The scatter plot shows individual fractal dimension (FD) values for the right cerebellar cortex.

The left panel shows FD values for typically developing (TD) children (N = 18) and the right panel FD for ASD children (N = 20). **denotes P<0.05, Bonferroni correction.</p

The scatter plot showing the association between individual FD values and PIQ>VIQ profiles for ASD boys.

Data shown are from ASD participants with higher PIQs relative to their VIQ scores (N = 17 out of 20 ASD participants have PIQ>VIQ scores). Overall FD is higher for ASD participants who have a higher, wider PIQ>VIQ spread compared to ASD participants with a lower or narrower PIQ>VIQ difference (P = 0.023). The size of the marker denotes the corresponding root mean square error (RMSE) value to the log-log line of best fit for each participant; a smaller marker indicates a lower error value to the line of best fit and a larger marker indicates a higher value. Note. PIQ: Performance Intelligence Quotient; VIQ: Verbal Intelligence Quotient. All IQ subtest scores are within the normal range, above 70.</p

Demographic characteristics of the sample.

<p>Demographic characteristics of the sample.</p

Reduced structural complexity of the right cerebellar cortex in male children with autism spectrum disorder - Fig 3

<p><b>Illustration of subtle surface non-linearities in the right cerebellar cortex of two individual participants (one TD and one ASD) as captured by the FD measure (<i>D</i></b>_<b>2</b><b>), (A)</b> TD male child (9.21 years old, UCLA 0051278 in ABIDE) and <b>(B)</b> ASD male child (10 years old, Yale 0050602 in ABIDE. The left panel shows the bilateral cerebellums in the coronal plane whereas the right panel shows a rendering of the right cerebellar cortex for each participant and its corresponding log-log plot. Note that the final slope estimate (i.e., the FD value) is higher for the TD participant, with higher <i>R</i>² and lower root mean square error (RMSE), indicating a better fit for TD individual relative to the participant with ASD.</p

Differences in fractal dimension for left hippocampus, right hippocampus and left thalamus between schizophrenia patients and healthy controls.

<p>Each data point represents <i>D</i>₁ 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 <i>U</i> test, <i>p</i>< 0.05; FDR correction). <i>Note</i>. SCZ: patients with schizophrenia; HC: healthy controls.</p

Fractal dimension values for subcortical structures.

<p>Fractal dimension values for subcortical structures.</p

Computing fractal dimension using 3D information measure.

<p>Information dimension, <i>D</i>₁, measure. In the scatterplot of log(1/<i>r</i>) versus log(<i>I</i>(<i>r</i>)), <i>r</i> is box size and <i>I</i>(<i>r</i>) is the information theoretic entropy for the box size <i>r</i>. <i>Note</i>. For information measure, the initial, pre-determined range of box sizes is <i>r</i> = 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 <i>r</i>. Red dotted line indicates the final fit (<i>R</i>²); the slope of this line corresponds to the fractal dimension, <i>D</i>₁. Breakpoint separates non-linear data points from the data used in the final regression analysis. ln denotes natural log. <i>Min r</i> is the new smallest box size and <i>Max r</i> is the new largest box size.</p

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).</p

Illustration of fractal self-similarity.

<p>(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).</p

Fractal dimension values and box size range of phantoms.

<p>Fractal dimension values and box size range of phantoms.</p