Fig. 6

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