2.1). A rotations test was used to determine if an optimal number of rotations was associated with the lowest possible entropy value for those circular-shaped taxa without distinct identifiable equally spaced surface features. For this test, Cyclotella meneghiniana was used.
Entropy is calculated as a cumulative quantity based on successive overlaying of rotated images while matching edges for each original image. After all rotations are completed, the final entropy value is the minimum amount of entropy left after successive subtractions of one image from the previous stack of rotated and overlain images. For entropy measurement, an equivalence between rotational and reflective symmetry also can be made. An equivalence between rotational and reflective symmetries was established as follows: two rotations represent one plane of reflective symmetry; three rotations represent two planes of reflective symmetry; and four rotations represent three planes of reflective symmetry. Rectangular taxa were rotated four times to represent four planes of reflective symmetry, and pentagonal taxa were rotated five times to represent five planes of reflective symmetry.
2.2.5 Image Tilt and Slant Measurement Correction for Entropy Values
Each SEM was taken at 0° stage angle. However, the diatom itself may be tilted or slanted in the stub mountant that may be present between the diatom and stub or sputter-coated medium that covers or infiltrates the diatom or other regions of the stub. Because of this, digital images need to be corrected for tilt and slant. Tilt and slant are measured via the gradient of the pixel gray-levels in each image. Tilt is the amount of change of the normal to a 0° plane or direction of slant [2.135], while slant is the angle of rotation different from the 0° plane [2.20]. The gradient measures the greatest amount of change in x- and y-directions over the entire image. Often, image changes are most evident at the boundaries or edges between objects or as changes in luminosity or gray scale value. This feature and the image pixel gray scale values contain information so that tilt and slant may be expressed as entropy values. The information contained in the digital image is measurable as the gradient entropy.
For an image with a range of pixel values, r,
The angular difference between tilt and normal to the x-y plane can be corrected by a shear transformation [2.140] with respect to the y-z plane. That is, pixel position in the original image is transformed via a tangent function of tilt to new pixel positions. After masking and prior to rotation, each image is corrected for tilt in this way. Angular rotation out of the x-y plane is used to correct slant by using n-rotations. That is, image rotation in the image preparation process is used to correct for slant.
2.2.6 Symmetry Analysis
The number of rotations to use per image was established using least-squares regression analysis for images without regularly spaced valve features. For each masked image, symmetry state was calculated from and plotted against tilt-corrected final entropy. Average taxon symmetry was measured for all images. Average taxon external valve symmetry was also measured. Comparison was made between symmetric Arachnoidiscus and Aulacodiscus valves and asymmetric Asteromphalus valves. Cyclotella meneghiniana was used in comparisons between normal and abnormal valves as well as comparison of initial to vegetative valves. For symmetric/asymmetric and normal/abnormal comparisons, least-squares regression analysis was accomplished. For initial and vegetative valves, a bar graph was used to depict symmetry differences.
Tests were conducted on symmetry changes with respect to valve formation. To model vertical change in morphology and symmetry, external and forming valves were compared. Entropy was measured for external and forming valves of Arachnoidiscus ehrenbergii, Asterolampra marylandica, Asteromphalus heptactis, Asteromphalus imbricatus, Asteromphalus vanheurckii, Cyclotella meneghiniana, Glyphodiscus stellatus, and Triceratium favus. Comparison among taxa was accomplished with a bar graph of external and forming valve symmetries.
For horizontal change in morphology and symmetry, 24 equally spaced concentric circles divided the valve face in a manner presumed to be similar to the time course of the first stage of valve growth [2.126]. Entropy was measured at each accumulative radius, and the resultant values were analyzed for symmetry of eight different circular-shaped taxon images: Actinoptychus senarius, Actinoptychus splendens, Arachnoidiscus ehrenbergii, Arachnoidiscus ornatus, Asteroplampra marylandica, Aulacodiscus oregonus, Coscinodiscus sp., and Cyclotella meneghiniana.
Entropy and symmetry were calculated for an elongated centric diatom as well. The image is masked then flipped along a bisecting horizontal axis, and then flipped along a bisecting vertical axis, illustrating the procedure for measurement of dihedral symmetry. The method of rotation and flipping along a bisecting axis enables entropy measurement of any organism with reflective, glide, or dihedral symmetry including pennate diatoms which develop via a sternum or sternum-raphe system [2.119]. Biddulphia sp. [2.145] was used for illustration (Figure 2.5).
Measurement error of entropy values for masked image matching after rotation and overlaying was calculated to determine accuracy and validity as
For analysis of all images and valve formation simulation, entropy values were characterized by their statistical distributions. A histogram of final entropies was plotted, and a probability density function (PDF) and a cumulative density function (CDF) were determined. Maximum entropy is determined from the PDF [2.65]. For a uniform PDF, maximum entropy indicates that every symmetry state has an equal chance of occurring. For a Gaussian PDF, the least amount of information available means having the lowest probability and is maximum entropy; the highest amount of information available means the lowest entropy and the highest probability of occurring. Maximum entropy is also determined via standardization of the PDF so that a prior probability distribution results [2.65, 2.67].
2.2.7 Entropy, Symmetry, and Stability
During morphogenesis, changes in symmetry may occur. These changes may be thought of as morphogenetic states with respect to the spatial and temporal displacement of cell components and morphological shape and surface structures. These changes occur over time and comprise a dynamical system. Symmetry is important in pattern formation in linear and non-linear dynamical systems [2.130]. The behavior of symmetry changes over time may be characterized in centric diatoms by step-wise valve formation patterns as a growth of distances [2.27] from central area to valve margin and may be construed to be characterizations of a dynamical system. In this system, stability at equilibrium with respect to symmetry state behavior may be assessed in a diatom morphogenetic dynamical system.
To model such a system, valve formation simulation as accretionary growth behavior is used. Centric diatom valve formation on the valve surface commences at an annulus and proceeds roughly as a sequential addition of surface ornamentation to the valve margin. This is consistent with the diffusion limited aggregation model in which most deposition occurs in a narrow zone behind the leading silica growth (Figure 9 in [2.47]). Although the initial state of the system may change slightly, valve formation proceeds to produce the same species-specific
