Vinh Thong Ta

Graph-based tools for microscopic cellular image segmentation

We propose a framework of graph-based tools for the segmentation of microscopic cellular images. This framework is based on an object oriented analysis of imaging problems in pathology. Our graph tools rely on a general formulation of discrete functional regularization on weighted graphs of arbitrary topology. It leads to a set of useful tools which can be combined together to address various image segmentation problems in pathology. To provide fast image segmentation algorithms, we also propose an image simplification based on graphs as a pre processing step. The abilities of this set of image processing discrete tools are illustrated through automatic and interactive segmentation schemes for color cytological and histological images segmentation problems.

Partial Difference Equations over Graphs: Morphological Processing of Arbitrary Discrete Data

Mathematical Morphology (MM) offers a wide range of operators to address various image processing problems. These processing can be defined in terms of algebraic set or as partial differential equations (PDEs). In this paper, a novel approach is formalized as a framework of partial difference equations (PdEs) on weighted graphs. We introduce and analyze morphological operators in local and nonlocal configurations. Our framework recovers classical local algebraic and PDEs-based morphological methods in image processing context; generalizes them for nonlocal configurations and extends them to the treatment of any arbitrary discrete data that can be represented by a graph. It leads to considering a new field of application of MM processing: the case of high-dimensional multivariate unorganized data.

Nonlocal graph regularization for image colorization

In this paper we present a simple colorization method that relies on nonlocal graph regularization. We introduce nonlocal discrete differential operators and a family of weighted p-Laplace operators. Then, p-Laplace regularization on weighted graphs problem is presented and the associated filter family. Image colorization is then considered as a graph regularization problem for a function mapping vertices to chrominances. Several results illustrate our framework and demonstrate the benefits of nonlocal graph regularization for image colorization.

Partial differences as tools for filtering data on graphs

High-dimensional feature spaces are often corrupted by noise. This is problematic for the processing of manifolds and data sets since most of reference methods (and especially graph-based ones) are sensitive to noise. This paper presents pre-processing methods for manifold denoising and simplification based on discrete analogues of continuous regularization and mathematical morphology. The proposed filtering methods provide a general discrete framework for the filtering of manifolds and data with p-Laplacian regularization and mathematical morphology. With our proposals, one obtains filters that can operate on any high-dimensional unorganized multivariate data. Experiments will show that the proposed approaches are efficient to denoise manifolds and data, to project initial noisy data onto a submanifold, and to ease dimensionality reduction, clustering and classification.

Adaptation of Eikonal Equation over Weighted Graph

In this paper, an adaptation of the eikonal equation is proposed by considering the latter on weighted graphs of arbitrary structure. This novel approach is based on a family of discrete morphological local and nonlocal gradients expressed by partial difference equations (PdEs). Our formulation of the eikonal equation on weighted graphs generalizes local and nonlocal configurations in the context of image processing and extends this equation for the processing of any unorganized high dimensional discrete data that can be represented by a graph. Our approach leads to a unified formulation for image segmentation and high dimensional irregular data processing.

Graph-based multi-resolution segmentation of histological whole slide images

In this paper, we present a graph-based multi-resolution approach for mitosis extraction in breast cancer histological whole slide images. The proposed segmentation uses a multi-resolution approach which reproduces the slide examination done by a pathologist. Each resolution level is analyzed with a focus of attention resulting from a coarser resolution level analysis. At each resolution level, a spatial refinement by semi-supervised clustering is performed to obtain more accurate segmentation around edges. The proposed segmentation is fully unsupervised by using domain specific knowledge.

Nonlocal multiscale hierarchical decomposition on graphs

The decomposition of images into their meaningful components is one of the major tasks in computer vision. Tadmor, Nezzar and Vese [1] have proposed a general approach for multiscale hierarchical decomposition of images. On the basis of this work, we propose a multiscale hierarchical decomposition of functions on graphs. The decomposition is based on a discrete variational framework that makes it possible to process arbitrary discrete data sets with the natural introduction of nonlocal interactions. This leads to an approach that can be used for the decomposition of images, meshes, or arbitrary data sets by taking advantage of the graph structure. To have a fully automatic decomposition, the issue of parameter selection is fully addressed. We illustrate our approach with numerous decomposition results on images, meshes, and point clouds and show the benefits.

Partial difference equations on graphs for Mathematical Morphology operators over images and manifolds

The main tools of mathematical morphology are a broad class of nonlinear image operators. They can be defined in terms of algebraic set operators or as partial differential equations (PDEs). We propose a framework of partial difference equations on arbitrary graphs for introducing and analyzing morphological operators in local and non local configurations. The proposed framework unifies the classical local PDEs-based morphology for image processing, generalizes them for non local configurations and extends them to the processing of any discrete data living on graphs.

Graph Based Semi and Unsupervised Classification and Segmentation of Microscopic Images

In this paper, we propose a general formulation of discrete functional regularization on weighted graphs. This framework can be used on any multi-dimensional data living on graphs of the arbitrary topologies. In this work, we focus on microscopic image segmentation and classification within semi and unsupervised schemes. Moreover, to provide a fast image segmentation we propose a graph based image simplification as a pre-processing step. Biological elements contained in images such as cells, cytoplasm and nuclei are segmented and classified with this image simplification and label diffusion processes on weighted graphs.

Nonlocal morphological levelings by partial difference equations over weighted graphs

In this paper, a novel approach to mathematical morphology operations is proposed. Morphological operators based on partial differential equations (PDEs) are extended to weighted graphs of the arbitrary topologies by considering partial difference equations. We focus on a general class of morphological filters, the levelings; and propose a novel approach of such filters. Indeed, our methodology recovers classical local PDEs-based levelings in image processing, generalizes them to nonlocal configurations and extends them to process any discrete data that can be represented by a graph. Experimental results show applications and the potential of our levelings to textured image processing, region adjacency graph based multiscale leveling and unorganized data set filtering.