Topological approaches for quantitative imaging: application to tissue and intracellular dynamic analysis

Our project focuses on the theoretical and practical development of new image segmentation methods based on the algebraic topology and mereotopology for the cancer diagnosis (automatic segmentation of unstained tissue sections) and the study of study of membrane dynamics and intracellular movements (dynamic intracellular segmentation). More specifically, we are interested in the extraction of local features from quantitative images, which should be invariant to rotations, translations, dilations, but also more general continuous transformations, by using the persistent homology. This first objective requires the construction of an abstract space suitable for understanding of quantitative phase images then its topological study. The computational aspect of this construction will be done thanks to the newly developed specific algorithms in algebraic topology. The second objective is the creation of new segmentation tools by using the topological approach and combining it with the mereotopological one in a unified framework. This includes the integration of the topological features within existing segmentation tools as well as the expression of the neighborhood relationships of the layers of cells or structures through mereotopological primitives. This project also aims to address two issues in the quantitative phase imaging field that emerges as a new biological imaging modality: the management of 2.5 D images, for which pixels’ absolute values have a significant interpretation, and the subcellular segmentation based on quantitative information, allowing the study of the movement of organelles and other vesicles.

 The analysis of quantitative phase images brings new problems in signal processing since these images include information at all scale levels (intracellular: nuclei and nucleoli, cellular: cell membrane, extracellular: glands, regions). To manage this multiplicity of scales, complementary approaches will be considered in parallel: create a multi-scale representation of the information provided by selected descriptors applied locally at intracellular levels; use of scale invariant descriptors by including homology groups that are invariant in all continuous transformations of the image; integrate information from local studies to construct global versions by combining the cohomology theory of bundles and mereotopological schemes.

 This project is particularly innovative because it uses notions which, although known in mathematics, are little used by the signal processing community. Indeed, the measurements provided by these approaches are not conventional in that they are often represented by purely algebraic objects which require the creation of abstract spaces that represent the local information and connectivity with the neighborhood. We do not intend to simply use notions as surface level, as has been already done before, but to study the information from quantitative phase images. Understanding these images by constructing ad-hoc spaces is also part of the innovative nature of this project.

 On the other hand, the implication of an industrial partner which develops quantitative phase imaging sensors, acquires and model the data, and provides physical interpretation of them, and of complementary teams including pathologists and bioinformatics analyst from one side and signal processing and mathematician from the other side, made this project an interface between arduous mathematical theories and practical applications.