favus and Triceratium pentacrinus fo. quadrata had higher forming valve symmetry than external valve symmetry. Arachnoidiscus ehrenbergii had approximately equal forming and external symmetry (Figure 2.14).
Arachnoidiscus and Aulacodiscus with symmetric valve surfaces were compared to Asteromphalus with asymmetric valve surfaces using least-squares regression (Figure 2.15). Best-fit curves for species, x, were wtaxon = 1.1878x + 27.27, wtaxon = 1.276x + 30.67, and wtaxon = 2.2925x + 23.238 for Asteromphalus, Arachnoidiscus, and Aulacodiscus, respectively. Coefficients of determination were 0.7622 for Asteromphalus and 0.9645 for Arachnoidiscus, and Aulacodiscus.
A number of analyses were conducted to illustrate symmetry differences in different valves of Cyclotella meneghiniana. Normal and abnormal Cyclotella meneghiniana valve symmetries were compared using least-squares regression (Figure 2.16). Slope for normal valves was 1.345 compared to a slope of 1.008 for abnormal valves. Coefficients of determination were 0.9737 and 0.9284 for normal and abnormal valves, respectively. Masked images of Cyclotella meneghiniana initial valves were measured for symmetry and compared to normal vegetative valve symmetry. Average initial valve symmetry had much higher symmetry than normal external and forming vegetative valves (Figure 2.17). Different valve surface morphologies revealed differences in symmetry that were discernable via analysis using entropy.
Figure 2.6 ProvBay5_12lx450, Arachnoidiscus ehrenbergii. (a) Original scanning electron micrograph image; (b) Entropy filtering applied to image; (c) Vectors showing gradient changes in magnitude and direction; (d) Tilt-corrected image. Original SEM by Mary Ann Tiffany.
Figure 2.7 Tilt-corrected final entropy vs. number of rotations for all images, omitting 64 rotations used for Cyclotella meneghiniana.
Measurement error for image matching rotation and overlaying was not a factor in symmetry analysis. Values ranged from 0.0000946 to 0.0184 measurement error with 0.0172 to 1.994% bias. The percentages fall within the range of values that do not affect the measurement of true variation [2.103]. For symmetry analysis, the entropy values measured are not biased by measurement error.
An image of Biddulphia sp. was used to illustrate measurement of dihedral symmetry. Entropy and symmetry values were 1.94041 and 6.96162 with a measurement error of 0.0000614 and a 0.00663% bias. Biddulphia sp. dihedral symmetry measurement was accomplished using rotation and flipping the masked image about the x and y axes in the x-y plane.
Valve formation was simulated in a simple model by sequentially adding concentric rings of valve structure from an annulus to the valve margin to cumulatively create a valve face surface. Although diatom valve formation is much more complicated than this, the simulation provided a glimpse into an intuitive way to view growth over time. Ten circular-shaped centric diatom images were chosen to illustrate the method for Actinoptychus senarius, Actinoptychus splendens, Arachnoidiscus ehrenbergii, Arachnoidiscus ornatus, Asterolampra marylandica, and Aulacodiscus oregonus. Twenty-four concentric rings were arbitrarily selected on each image. Each ring was of an arbitrary width and added cumulatively, starting with an annulus at the center and ending with a ring at the valve margin (Figure 2.18).
Figure 2.8 Typical entropy vs. symmetry plot. The example is image ProvBay5_12lx450, Arachnoidiscus ehrenbergii. The first datum is the original masked tilt-corrected image.
Figure 2.9 Number of rotations vs. tilt-corrected final entropies for Cyclotella meneghiniana external valves using 3 to 64 rotations.
Figure 2.10 Histogram and overlain probability density function for all tilt-corrected final entropies resulting in a Gaussian distribution.
Figure 2.11 Histogram and overlain cumulative density function for all tilt-corrected final entropies.
Figure 2.12 For all images, average symmetry per taxon ranging from Triceratium crenulatum with the lowest symmetry to Actinoptychus senarius with the highest symmetry.
Valve formation simulation for the eight taxa was plotted against symmetry, and the resultant curves were exponential depicting the change in symmetry from annulus to valve margin (Figure 2.19). The plot includes a second Arachnoidiscus ehrenbergii external valve and a forming valve of this taxon. For valve formation simulation, the distribution of average entropy values was depicted in a histogram and overlain PDF (Figure 2.20) as a gamma distribution and is a maximum entropy distribution. A histogram and overlain CDF of average entropy values were also constructed (Figure 2.21). Standardizing the PDF produced a skewed-right normal prior probability distribution (Figure 2.22), establishing the relation between available information and its probability of occurrence. The prior distribution is also a maximum entropy distribution as a minimization of prior information given via the PDF.
Figure 2.13 Average taxon symmetry for all external centric diatom valves ranging from Triceratium crenulatum with the lowest symmetry to Glyphodiscus stellatus with the highest symmetry.
Figure 2.14 Vertical valve formation comparison: average external and forming valve symmetry for ten centric diatoms.
Figure 2.15 Symmetry values for Asteromphalus, Arachnoidiscus, and Aulacodiscus external valves.
2.3.2 Valve Formation—Stability and Instability Analyses
Stability analysis was performed using symmetries calculated from the valve formation simulation of cumulative concentric rings from annulus to valve margin for eight taxa. Lyapunov exponents were calculated to determine the behavior of symmetry changes. Initial conditions for valve formation are vValve formation (0) = vannulus. Each valve formation simulation produced a series of Lyapunov exponents comprising a Lyapunov