space.
• All three algorithms significantly improved segmentation smoothness over independent mixture models (ML), confirming the benefit of modeling spatial dependencies via Markov priors.