Segmentation of multicorrelated images with copula models and conditionally random fields

Abstract

Purpose: Multisource images are interesting in medical imaging. Indeed, multisource images enable the use of complementary information of different sources such as for T1 and T2 modalities in MRI imaging. However, such multisource data can also be subject to redundancy and correlation. The question is how to efficiently fuse the multisource information without reinforcing the redundancy. We propose a method for segmenting multisource images that are statistically correlated. Approach: The method that we propose is the continuation of a prior work in which we introduce the copula model in hidden Markov fields (HMF). To achieve the multisource segmentations, we use a functional measure of dependency called "copula." This copula is incorporated in the conditionally random fields (CRF). Contrary to HMF, where we consider a prior knowledge on the hidden states modeled by an HMF, in CRF, there is no prior information and only the distribution of the hidden states conditionally to the observations can be known. This conditional distribution depends on the data and can be modeled by an energy function composed of two terms. The first one groups the voxels having similar intensities in the same class. As for the second term, it encourages a pair of voxels to be in the same class if the difference between their intensities is not too big. Results: A comparison between HMF and CRF is performed via theory and experimentations using both simulated and real data from BRATS 2013. Moreover, our method is compared with different state-of-the-art methods, which include supervised (convolutional neural networks) and unsupervised (hierarchical MRF). Our unsupervised method gives similar results as decision trees for synthetic images and as convolutional neural networks for real images; both methods are supervised. Conclusions: We compare two statistical methods using the copula: HMF and CRF to deal with multicorrelated images. We demonstrate the interest of using copula. In both models, the copula considerably improves the results compared with individual segmentations.

Keywords: Bayesian inference; brain MRI; conditional random fields; copulas; data fusion; hidden Markov fields; multisource images; tumor segmentation.