Fig 1.
Illustration of fractal self-similarity.
(a) A Sierpinski triangle is an example of a pure fractal. A small portion of the triangle looks exactly like the whole triangle. (b) Self-similarity holds across a limited range of spatial scales for a natural object such as this Romanesco Broccoli (Photos courtesy of Live Earth Farm).
Fig 2.
Computing fractal dimension using 3D information measure.
Information dimension, D1, measure. In the scatterplot of log(1/r) versus log(I(r)), r is box size and I(r) is the information theoretic entropy for the box size r. Note. For information measure, the initial, pre-determined range of box sizes is r = 2… 30 voxels (in increments of 1 voxel). Data shown are for left hippocampus from one healthy control participant. Linear regression analysis is performed iteratively. Blue line indicates the linear fit over the entire range of r. Red dotted line indicates the final fit (R2); the slope of this line corresponds to the fractal dimension, D1. Breakpoint separates non-linear data points from the data used in the final regression analysis. ln denotes natural log. Min r is the new smallest box size and Max r is the new largest box size.
Fig 3.
Simulated phantoms used in algorithm validation, with theoretical fractal dimensions ranging between 1 and 3.
(a) Circle: radius = 8, image size: 120 x 120, line width is 1 pixel (theoretical FD = 1). (b) Fourth-iteration Koch, image size: 283 x 84, line width is 1 pixel (theoretical FD = 1.2619). (c) 3D random Cantor set with p = 0.7, image size: 128 x 128 x 128, Voxels set to 1 (theoretical FD = 2.485).
Table 1.
Fractal dimension values and box size range of phantoms.
Fig 4.
Differences in fractal dimension for left hippocampus, right hippocampus and left thalamus between schizophrenia patients and healthy controls.
Each data point represents D1 information dimension value for each participant for (a) Left hippocampus, (b) Right hippocampus, and (c) Left thalamus. The black dash-dot line and the magenta dash-dash line denote median fractal dimension values, for schizophrenia patients and healthy control groups, respectively. Significantly lower FD values were found for schizophrenia patients relative to healthy controls (Mann-Whitney U test, p< 0.05; FDR correction). Note. SCZ: patients with schizophrenia; HC: healthy controls.
Table 2.
Fractal dimension values for subcortical structures.
Fig 5.
Illustration of subtle surface non-linearities in schizophrenia as captured by the FD measure, using individual participants’ data for left and right hippocampus.
(a) Representative healthy control, and (b) Patient with schizophrenia. Left panel shows left hippocampus, right panel shows right hippocampus. In (b), shadow highlight indicates data points used in the final fit. Subtle deviations from linearity are seen in (b), which shows data points whose I(r) counts deviate relative to the line of best fit. In the left panel, an example of this can be observed at a point with x and y coordinates [-1.946, 2.652] (fourth from the bottom of shaded area) and at [-1.386, 3.608] (third from the top), and in the right panel, at [-2.079, 2.445] (third from the bottom) (please see main text for detailed explanation). Insets show reconstructions of left and right hippocampi for these participants; 1 cube represents 1 voxel (1.5 x 1.5 x 1.5 mm).