This work presents the general form solution of Koopman Partial Differential Equation and shows t... more This work presents the general form solution of Koopman Partial Differential Equation and shows that its functional dimensionality is finite. The dimensionality is as the dimensionality of the dynamics. Thus, the representation of nonlinear dynamics as a linear one with a finite set of Koopman eigenfunctions without error is possible. This formulation justifies the flowbox statement and provides a simple numerical method to find such representation.
This work presents the general form solution of Koopman Partial Differential Equation and shows t... more This work presents the general form solution of Koopman Partial Differential Equation and shows that its functional dimensionality is finite. The dimensionality is as the dimensionality of the dynamics. Thus, the representation of nonlinear dynamics as a linear one with a finite set of Koopman eigenfunctions without error is possible. This formulation justifies the flowbox statement and provides a simple numerical method to find such representation.
Inpainting of images, represented on surfaces, was previously considered successfully by the appl... more Inpainting of images, represented on surfaces, was previously considered successfully by the application of a two-dimensional Laplacian operator, including in the context of the discrete representation on surfaces. Considering the shortcoming of errors and computational complexity of the strictly-two-dimensional approaches, we propose a one-dimensional-based multi-line approach, wherein the set of lines cover the region of inpainting and its boundaries. The multi-line framework is implemented on smooth and non-smooth surfaces and on images and the results are found to be superior to the previously published results. Additional possible one-dimensional representations are discussed.
European Signal Processing Conference, Oct 18, 2012
A discrete version of the Ricci flow, applicable to images, is introduced and applied in image de... more A discrete version of the Ricci flow, applicable to images, is introduced and applied in image denoising and in singleimage-based enhancement and super-resolution. This flow is unique among the geometric flows that have been applied in image processing, in that it is the only flow wherein the metric of an image evolves rather than the image itself as is the case in other geometric flows applicable in image processing. The flow is based on the combinatorial Ricci curvature defined by Forman, that was previously introduced by the authors in the context of image processing. It is shown that the Ricci flow preserves image structure much better than the Beltrami flow and other state-of-the-art image enhancement schemes. Implementation of the Ricci flow is applicable also to general surfaces such as required in computer graphics and other applications.
In this paper we present a simple method for minimal distortion development of triangulated surfa... more In this paper we present a simple method for minimal distortion development of triangulated surfaces for mapping and imaging. The method is based on classical results of F. Gehring and Y. Väisälä regarding the existence of quasi-conformal and quasi-isometric mappings between Riemannian manifolds. A random starting triangle version of the algorithm is presented. A curvature based version is also applicable. In addition the algorithm enables the user to compute the maximal distortion errors. Moreover, the algorithm makes no use to derivatives, hence it is suitable for analysis of noisy data. The algorithm is tested on data obtained from real CT images of the human brain cortex.
Journal of Mathematical Imaging and Vision, Nov 30, 2007
We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the p... more We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the proofs resides in triangulation results for manifolds with boundary, not necessarily bounded. The method is based upon geometric considerations that are further augmented for 2-dimensional manifolds (i.e surfaces). In addition, we show how to apply the main results to obtain a new, geometric proof of the classical Shannon sampling theorem, and also to image analysis.
In this paper we present a simple method for minimal distortion development of triangulated surfa... more In this paper we present a simple method for minimal distortion development of triangulated surfaces for colon mapping and general analysis of medical images. The method is based on classical results of Gehring and Väisalä regarding the existence of quasi-comformal and quasi-isometric mappings between Riemannian manifolds. Random and curvature based variations of the algorithm are presented. In addition the algorithm enables the user to compute the maximal distortion errors. The algorithm was tested both on synthetic images of the human skull and on real CT images of the human colon.
In this paper we introduce two theories of finite type invariants for framed links with a fixed l... more In this paper we introduce two theories of finite type invariants for framed links with a fixed linking matrix. We show that these theories are different from, but related to, the theory of Vassiliev invariants of knots and links. We will take special note of the case of zero linking matrix. i.e., zero-framed algebraically split links. We also study the corresponding spaces of "chord diagrams".
We present a novel scheme to the coverage problem, introducing a quantitative way to estimate the... more We present a novel scheme to the coverage problem, introducing a quantitative way to estimate the interaction between a block an its environment. This is achieved by setting a discrete version of Green's Theorem, specially adapted for Model Checking based verification of integrated circuits. This method is best suited for the coverage problem since it enables one to quantify the incompleteness or, on the other hand, the redundancy of a set of rules, describing the model under verification. Moreover this can be done continuously throughout the verification process, thus enabling the user to pinpoint the stages at which incompleteness/redundancy occurs. Although the method is presented locally on a small hardware example, we additionally show its possibility to provide precise coverage estimation also for large scale systems. We compare this method to others by checking it on the same test-cases.
Relationships that exist between the classical, Shannontype, and geometric-based approaches to sa... more Relationships that exist between the classical, Shannontype, and geometric-based approaches to sampling are investigated. Some aspects of coding and communication through a Gaussian channel are considered. In particular, a constructive method to determine the quantizing dimension in Zador's theorem is provided. A geometric version of Shannon's Second Theorem is introduced. Applications to Pulse Code Modulation and Vector Quantization of Images are addressed.
European Signal Processing Conference, Sep 1, 2006
The general problem of sampling and flattening of folded surfaces for the purpose of their two-di... more The general problem of sampling and flattening of folded surfaces for the purpose of their two-dimensional representation and analysis as images is addressed. We present a method and algorithm based on extension of the classical results of Gehring and Väisalä regarding the existence of quasi-conformal and quasi-isometric mappings between Riemannian manifolds. Proper surface sampling, based on maximal curvature is first discussed. We then develop the algorithm for mapping of this surface triangulation into the corresponding flat triangulated representation. The proposed algorithm is basically local and, therefore, suitable for extensively folded surfaces such as encountered in medical imaging. The theory and algorithm guarantee minimal metric, angular and area distortion. Yet, it is relatively simple, robust and computationally efficient, since it does not require computational derivatives. In this paper we present the sampling and flattening only, without complementing them by proper interpolation. We demonstrate the algorithm using medical and synthetic data.
Journal of Knot Theory and Its Ramifications, Nov 1, 2002
In this paper we introduce two theories of finite type invariants for framed links with a fixed l... more In this paper we introduce two theories of finite type invariants for framed links with a fixed linking matrix. We show that these theories are different from, but related to, the theory of Vassiliev invariants of knots and links. We will take special note of the case of zero linking matrix. i.e., zero-framed algebraically split links. We also study the corresponding spaces of "chord diagrams".
Research on Koopman operator theory has focused on three key areas for several decades: the mathe... more Research on Koopman operator theory has focused on three key areas for several decades: the mathematical structure of the Koopman eigenfunction space, the basis of this space, and the ability to represent nonlinear dynamics as linear. This study provides a thorough and comprehensive framework for these topics, including theoretical, analytical, and numerical approaches. A novel mathematical structure is introduced, which outlines permissible actions on the infinite set of Koopman Eigenfunction, under which this set is closed. Notions of generating and independent sets of Koopman eigenfunctions are defined. In addition, notions of a minimal generating set, and a maximal independent set are defined and are shown to be equivalent. This structure defines conditions for independence within the set of Koopman eigenfunctions. This independent set can be interpreted as a new coordinate system in which the dynamical system is linear. The theory also highlights the equivalence of a minimal set, flowbox representation, and conservation laws. Finally, the presented theory is supported by numerical experiments. Given an N dimensional dynamical system, the conclusions from this work are as follows. Cardinality: There are only N independent solutions for Koopman PDE that generate the space of these solutions termed as a minimal set. System Reconstruction: From them, the governing laws are revealed as well as the conservation laws. Equivalency: Flowbox representation, conservation laws, and a minimal set are found to be equivalent to each other. Precise and Global Linearity: A minimal set is a coordinate system in which this dynamic is linear. I. INTRODUCTION The Koopman spectrum is a commonly used tool for the analysis of dynamical systems. Treating Koopman Eigenfunction space as an infinite dimensional space yields various techniques to represent the system as a linear one with truncated dimensionality 1-5. Naturally, these methods occasionally result in a redundant and overly large decomposition 6,7 , which can sometimes be inaccurate 8,9. Thus, the challenge of extracting meaningful information about the dynamics from samples remains open 10. In this study, we present analytical and numerical frameworks to identify the minimal set of Koopman Eigenfunctions required to perfectly recover the dynamic from samples. Regrettably, the mathematical framework of the Koopman spectrum has received little attention, resulting in a limited understanding of how to efficiently extract a representation based on the underlying geometry of this space from samples 11. Due to this lack of knowledge, unsophisticated and exhaustive algorithms 7,12,13 have been employed, but their assumptions have led to intrinsic flaws in dynamical representation and prediction, such as in highly nonlinear time-variant systems 14 , homogeneous flows 8 , or even linear systems with non-zero inputs 15. This study aims to bridge this knowledge gap by presenting a comprehensive theory on the mathematical structure of the set of Koopman eigenfunctions.
2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO), 2012
A discrete version of the Ricci flow, applicable to images, is introduced and applied in image de... more A discrete version of the Ricci flow, applicable to images, is introduced and applied in image denoising and in single-image-based enhancement and super-resolution. This flow is unique among the geometric flows that have been applied in image processing, in that it is the only flow wherein the metric of an image evolves rather than the image itself as is the case in other geometric flows applicable in image processing. The flow is based on the combinatorial Ricci curvature defined by Forman, that was previously introduced by the authors in the context of image processing. It is shown that the Ricci flow preserves image structure much better than the Beltrami flow and other state-of-the-art image enhancement schemes. Implementation of the Ricci flow is applicable also to general surfaces such as required in computer graphics and other applications.
In this work we establish a theoretical relation between the notions of scale and a discrete Fins... more In this work we establish a theoretical relation between the notions of scale and a discrete Finsler-Haantjes curvature. Based on this connection we demonstrate the applicability of the interpretation of scale in terms of curvature, to signal processing in the context of analysis and segmentation of textures in images. The outcome of this procedure is a novel scheme for texture segmentation that is based on scaled metric curvature. The presented method proves itself to be efficient even when the multiscale analysis is done up to scales of 19 and more. Our main conclusions are that the discrete curvature calculated on sampled images can give us an indication on the local scale within the image, and therefore can be used for many additional tasks in image analysis.
A method and algorithm of flattening folded surfaces, for two-dimensional representation and anal... more A method and algorithm of flattening folded surfaces, for two-dimensional representation and analysis of medical images, are presented. The method is based on an application to triangular meshes of classical results of Gehring and Väisälä regarding the existence of quasiconformal and quasi-isometric mappings. The proposed algorithm is basically local and, therefore, suitable for extensively folded surfaces encountered in medical imaging. The theory and algorithm guarantee minimal distance, angle and area distortion. Yet, the algorithm is relatively simple, robust and computationally efficient, since it does not require computational derivatives. Both randomstarting-point and curvature-based versions of the algorithm are presented. We demonstrate the algorithm using medical data obtained from real CT images of the colon and MRI scans of the human cortex. Further applications of the algorithm, for
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Papers by Eli Appleboim