Limits...
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.


A line, square or cube all exist in Euclidean space with a certain number of dimensions described classically by DE = 0 for a single point, 1 for a line (a), 2 for a plane (b) and 3 for a 3D object (d) [38]. The concept of topology is rooted in the idea of connectedness among points in a set. The  (empty) set in topology (∅) has no points and its DT is by definition ‘-1’. A single point or a number of points makes up a ‘countable set’. In topology, a set’s DT is always 1 integer value greater than the particular DT of the simplest form that can be used to ‘cut’ the set into two parts [42]. A single point or a few points (provided they are not connected) are already separated, so it takes ‘nothing’ (∅) to separate them. Thus the DT of a point is 0 (−1 + 1 = 0). A line (a) or an open curve can be severed by the removal of a point so it has DT = 1. A topological subset such as b can have an interior, boundary and exterior. b has a closed boundary of points (like y). When its interior is empty, b is referred to as a boundary set. Its interior may instead be full of points (like x) that are not boundary points because separating them from the exterior is a neighbourhood of other points also contained in b. All points of the subset that are neither interior nor boundary will form the exterior of b. A line of DT = 1 is required to split this topological set into 2 parts, therefore the DT of b = 2. Flat disks (c) have DT = 2 because they can be cut by a line with a DT = 1. A warped surface can be cut by a curved open line (of DT = 1) so its DT = 2 although its DE = 3. Therefore, while lines and disks have DT = DE, warped surfaces have DT one less than DE. DE = Euclidean dimension; DT = topological dimension
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License 1 - License 2
getmorefigures.php?uid=PMC4562373&req=5

Fig3: A line, square or cube all exist in Euclidean space with a certain number of dimensions described classically by DE = 0 for a single point, 1 for a line (a), 2 for a plane (b) and 3 for a 3D object (d) [38]. The concept of topology is rooted in the idea of connectedness among points in a set. The (empty) set in topology (∅) has no points and its DT is by definition ‘-1’. A single point or a number of points makes up a ‘countable set’. In topology, a set’s DT is always 1 integer value greater than the particular DT of the simplest form that can be used to ‘cut’ the set into two parts [42]. A single point or a few points (provided they are not connected) are already separated, so it takes ‘nothing’ (∅) to separate them. Thus the DT of a point is 0 (−1 + 1 = 0). A line (a) or an open curve can be severed by the removal of a point so it has DT = 1. A topological subset such as b can have an interior, boundary and exterior. b has a closed boundary of points (like y). When its interior is empty, b is referred to as a boundary set. Its interior may instead be full of points (like x) that are not boundary points because separating them from the exterior is a neighbourhood of other points also contained in b. All points of the subset that are neither interior nor boundary will form the exterior of b. A line of DT = 1 is required to split this topological set into 2 parts, therefore the DT of b = 2. Flat disks (c) have DT = 2 because they can be cut by a line with a DT = 1. A warped surface can be cut by a curved open line (of DT = 1) so its DT = 2 although its DE = 3. Therefore, while lines and disks have DT = DE, warped surfaces have DT one less than DE. DE = Euclidean dimension; DT = topological dimension

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


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)

A line, square or cube all exist in Euclidean space with a certain number of dimensions described classically by DE = 0 for a single point, 1 for a line (a), 2 for a plane (b) and 3 for a 3D object (d) [38]. The concept of topology is rooted in the idea of connectedness among points in a set. The  (empty) set in topology (∅) has no points and its DT is by definition ‘-1’. A single point or a number of points makes up a ‘countable set’. In topology, a set’s DT is always 1 integer value greater than the particular DT of the simplest form that can be used to ‘cut’ the set into two parts [42]. A single point or a few points (provided they are not connected) are already separated, so it takes ‘nothing’ (∅) to separate them. Thus the DT of a point is 0 (−1 + 1 = 0). A line (a) or an open curve can be severed by the removal of a point so it has DT = 1. A topological subset such as b can have an interior, boundary and exterior. b has a closed boundary of points (like y). When its interior is empty, b is referred to as a boundary set. Its interior may instead be full of points (like x) that are not boundary points because separating them from the exterior is a neighbourhood of other points also contained in b. All points of the subset that are neither interior nor boundary will form the exterior of b. A line of DT = 1 is required to split this topological set into 2 parts, therefore the DT of b = 2. Flat disks (c) have DT = 2 because they can be cut by a line with a DT = 1. A warped surface can be cut by a curved open line (of DT = 1) so its DT = 2 although its DE = 3. Therefore, while lines and disks have DT = DE, warped surfaces have DT one less than DE. DE = Euclidean dimension; DT = topological dimension
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4562373&req=5

Fig3: A line, square or cube all exist in Euclidean space with a certain number of dimensions described classically by DE = 0 for a single point, 1 for a line (a), 2 for a plane (b) and 3 for a 3D object (d) [38]. The concept of topology is rooted in the idea of connectedness among points in a set. The (empty) set in topology (∅) has no points and its DT is by definition ‘-1’. A single point or a number of points makes up a ‘countable set’. In topology, a set’s DT is always 1 integer value greater than the particular DT of the simplest form that can be used to ‘cut’ the set into two parts [42]. A single point or a few points (provided they are not connected) are already separated, so it takes ‘nothing’ (∅) to separate them. Thus the DT of a point is 0 (−1 + 1 = 0). A line (a) or an open curve can be severed by the removal of a point so it has DT = 1. A topological subset such as b can have an interior, boundary and exterior. b has a closed boundary of points (like y). When its interior is empty, b is referred to as a boundary set. Its interior may instead be full of points (like x) that are not boundary points because separating them from the exterior is a neighbourhood of other points also contained in b. All points of the subset that are neither interior nor boundary will form the exterior of b. A line of DT = 1 is required to split this topological set into 2 parts, therefore the DT of b = 2. Flat disks (c) have DT = 2 because they can be cut by a line with a DT = 1. A warped surface can be cut by a curved open line (of DT = 1) so its DT = 2 although its DE = 3. Therefore, while lines and disks have DT = DE, warped surfaces have DT one less than DE. DE = Euclidean dimension; DT = topological dimension
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

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.