A Computational Approach to Edge Detection
IEEE Transactions on Pattern Analysis and Machine Intelligence
Principal Warps: Thin-Plate Splines and the Decomposition of Deformations
IEEE Transactions on Pattern Analysis and Machine Intelligence
Scale-Space and Edge Detection Using Anisotropic Diffusion
IEEE Transactions on Pattern Analysis and Machine Intelligence
Multichannel Texture Analysis Using Localized Spatial Filters
IEEE Transactions on Pattern Analysis and Machine Intelligence
Intrinsic scale space for images on surfaces: The Geodesic Curvature Flow
Graphical Models and Image Processing
Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures
Geometric Multiscale Representation of Numerical Images
SCALE-SPACE '99 Proceedings of the Second International Conference on Scale-Space Theories in Computer Vision
An EMD-based recognition method for Chinese fonts and styles
Pattern Recognition Letters
A PDE characterization of the intrinsic mode functions
ICASSP '09 Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing
IJCAI'83 Proceedings of the Eighth international joint conference on Artificial intelligence - Volume 2
Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations
Evolution equations for continuous-scale morphological filtering
IEEE Transactions on Signal Processing
Localized measurement of emergent image frequencies by Gabor wavelets
IEEE Transactions on Information Theory - Part 2
An axiomatic approach to image interpolation
IEEE Transactions on Image Processing
A PDE based and interpolation-free framework for modeling the sifting process in a continuous domain
Advances in Computational Mathematics
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Many works have been achieved for analyzing images with a multiscale approach. In this paper, an intrinsic and nonlinear multiscale image decomposition is proposed, based on partial differential equations (PDEs) and the image frequency contents. Our model is inspired from the 2D empirical mode decomposition (EMD) for which a theoretical study is quite nonexistent, mainly because the algorithm is based on heuristic and ad hoc elements making its mathematical study hard. This work has three main advantages. Firstly, we prove that the 2D sifting process iterations are consistent with the resolution of a nonlinear PDE, by considering continuous morphological operators to build local upper and lower envelopes of the image extrema. In addition to the fact that now differential calculus can be performed on envelopes, the introduction of such morphological filters eliminates the interpolation dependency that also terribly suffers the method. Also, contrary to former 2D empirical modes, precise mathematical definition for a class of functions are now introduced thanks to the nonlinear PDE derived from the consistency result, and their characterization on the basis of Meyer spaces. Secondly, an intrinsic multiscale image decomposition is introduced based on the image frequency contents; the proposed approach almost captures the essence and philosophy of the 2D EMD and is linked to the well known Absolutely Minimizing Lipschitz Extension model. Lastly, the proposed multiscale decomposition allows a reconstruction of images. The filterbank capability of the new multiscale decomposition algorithm is shown both on synthetic and real images, and results show that our proposed approach improves a lot on the 2D EMD. Moreover, the complexity of the proposed multiscale decomposition is very reduced compared to the 2D EMD by avoiding the surface interpolation approach, which is the core of all 2D EMD algorithms and is very time consuming. For that purpose also, our work will then be a great benefit; especially, in higher dimension spaces.