Visual Point Set Processing with Lattice Structures: Application to parsimonious representations of digital histopathology images

28/08/2013
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Visual Point Set Processing with Lattice Structures: Application to parsimonious representations of digital histopathology images

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Visual Point Set Processing with Lattice Structures : Application to Parsimonious Representations of Digital Histopathology Images Nicolas Lom´enie Universit´e Paris Descartes, LIPADE, SIP Group nicolas.lomenie@parisdescartes.fr Digital tissue images are too big to be processed with traditional image processing pipelines. We resort to the nuclear architecture within the tissue to explore such big images with geometrical and topological representations based on Delaunay triangulations of seed points. Then, we relate this representation to the parsimonious paradigm. Finally, we develop specific mathematical morphology operators to analyze any point set and contribute to the exploration of these huge medical images. Preliminary results proved good performance for both focusing on areas of interest and discrimination between slightly but significantly varying nuclear geometric configurations. Keywords : Digital Histopathology ; Point Set Processing ; Mathematical Morphology Sparsity and Digital Histopathology The rationale : 1. Shape as a geometric visual point set vs. an assembly of radiometric pixels ; 2. Image Analysis/Pattern Recognition Issues over Geometric and hence Sparse repre- sentations ; 3. Versatile nature of digital high-content histopathological images : staining procedure, biopsy techniques → structural analysis. The statement : Promoting new representations for the exploration of Whole Slide Images (WSIs) by using the recently acknowledged sparsity paradigm based on geometric representations. In [Chen et al. 2001], Chen et al. relates Huo’s findings about general image analysis : ”In one experiment, Huo analyzed a digitized image and found that the humanly interpretable information was really carried by the edgelet component of the de- composition. This surprising finding shows that, in a certain sense, images are not made of wavelets, but instead, the perceptually important components of the image are carried by edgelets. This contradicts the frequent claim that wavelets are the optimal basis for image representation, which may stimulate discussion.” We propose a sparse representation of a WSI based on a codebook of representative cells that are translated over the seed points detected by a low level processing operator as illustrated below. We use a semantic sparse representation relying on the most robustly detected significant tissue elements : the nuclei. WSInuclear(x, y) = (i,j)∈S δi,j(x, y) ∗ Cell Atom where S is a geometric point set corresponding to the nucleus seeds and Cell Atom is an atomic cell element image in the specific case of a 1-cell dictionary. S can be considered as a sparse representation of a WSI according to the given definition of a s-sparse vector x ∈ ℜd as given in [Needell & Ward 2012] : ||x||0 = |supp(x)| ≤ d << s (a) (b) (c) (d) (e) (f) (g) (h) (i) Sparse representation of a WSI illustrated with a tubule/gland structure ; (a) based on the (b) 1- atomic cell dictionary and the sparse represen- tation in (c) as a point set binary matrix S1 ; (d) Reconstruction of the tubule by convolution with a point set S1 obtained with a specific seed extractor ; (e) Superimposed with the gland ; (f) Reconstruction of the tubule by convolution with a point set S2 obtained with another specific seed extractor ; (g) superimposed with the gland struc- ture ; (h)(i) Sparse representations over a 1024 × 1024 sub-image of a more complex view out of a WSI (about 50 000 × 70 000 pixels size). In the field of computational pathology, graph-based representations and geometric science of information are gaining momentum [Doyle et al. 2008]. R´ef´erences [Chen et al. 2001] Chen SS, Donoho DL, Saunders MA. (2001) Atomic Decomposition Basis Pursuit, SIAM Review, 3(1), 129-159. [Doyle et al. 2008] Doyle, S., Agner, S., Madabhushi, A., Feldman, M. and Tomaszewski, J. (2008). Automated Grading of Breast Cancer Histopathology Using Spectral Clustering with Textural and Architectural Image Features, 5th IEEE Interna- tional Symposium on Biomedical Imaging, 29 :496-499. [Needell & Ward 2012] Needell D, Ward, R. (2012) Stable image reconstruction using total variation minimization http://arxiv.org/abs/1202 Point Set Processing Point set processing in the manner of image processing is gaining momentum in the computer graphics community [Rusu & Cousins 2011] with the example of the Point Cloud Library (PCL : http://www.pointclouds.org) inspired by the GNU Image Manipulation Program (GIMP : http://www.gimp.org). At the same time, in the field of applied mathematics, a new trend consists in adapting mature image analysis algorithms working on regular grids to parsimonious representa- tions like graphs of interest points or superpixels [Ta et al. 2009]. Applying mathematical morphology to graphs was first suggested in [Heijmans et al. 1992] but never really came up with tractable applications. Nevertheless, the idea is emerging again with recent works by the mathematical morphology pioneers [Cousty et al. 2009] and was also related to the concept of α-objects in [Lom´enie & Stamon 2008] based on seminal ideas in [Lom´enie et al. 2000] and then applied to the modeling of spatial relations and histopathology in [Lom´enie & Racoceanu 2012]. Lattice Structures for Point Set Processing We refer the reader to [Lom´enie & Stamon 2011] for a detailed presentation of the ma- thematical morphology framework operating on point sets. But formally it is enough to define a lattice structure operating on unorganized point sets, or more precisely, on a tessellation of the space that embeds any point set S in a neighborhood system. For any point set S ⊂ ℜ2, it exists a Delaunay triangulation Del(S) defining the aforementioned topology of the workspace. This mesh acts as the regular grid for a radiometric image. Then we define the complete lattice algebraic structure called L = (M(Del), ≤), where M(Del) is the set of meshes defined on Del, that means the set of mappings from a triangle T in Del(S) to a φT value in ℜ that is M ∈ M(Del) = {(T, φ)}T∈Del, and where the partial ordering relation ≤ is defined as follows : ∀M1 et M2 ∈ M(Del), M1 ≤ M2 ⇐⇒ ∀T ∈ Del, φ1 T ≤ φ2 T where φT is a positive measure of the k-simplex T in Del(S) related to the size, shape, area or visibility of the triangle [Lom´enie & Racoceanu 2012].The infimum operators are defined as follows : ∀M1 et M2 ∈ M(Del), inf(M1, M2) = {T ∈ Del, min(φ1 T , φ2 T )} sup(M1, M2) = {T ∈ Del, max(φ1 T , φ2 T )} Then, given the basic definition of an erosion and an involution c operators, we inherit the infinite spectrum of theoretical well-sounded range of operators from mathematical mor- phology : ∀M ∈ M(Del), e(M) = {T ∈ Del, eT } and Mc = {T ∈ Del, 1 − φT } Left : The pyramid of structural operators we can obtain ranging from the fundamen- tal low-level erosion operator to the se- mantic high-level Ductal Carcinoma In Situ (DCIS) characterization and the represen- tation of spatial relationships like ’between’. Structural Analysis for Digital Histopathology Focusing Operators : (Top) Focusing on a tumorous area at magnification ×1 of the WSI ; (Down) Focusing on a small part of the WSI at ×20 Pattern Recognition Operators : (Above) Characterizing a DCIS structure with a structural bio-code’110’ based on our operators with a precise (Method 1) and a coarse seed nu- clei extractor (Method 2) at magnification ×40. (Below) New results on a small database. Type Nb samples Correct Biocodes Method 1 Method 2 DCIS(S) =′ 110′ 10 9 8 DCISpost(S) =′ 110′ 10 9 9 Tubule(S) =′ 101′ 10 10 10 Digital Histopathology and Geometric Information Science : great challenges to tackle in the coming decade [GE Healthcare 2012]. R´ef´erences [Cousty et al. 2009] Cousty, J., Najman, L., and Serra, J. (2009). Some morphological operators in graph spaces, Lecture Notes in Computer Science, Mathemati- cal Morphology and Its Application to Signal and Image Processing, Springer, 5720 :149-160. [GE Healthcare 2012] Pathology Innovation Centre of Excellence (PICOE). Digital Histopathology : A New Frontier in Canadian Healthcare. White Paper. Ja- nuary 2012. GE Healthcare. http://www.gehealthcare.com/canada/it/ downloads/digitalpathology/GE_PICOE_Digital_Pathology_A_New_ Frontier_in_Canadian_Healthcare.pdf . Accessed December 2012. [Heijmans et al. 1992] Heijmans, H., Nacken, P., Toet, A., & Vincent, L. (1992). Graph Morphology. Journal of Visual Communication and Image Representation, 3(1) :24-38. [Lom´enie et al. 2000] Lom´enie, N., Gallo, L., Cambou, N. & Stamon, G. (2000). Mor- phological Operations on Delaunay Triangulations. International Conference on Pattern Recognition, 556-59. [Lom´enie & Stamon 2008] Lom´enie, N. and Stamon, G. (2008). Morphological Mesh fil- tering and alpha-objects, Pattern Recognition Letters, 29(10) :1571-79. [Lom´enie & Stamon 2011] Lom´enie, N. and Stamon, G. (2011). Point Set Analysis, Ad- vances in Imaging and Electron Physics, Peter W. Hawkes, San Diego : Academic Press, vol. 167, pp. 255-294. [Lom´enie & Racoceanu 2012] Lom´enie, N. and Racoceanu, D. (2012). Point set morpho- logical filtering and semantic spatial configuration modeling : application to micro- scopic image and bio-structure analysis, Pattern Recognition, 45(8) :2894-2911. [Rusu & Cousins 2011] , Rusu, R.B. and Cousins S. (2011) 3D is here : Point Cloud Library (PCL), IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China. [Ta et al. 2009] Ta, V.T., Lezoray, O., Elmoataz, A. and Schupp, S. (2009). Graph-based Tools for Microscopic Cellular Image Segmentation, Pattern Recognition, Special Issue on Digital Image Processing and Pattern Recognition Techniques for the Detection of Cancer, 42(6) :1113-25. Acknowledgement : This work is part of the SPIRIT project, program JCJC 2011 - ref : ANR-11-JS02-008-01 and of the MICO project, program TecSan 2010 - ref : ANR- 10-TECS-015. A free demonstrator can be downloaded at http://www.math-info. univ-paris5.fr/~lomn/Data/MorphoMesh.zip as an imageJ plugin.