The authors present an overview of part of their work on graph-based regularization. Introduced first in order to smooth and filter images, the authors have extended these methods to address semi-supervised clustering and segmentation of any discrete domain that can be represented by a graph of arbitrary structure. This framework unifies, within a same formulation, methods from machine learning and image processing communities. In this chapter, the authors propose to show how these graph-based approaches can lead to a useful set of tools that can be combined altogether to address various image processing problems in pathology such as cytological and histological image filtering, segmentation and classification.
TopIntroduction
In microscopic cellular imaging, the objective of segmentation is the extraction of cellular or tissue components. This problem is a difficult problem due to the large variations of the structures’ features. There are several strategies for segmenting images and a lot of different segmentation methods can be found in literature. For instance, histogram analysis, pixel classification, region growing, morphological segmentation or methods based on partial differential equations (PDEs) can be mentioned. PDEs-based methods are very effective tools that enable to perform a lot of different image processing tasks under a unified formalism, see for instance (Malladi & Sethian, 1996; Adiga & Malladi & Fernandez-Gonzalez & Solorzano, 2006; Chan & Shen, 2005; Aubert & Kornprobst, 2006) and references therein. Recently, data sets analysis and machine learning methods have received a lot of attention. They are based on graph Laplacian diffusion processes and have been used to perform data sets dimensionality reduction or classification (Belkin & Niyogi & Sindhwani, 2006; Zhou & Scholkopf, 2005; Lafon & Lee, 2006).
In this chapter, we use our recently proposed discrete regularization framework based on weighted graphs (Elmoataz & Lézoray & Bougleux, 2008) to address both the microscopic cellular image segmentation and classification problems. This framework is inspired by continuous regularization and data-dependent function analysis methods. It provides a unified formulation of functionals regularization between PDEs-based methods from image processing and data analysis from machine learning community. These tools constitute a framework. Within this framework, a large variety of operations can be performed, combined or derived, to produce a specific segmentation scheme for a given problem. This framework leads, on the one hand, to a family of linear and nonlinear filters. On the other hand, it provides label-based diffusion processes for image automatic and interactive segmentation. One of the specificity of the proposed framework is to use graphs as a discrete modeling of images at different levels (pixels or regions) and different component relationships (grid graph, proximity graph, etc.). Working on graphs, our framework leads to a set of flexible tools for image segmentation, image regularization or clusters extraction. The main purpose of this chapter is not to solve a particular class of cytology or histology problems but to show how, with our graph-based methodology, we can address particular image segmentation problems.
The chapter is organized as follows. Next section presents a modeling of microscopic imaging problems in pathology. It also defines the core structure of our approach, the weighted graphs and the discrete calculus performed on such structures. Section called “Graph-based regularization models” introduces our discrete regularization framework. From this formulation, we derive our discrete label diffusion method (semi-supervised clustering) and show how we can improve classical pixel-based semi-supervised segmentation by using region-based graphs. Section entitled “Experiments” presents applications for automatic and interactive color cytological and histological image segmentation and finally compares our approach with other methods. All the experiments or illustrations presented in this chapter are obtained with a standard Linux computer equipped with quadri 2.4 GHz Intel Xeon processors and 16 GB of RAM. Finally, last sections describe future works and conclusion.