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Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation.

Captur G, Karperien AL, Li C, Zemrak F, Tobon-Gomez C, Gao X, Bluemke DA, Elliott PM, Petersen SE, Moon JC - J Cardiovasc Magn Reson (2015)

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.

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.

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.


It is possible to construct a family of fractals that share the same FD, but differ sharply in their overall texture so they have uncorrelated values for λ —likewise two objects may have the same λ but very different FD. In a, two 2D binary sets are presented that share the same λ but have different FD. For quantifying myocardial trabecular complexity in CMR cines, FD was chosen over λ for a number of reasons: 1) experiments on grayscale short-axis imaging sequences showed λ was confounded by signal from the central blood pool; 2) as λ measures translational invariance (imagine the binary edge-image rotated clockwise as per curved arrow in b), it is theoretically possible for a heavily but symmetrically trabeculated heart (b, left image) to have a lower value for λ than one with fewer, more irregularly spaced trabeculae (b, right image). On the contrary, if there are more trabeculae, whether regularly or irregularly spaced, FD will always be higher. As the sole objective of this tool was to quantify trabeculae, the extra information on spatial heterogeneity encoded in λ could only have distracted from the biological signal of interest; 3) λ is a very scale-dependent meter and potentially more susceptible to differences in image resolution across vendors and CMR centres compared to FD. λ = lacunarity. Other abbreviations as in Figs. 2 and 3
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Fig6: It is possible to construct a family of fractals that share the same FD, but differ sharply in their overall texture so they have uncorrelated values for λ —likewise two objects may have the same λ but very different FD. In a, two 2D binary sets are presented that share the same λ but have different FD. For quantifying myocardial trabecular complexity in CMR cines, FD was chosen over λ for a number of reasons: 1) experiments on grayscale short-axis imaging sequences showed λ was confounded by signal from the central blood pool; 2) as λ measures translational invariance (imagine the binary edge-image rotated clockwise as per curved arrow in b), it is theoretically possible for a heavily but symmetrically trabeculated heart (b, left image) to have a lower value for λ than one with fewer, more irregularly spaced trabeculae (b, right image). On the contrary, if there are more trabeculae, whether regularly or irregularly spaced, FD will always be higher. As the sole objective of this tool was to quantify trabeculae, the extra information on spatial heterogeneity encoded in λ could only have distracted from the biological signal of interest; 3) λ is a very scale-dependent meter and potentially more susceptible to differences in image resolution across vendors and CMR centres compared to FD. λ = lacunarity. Other abbreviations as in Figs. 2 and 3

Mentions: The FD is not the only tool available in fractal geometry—others such as lacunarity also exist that provide a different layer of information relating more to the texture of objects [17]. Lacunarity (λ) mesures the size distribution of gaps (lacunae) in an image, providing a measure of heterogeneity [18]. It is the counterpart to the FD but the two are non-identical (Fig. 6). If an image has few, small, and regular gaps and is translationally and rotationally invariant, it will have low λ; if it has many large and irregular gaps with notable translational and rotational variance, it will have high λ. The translational invariance (spatial heterogeneity [19]) that is measured by lacunarity implies that: 1) λ is highly scale-dependent, meaning an image that appears highly heterogenous at low scale may appear much more homogenous at large scale producing two very different values of λ; and 2) λ (like the related box-counting fractal analysis) may be used to study non-fractal objects. λ and the FD are usually used complementarily, but for some biomedical applications lacunarity may be preferred (e.g., quantification of trabecular bone by MR [20] where the widely varying pattern of emptiness between spicules is the feature of interest, Fig. 1c), and in others the FD is preferred (e.g., endocardial contours with large central emptiness and edge detail, Fig. 5).Fig. 6


Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation.

Captur G, Karperien AL, Li C, Zemrak F, Tobon-Gomez C, Gao X, Bluemke DA, Elliott PM, Petersen SE, Moon JC - J Cardiovasc Magn Reson (2015)

It is possible to construct a family of fractals that share the same FD, but differ sharply in their overall texture so they have uncorrelated values for λ —likewise two objects may have the same λ but very different FD. In a, two 2D binary sets are presented that share the same λ but have different FD. For quantifying myocardial trabecular complexity in CMR cines, FD was chosen over λ for a number of reasons: 1) experiments on grayscale short-axis imaging sequences showed λ was confounded by signal from the central blood pool; 2) as λ measures translational invariance (imagine the binary edge-image rotated clockwise as per curved arrow in b), it is theoretically possible for a heavily but symmetrically trabeculated heart (b, left image) to have a lower value for λ than one with fewer, more irregularly spaced trabeculae (b, right image). On the contrary, if there are more trabeculae, whether regularly or irregularly spaced, FD will always be higher. As the sole objective of this tool was to quantify trabeculae, the extra information on spatial heterogeneity encoded in λ could only have distracted from the biological signal of interest; 3) λ is a very scale-dependent meter and potentially more susceptible to differences in image resolution across vendors and CMR centres compared to FD. λ = lacunarity. Other abbreviations as in Figs. 2 and 3
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Related In: Results  -  Collection

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Fig6: It is possible to construct a family of fractals that share the same FD, but differ sharply in their overall texture so they have uncorrelated values for λ —likewise two objects may have the same λ but very different FD. In a, two 2D binary sets are presented that share the same λ but have different FD. For quantifying myocardial trabecular complexity in CMR cines, FD was chosen over λ for a number of reasons: 1) experiments on grayscale short-axis imaging sequences showed λ was confounded by signal from the central blood pool; 2) as λ measures translational invariance (imagine the binary edge-image rotated clockwise as per curved arrow in b), it is theoretically possible for a heavily but symmetrically trabeculated heart (b, left image) to have a lower value for λ than one with fewer, more irregularly spaced trabeculae (b, right image). On the contrary, if there are more trabeculae, whether regularly or irregularly spaced, FD will always be higher. As the sole objective of this tool was to quantify trabeculae, the extra information on spatial heterogeneity encoded in λ could only have distracted from the biological signal of interest; 3) λ is a very scale-dependent meter and potentially more susceptible to differences in image resolution across vendors and CMR centres compared to FD. λ = lacunarity. Other abbreviations as in Figs. 2 and 3
Mentions: The FD is not the only tool available in fractal geometry—others such as lacunarity also exist that provide a different layer of information relating more to the texture of objects [17]. Lacunarity (λ) mesures the size distribution of gaps (lacunae) in an image, providing a measure of heterogeneity [18]. It is the counterpart to the FD but the two are non-identical (Fig. 6). If an image has few, small, and regular gaps and is translationally and rotationally invariant, it will have low λ; if it has many large and irregular gaps with notable translational and rotational variance, it will have high λ. The translational invariance (spatial heterogeneity [19]) that is measured by lacunarity implies that: 1) λ is highly scale-dependent, meaning an image that appears highly heterogenous at low scale may appear much more homogenous at large scale producing two very different values of λ; and 2) λ (like the related box-counting fractal analysis) may be used to study non-fractal objects. λ and the FD are usually used complementarily, but for some biomedical applications lacunarity may be preferred (e.g., quantification of trabecular bone by MR [20] where the widely varying pattern of emptiness between spicules is the feature of interest, Fig. 1c), and in others the FD is preferred (e.g., endocardial contours with large central emptiness and edge detail, Fig. 5).Fig. 6

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.

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.

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.