Here you can find some of my recent research works "topic"-wise.
Multi-Field Topological Analysis
Over the last two decades, scalar topology has been used to support scientific data analysis and visualization, in particular through the use of the Reeb graph and its specialization, the contour tree. For multiple scalar fields (or multi-fields), the Reeb graph can be generalized with a structure called the Reeb space which captures the topological relationships of all fields simultaneously. However, working with the Reeb spaces needs further development both mathematically and computationally.
In META-project:
- Multivariate Topology Simplification. Amit Chattopadhyay, Hamish Carr, David Duke, Zhao Geng and Osamu Saeki. Computational Geometry: Theory and Application, Elsevier, 2016 (doi:10.1016/j.comgeo.2016.05.006).
- Fiber Surfaces: Generalizing Isosurfaces to Bivariate Data. Hamish Carr, Zhao Geng, Julien Tierny, Amit Chattopadhyay, Aaron Knoll. (EuroVis 2015), Computer Graphics Forum, Vol. 34, Issue 3, 2015.
- Extracting Jacobi Structures in Reeb Spaces. Amit Chattopadhyay, Hamish Carr, David Duke and Zhao Geng. The Eurographics Conference on Visualization, Eurovis 2014, 9-13 June, Swansea, UK, Short Paper.
- Simplifying Multivariate Topology (Extended Abstract). Amit Chattopadhyay, Hamish Carr, David Duke and Zhao Geng. In Computer Graphics & Visual Computing, CGVC 2014, 10-11 September, Leeds, UK, 2 pages.
- Visual Analysis of Hurricane Data Using Joint Contour Net. Zhao Geng, David Duke, Hamish Carr, Amit Chattopadhyay. In Computer Graphics & Visual Computing, CGVC 2014, 10-11 September, Leeds, UK.
Optimization on Matrix Manifolds and Applications
The problem of optimizing a real-valued function on a matrix manifold appears in a wide variety of computational problems, such as independent component analysis or blind source separation in signal processing, eigen value problem, matrix approximation and so on. From an optimization point of view, this problem could be formulated as finding a minimum of a real-valued non-smooth function defined on a matrix manifold. This project aims to extend different optimization methods from Euclidean set-up to Riemannian manifolds along with the convergence analysis.
- A Derivative-Free Riemannian Powell's Method Minimizing Hartley-Entropy-Based ICA Contrast. Amit Chattopadhyay, S. Easter Selvan and Umberto Amato. IEEE Transactions on Neural Networks and Learning Systems, Vol. 27, No. 9, Sept. 2016.
- Spherical mesh adaptive direct search for separating quasi-uncorrelated sources by range-based independent component analysis. S. Easter Selvan, Pierre B. Borckmans, Amit Chattopadhyay, P.-A. Absil. Neural Computation, MIT Press, Volume 25, Issue 9, Pages 2486-2522, 2013.
- Range-based non-orthogonal ICA using cross-entropy method. S. Easter Selvan, Amit Chattopadhyay, Umberto Amato, P.-A. Absil. European Symposium on Artificial Neural Networks (ESANN 2012), Computational Intelligence and Machine Learning, Bruges, Belgium, pages 519-524, 25-27 April 2012.
PhD thesis
- Certified Geometric Computation: Radial Basis Function based Isosurfaces and Morse-Smale Complexes. Amit Chattopadhyay, Advisor: Prof. Gert Vegter. PhD thesis, University of Groningen, ISBN: 978-90-367-4757-8. 172 pages. (English and Dutch summary)
Certified Computation of Morse-Smale Complexes
The Morse-Smale complex is an important tool for the global topological analysis of complex geometrical shapes or data. We consider the problem of computing certified separatrices of a Morse-Smale system connecting a saddle to a source or a sink, and separating attracting regions of sinks, and repelling regions of sources of the gradient field. Computing the Morse-Smale complex, i.e., the configurations of singular points and separatrices of a Morse-Smale gradient field, can be extremely challenging because of the arbitrarily complex nature of the input data, whereas available computational resources are limited.
- Certified Computation of planar Morse-Smale Complexes of Smooth Functions. Amit Chattopadhyay, Gert Vegter and Chee K. Yap. Journal of Symbolic Computation: Special issue on Algorithms and Software for Computational Topology, Elsevier, Sept, 2015, (doi:10.1016/j.jsc.2016.03.006).
- Certified Computation of planar Morse-Smale Complexes of Smooth Functions. Amit Chattopadhyay, Gert Vegter and Chee K. Yap. In Proceedings 27th ACM Symposium on Computational Geometry. SoCG 2012: Chapel Hill, NC, USA, Pages 259–268, June 16-20, 2012.
- Certified Computation of Morse-Smale Complexes on Implicit Surfaces. Amit Chattopadhyay and Gert Vegter. Abstract in 27th European Workshop on Computational Geometry, Morschach, Switzerland, pages 221-224, March 28-31, 2011.
- Certified Computation of planar Morse-Smale Complexes. Amit Chattopadhyay, Sijbo Holtman and Gert Vegter. Abstract in 26th European Workshop on Computational Geometry, Dortmund, Germany, pages 105-108 , March 22-24, 2010.
Certified Shape Reconstruction using Radial Basis Function (RBF) Method
Radial Basis Functions are widely used in scattered data interpolation. The surface-reconstruction method using radial basis functions consists of two steps: (i) computing an interpolating implicit function the zero set of which contains the points in the data set, followed by (ii) extraction of isocurves or isosurfaces. The problem of `proving a topological guarantee' of the reconstructed shapes is a challenging open problem.
- Certified Meshing of RBF-based Isosurfaces. Amit Chattopadhyay, Simon Plantinga and Gert Vegter. The Visual Computer, Springer, Vol. 28, Number 5, Pages 445-462, 2011.
- Poster: Certified Meshing of RBF-based Isosurfaces. Amit Chattopadhyay, Simon Plantinga and Gert Vegter. Workshop: Subdivide and Tile (Triangulating spaces for understanding the world), Lorentz Center, the Netherlands, Nov 16-20, 2009.
- Certified Meshing of RBF-based Isosurfaces. Amit Chattopadhyay, Simon Plantinga and Gert Vegter. Abstract in 25th European Workshop on Computational Geometry, Brussels, Belgium, pages 101-104, March 16-18, 2009.
Level Set Based Curve Evolution
Level set based curve evolution is a well-known research topic and has wide application ranging from image restoration to image segmentation, object tracking etc. The goal of my master project was to develop a method for pattern generation using level set based curve evolution. We analyzed two pattern generation models one from biology (Reaction-Diffusion model) and another one from physics (structural optimization model) and implemented these models under level set based curve evolution paradigm.
- Pattern Generation Using Level Set Based Curve Evolution. Amit Chattopadhyay and Dipti Prasad Mukherjee. Advances in Intelligent Information Processing: Tools and Applications, B. Chanda and C. A. Murthy (Eds.), Vol. 2, pp. 19-36, World Scientific, 2008.
On Zeros of Polynomials and Analytic Functions
- Certain Generalizations of Enestrom-Kakeya Theorem. Amit Chattopadhyay, S. Das, V. K. Jain and H. Konwar. Journal of Indian Mathematical Society, 72,147-156 (2005).