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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.


Related in: MedlinePlus

Applying fractal analysis to a 2D cine CMR slice (a) at the mid-left ventricular level [9]. Trabecular detail is extracted by a region-based level-set segmentation [40], followed by binarisation (b) and edge-detection (c). Binarisation eliminates pixel detail originating from the blood pool. The edge image is covered by a series of grids (d). The total number of sized d boxes making up this exemplar grid is 16, and the number of boxes N(d) required to completely cover the contour, 14 (2 boxes overlie blank space). For this set, box-counting will involve the application of 86 grid sizes. The minimum size is set to 2 pixels. The maximum size of the grid series is dictated by the dimensions of the bounding box (discontinuous red line) where ‘bounding box’ refers to the smallest rectangle that encloses the foreground pixels. The box diameter for each successive grid is set to drop by d-1 pixels each time. Through the implementation of this 2D box-counting approach, a fractal output of between 1 and 2 is expected. The log-lot plot (e) produces a good fit using linear regression and yields a gradient equivalent to - FD (1.363). d = box dimension; Ln = natural logarithm; N(d) = number of boxes. Other abbreviations as in Figs. 1 and 2
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Fig5: Applying fractal analysis to a 2D cine CMR slice (a) at the mid-left ventricular level [9]. Trabecular detail is extracted by a region-based level-set segmentation [40], followed by binarisation (b) and edge-detection (c). Binarisation eliminates pixel detail originating from the blood pool. The edge image is covered by a series of grids (d). The total number of sized d boxes making up this exemplar grid is 16, and the number of boxes N(d) required to completely cover the contour, 14 (2 boxes overlie blank space). For this set, box-counting will involve the application of 86 grid sizes. The minimum size is set to 2 pixels. The maximum size of the grid series is dictated by the dimensions of the bounding box (discontinuous red line) where ‘bounding box’ refers to the smallest rectangle that encloses the foreground pixels. The box diameter for each successive grid is set to drop by d-1 pixels each time. Through the implementation of this 2D box-counting approach, a fractal output of between 1 and 2 is expected. The log-lot plot (e) produces a good fit using linear regression and yields a gradient equivalent to - FD (1.363). d = box dimension; Ln = natural logarithm; N(d) = number of boxes. Other abbreviations as in Figs. 1 and 2

Mentions: Once the ROI is extracted, the FD can be calculated using many analysis methods (Table 1). Each will compute a different type of FD but fundamentally they all measure the same property of the ROI—they are all meters of complexity. Even for a single method (e.g., box-counting) multiple algorithmic variants may exist (box-counting may use either a conventional, overlapping, folded or symmetric surface scanning approach [15]). The conventional procedure for box-counting (Fig. 5) rests on simple arbitrary scaling and can be applied to structures lacking strictly self-similar patterns. It works by systematically laying a series of grids of boxes of decreasing calibre onto the ROI and counting (at each level) the number of boxes that overlies pixel detail. The FD is derived from the slope of the logarithmic regression line graphing the relationship of box count and scale. The number of data points used to generate these log-log plots is related to the number of measuring steps. Theoretically, given a priori knowledge of the scaling rules, a mathematical fractal would generate data points that lie along a perfect straight line. The point of practical analysis, however, is to find the scaling rule in the first place. For anisotropic biological objects (like left ventricular endocardial contours) as well as for precisely generated fractal images analysed without knowledge of the scaling rule, the data points do not generally lie on a straight line, reflecting sampling limitations as well as limited self-similarity [16], thus the slope is estimated from the regression line for the log-log plot. The choice of image preparation routine and the details of the method used to gather the data for fractal analysis are important as they can either increase or decrease the correlation coefficient of the double logarithmic plot (more linear or more sigmoid fit respectively).Table 1


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)

Applying fractal analysis to a 2D cine CMR slice (a) at the mid-left ventricular level [9]. Trabecular detail is extracted by a region-based level-set segmentation [40], followed by binarisation (b) and edge-detection (c). Binarisation eliminates pixel detail originating from the blood pool. The edge image is covered by a series of grids (d). The total number of sized d boxes making up this exemplar grid is 16, and the number of boxes N(d) required to completely cover the contour, 14 (2 boxes overlie blank space). For this set, box-counting will involve the application of 86 grid sizes. The minimum size is set to 2 pixels. The maximum size of the grid series is dictated by the dimensions of the bounding box (discontinuous red line) where ‘bounding box’ refers to the smallest rectangle that encloses the foreground pixels. The box diameter for each successive grid is set to drop by d-1 pixels each time. Through the implementation of this 2D box-counting approach, a fractal output of between 1 and 2 is expected. The log-lot plot (e) produces a good fit using linear regression and yields a gradient equivalent to - FD (1.363). d = box dimension; Ln = natural logarithm; N(d) = number of boxes. Other abbreviations as in Figs. 1 and 2
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Fig5: Applying fractal analysis to a 2D cine CMR slice (a) at the mid-left ventricular level [9]. Trabecular detail is extracted by a region-based level-set segmentation [40], followed by binarisation (b) and edge-detection (c). Binarisation eliminates pixel detail originating from the blood pool. The edge image is covered by a series of grids (d). The total number of sized d boxes making up this exemplar grid is 16, and the number of boxes N(d) required to completely cover the contour, 14 (2 boxes overlie blank space). For this set, box-counting will involve the application of 86 grid sizes. The minimum size is set to 2 pixels. The maximum size of the grid series is dictated by the dimensions of the bounding box (discontinuous red line) where ‘bounding box’ refers to the smallest rectangle that encloses the foreground pixels. The box diameter for each successive grid is set to drop by d-1 pixels each time. Through the implementation of this 2D box-counting approach, a fractal output of between 1 and 2 is expected. The log-lot plot (e) produces a good fit using linear regression and yields a gradient equivalent to - FD (1.363). d = box dimension; Ln = natural logarithm; N(d) = number of boxes. Other abbreviations as in Figs. 1 and 2
Mentions: Once the ROI is extracted, the FD can be calculated using many analysis methods (Table 1). Each will compute a different type of FD but fundamentally they all measure the same property of the ROI—they are all meters of complexity. Even for a single method (e.g., box-counting) multiple algorithmic variants may exist (box-counting may use either a conventional, overlapping, folded or symmetric surface scanning approach [15]). The conventional procedure for box-counting (Fig. 5) rests on simple arbitrary scaling and can be applied to structures lacking strictly self-similar patterns. It works by systematically laying a series of grids of boxes of decreasing calibre onto the ROI and counting (at each level) the number of boxes that overlies pixel detail. The FD is derived from the slope of the logarithmic regression line graphing the relationship of box count and scale. The number of data points used to generate these log-log plots is related to the number of measuring steps. Theoretically, given a priori knowledge of the scaling rules, a mathematical fractal would generate data points that lie along a perfect straight line. The point of practical analysis, however, is to find the scaling rule in the first place. For anisotropic biological objects (like left ventricular endocardial contours) as well as for precisely generated fractal images analysed without knowledge of the scaling rule, the data points do not generally lie on a straight line, reflecting sampling limitations as well as limited self-similarity [16], thus the slope is estimated from the regression line for the log-log plot. The choice of image preparation routine and the details of the method used to gather the data for fractal analysis are important as they can either increase or decrease the correlation coefficient of the double logarithmic plot (more linear or more sigmoid fit respectively).Table 1

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.


Related in: MedlinePlus