various structures are created at various times in a sequence. Major structures common to all species of a given genus, particular valve structures that are important morphologically, and concavity or undulations associated with the valve face (perhaps due to buckling [2.52]) are associated with symmetries of the diatom valve. The species-specific differences in valve structures and their location and timing of formation are indicators of symmetry changes during valve formation.
Sexual reproduction and auxosporulation result in a cell wall that is similar to a diatom spore and takes a few generations to “restore” the vegetative shape [2.69]. The initial valve that is formed, by an unknown mechanism, morphogenetically determines vegetative valve shape via protoplast adhesion and contraction within the spherical auxospore [2.122, 2.156]. Heterovalvy is apparent even within the initial valve [2.125], having implications for symmetry.
2.2.3 Image Entropy and Symmetry Measurement
Symmetry measurement has been accomplished in a variety of ways. For example, rotational symmetry has been measured as a comparison of equal length five-petalled flowers to pentagons [2.33], point-spread function of optical properties within circular shapes exclusively [2.15], landmarks [2.121] or feature points [2.86] used for intraspecies shape comparison, and cylindrical helical diffraction grid patterns via trigonometric polynomials to measure helices [2.136]. All of these methods are inadequate for measuring symmetry in centric diatoms for a number of reasons. Because symmetry has been primarily applied to the understanding of developmental stability, fitness and adaptation [2.54], measurement needs to reflect the connection among all changes during development and should not be merely a matter of geometric convenience. Centric diatoms exhibit multiple symmetries (Figure 2.3) which necessarily requires a flexible measurement tool. From the spherical auxospore to the variably symmetric initial cell to the rotationally and/or reflective symmetric vegetative cell, different symmetries are expressed throughout the diatom life cycle. The sources of symmetry that contribute to pattern formation include perturbations as symmetry breaking instances to preservation of structural elements throughout development [2.40], and these structural elements may have different symmetries. None of the aforementioned methods have utility in comparing multiple symmetries concurrently or have been used in symmetry measurement for multiple genera or species at the same time. Additionally, these methods are used only in intraspecies shape comparisons without surface feature measurement in symmetry assessment.
Alternatively, a more broad-based methodology is necessary to use in symmetry measurement in centric diatoms. Valve shape and surface feature pattern variation as symmetry changes are measurable simultaneously via the amount of information available in an image. Measurement of information is not reliant on specific geometries or the spatial frequency or position of morphological features or scale. Multiple symmetries may be measured based on available information of shape and surface.
Symmetry may be defined as the micro-states of a macro-state dynamical system of an organism which undergoes morphogenetic, embryogenetic, cytogenetic, epigenetic, or developmental changes. When information is known only about the macro-state, probable micro-states are measures of the lack of this information and what is unknown about the actual micro-states [2.100]. Entropy is a measure of the lack of this information [2.66], and the behavior of symmetry changes is measurable as changes in the amount of information available about the arrangement of and distribution of all symmetry micro-states.
Image processing techniques are useful in characterizing information contained in pixels in digital images. The geometry of a diatom is related to pixel intensities in a digital image, and that geometry can be translated into information about diatom shape and surface features. Image entropy is the amount of compression of an image as it relates to the amount of contrast. Low image entropy indicates that there is low contrast with a lot of similar valued pixels in the image, i.e., black dominates the image, and the image is compressible to a small size. High image entropy indicates that there is high contrast, and the image requires a high amount of compression. Entropy is invariant to translation, rotation and scaling [2.5, 2.70].
Image entropy is
In an image, the probable information in gray-level pixels is equated with the state of the gray-level pixels associated with a given outcome. One such outcome is a symmetry state of an ensemble of gray-level pixels. For an image, information as states is calculated as a Boltzmann entropy [2.16–2.18, 2.84], S = -kB\n w, where micro-state w is a symmetry state and kB is the Boltzmann constant [2.16-2.18, 2.84]. For
Using the natural logarithm,
Rotational symmetry of an image may be dissected iteratively as angular segments representing repeating regular features in a 2D plane whether the shape of the planar figure is circular or an n-polygon [2.160]. The 3D surface contained in a 2D boundary shape has information content that plays into symmetry measurement. For each image, entropy is a measure of information about how well the rotated images are matched after being overlain, i.e., how rotationally symmetrical an image is after subtracting the rotated image from its previously state. The more uniform in grayscale endpoint, i.e., degree of blackness, the better the match, and the entropy is the lowest possible number with the symmetry at its the highest value. A black image will have zero entropy, so that uncanny symmetry is calculated to be the value closest to this perfect symmetry via
2.2.4 Image Preparation for Measurement
Each original image is masked to eliminate the background. From this, the masked image is used in entropy analysis. The number of rotations per image depends on identifiable regular occurring equally angularly spaced surface features such as rays, ocelli, rimoportulae, areolae, costae, chambers, ribbing, or slits. For example, in Asterolampra marylandica, there are large extended chambers on the surface resulting in a star-like shaped central area, and the number of these chambers dictates the number of rotations. Or, for Aulacodiscus, the number of rotations is equal to 360° divided by the number of ocelli or rimoportulae on the valve surface. If there are multiple images of the same species with a different number of the same identifiable feature, then the number of rotations is determined for each image, not each species. The number of rotations is based also on shape for non-circular valves. For example, Triceratium and Amphitetras would receive three and four rotations, respectively.
Number of rotations per image ranged from 3 to 64 as determined by the taxa used in this study (Table
