Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation.
Bottom Line:
Many of the structures and parameters that are detected, measured and reported in cardiovascular magnetic resonance (CMR) have at least some properties that are fractal, meaning complex and self-similar at different scales.To date however, there has been little use of fractal geometry in CMR; by comparison, many more applications of fractal analysis have been published in MR imaging of the brain.This review explains the fundamental principles of fractal geometry, places the fractal dimension into a meaningful context within the realms of Euclidean and topological space, and defines its role in digital image processing.It summarises the basic mathematics, highlights strengths and potential limitations of its application to biomedical imaging, shows key current examples and suggests a simple route for its successful clinical implementation by the CMR community.By simplifying some of the more abstract concepts of deterministic fractals, this review invites CMR scientists (clinicians, technologists, physicists) to experiment with fractal analysis as a means of developing the next generation of intelligent quantitative cardiac imaging tools.
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Affiliation: UCL Institute of Cardiovascular Science, University College London, Gower Street, London, WC1E 6BT, UK. gabriella.captur.11@ucl.ac.uk.
ABSTRACT
Many of the structures and parameters that are detected, measured and reported in cardiovascular magnetic resonance (CMR) have at least some properties that are fractal, meaning complex and self-similar at different scales. To date however, there has been little use of fractal geometry in CMR; by comparison, many more applications of fractal analysis have been published in MR imaging of the brain.This review explains the fundamental principles of fractal geometry, places the fractal dimension into a meaningful context within the realms of Euclidean and topological space, and defines its role in digital image processing. It summarises the basic mathematics, highlights strengths and potential limitations of its application to biomedical imaging, shows key current examples and suggests a simple route for its successful clinical implementation by the CMR community.By simplifying some of the more abstract concepts of deterministic fractals, this review invites CMR scientists (clinicians, technologists, physicists) to experiment with fractal analysis as a means of developing the next generation of intelligent quantitative cardiac imaging tools. No MeSH data available. |
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Mentions: Geometrically a fractal would exist in between our more familiar topological dimensions (DT): between the 1st and 2nd DT, or between the 2nd and 3rd, etc. An understanding of the concept of fractal dimensionality begins therefore with at least some understanding of DT and Euclidean dimensionality (DE) (Fig. 3). Euclidean space refers to an object’s embedding space and encompasses dimensions that we define using Cartesian coordinates (real numbers e.g., x, y and z). Figure 3 explains why some objects will have DT = DE, while others will have DT < DE. Unlike the topological and Euclidean dimensions, the fractal dimension (FD) measures the detailed self-similarity of fractals—the space-filling capacity of a set of points embedded in space or its complexity. It is related to DE and DT by Eq 1:Fig. 3 |
View Article: PubMed Central - PubMed
Affiliation: UCL Institute of Cardiovascular Science, University College London, Gower Street, London, WC1E 6BT, UK. gabriella.captur.11@ucl.ac.uk.
No MeSH data available.