Wasserstein Image Local Analysis: Histogram of Orientations, Smoothing and Edge Detection

TL;DR

This paper introduces a novel approach using optimal transport maps for local image analysis, improving robustness in directionality, smoothing, and edge detection, with applications in tumor heterogeneity assessment and image preprocessing.

Contribution

It presents a new method leveraging optimal transport for local image analysis, enhancing robustness over traditional techniques, and demonstrates its utility in medical imaging and classical image processing tasks.

Findings

Higher entropy in tumor images correlates with worse survival.

Optimal transport-based smoothing and edge detection preserve image features.

The method outperforms classical approaches in noisy medical images.

Abstract

The Histogram of Oriented Gradient is a widely used image feature, which describes local image directionality based on numerical differentiation. Due to its ill-posed nature, small noise may lead to large errors. Conventional HOG may fail to produce meaningful directionality results in the presence of noise, which is common in medical radiographic imaging. We approach the directionality problem from a novel perspective by the use of the optimal transport map of a local image patch to a uni-color patch of its mean. We decompose the transport map into sub-work costs in different directions. We evaluated the ability of the optimal transport to quantify tumor heterogeneity from brain MRI images of patients with glioblastoma multiforme from the TCIA. By considering the entropy difference of the extracted local directionality within tumor regions, we found that patients with higher entropy in their images, had statistically significant worse overall survival (p $=0.008$), which indicates that tumors exhibiting flows in many directions may be more malignant, perhaps reflecting high tumor histologic grade, a reflection of histologic disorganization. We also explored the possibility of solving classical image processing problems such as smoothing and edge detection via optimal transport. By looking for a 2-color patch with minimum transport distance to a local patch, we derive a nonlinear shock filter, which preserves edges. Moreover, we found that the color difference of the computed 2-color patch indicates whether there is a large change in color, i.e., an edge in the given patch. In summary, we expand the usefulness of optimal transport as an image local analysis tool, to extract robust measures of imaging tumor heterogeneity for outcomes prediction as well as image pre-processing. Because of its robust nature, we find it offers several advantages over the classical approaches.