Fig. 2

a The first 3 iterations of the Koch coastline, an exact geometrical fractal. It can be quantified by its perimeter, its AUC or its FD. With each successive iteration of the Koch coastline the original pattern is repeated at a finer level, corresponding to how with increasingly greater magnification increasingly fine detail is revealed in fractals. By traditional methods, the AUC will converge on \( \frac{8}{5} \) and the perimeter of the curve after n iterations will be \( \left(\frac{4}{3}\right)n \) times the original perimeter (4 times more lines, \( \frac{1}{3} \) greater length per iteration), and since \( \left|\frac{4}{3}\right|>1, \) perimeter will tend to infinity. These exemplify the inherent problem with traditional mathematics: it is capable of providing only scale-dependent descriptors that give limited insight into the motif’s overarching complexity. The FD of the Koch curve, on the other hand, summarises its complexity independently of scale. At every iteration (from 1 to infinity) the FD is invariant at \( \frac{ \log 4}{ \log 3}\approx 1.26186 \). Biological quasi-fractals are measured by ‘sampling’ them with an imaging ‘camera’ relevant to a particular imaging modality. Different cameras have different resolutions, but in all cases increasing resolution is similar to accumulating iterations on a mathematical fractal. Natural quasi-fractals are self-similar across a finite number of scales only—a lower limit of representation is imposed by the limit of the screen (pixel resolution). For CMR cines, blurring (quite extreme in b) has the same effect as setting a lower resolution for the particular sequence, and this is equivalent to having fewer fractal iterations. With such manipulation, it can be seen that the area of the set changes little (here by 2 %), the perimeter a lot (by 43 %) and the FD less (by 8 %). This implies that high image resolution (and a fractal approach) may not add much value when attempting to measure the left ventricular volume; but image resolution (and a fractal approach) will make a considerable difference when intricate features like trabeculae are the features of interest: the perimeter length or other 1D approach will be less robust than the FD. AUC = area under the curve; d = length of segment; 1D = one-dimension/al; FD = fractal dimension; px = pixels. Other abbreviation as in Fig. 1