Lampriscus shadboltianum has pseudocelli [2.30], and Glyphodiscus stellatus has four-part symmetry in which the valve is concentrically undulate. Triceratium pentacrinus fo. quadrata and T. bicorne are uncommon four-part symmetrical species, as most are three-part symmetrical valves [2.31].
Pseudoloculi or loculi are variably present in Triceratium, e.g., as in pseudoloculi surrounding areolar clusters on the valve surface of Triceratium dubium [2.31]. Rimoportulae openings differentiate species, such that Triceratium dubium having an elongated tube opening with apical spinules is different from T. favus having a spatulate opening that is hemispherical internally [2.31].
Trigonium have pseudocelli, centrally located rimoportulae with oval openings, and honeycombed loculate areolae [2.143] in a radial arrangement [2.38, 2.143]. In Trigonium arcticum, areolae are covered by rota-type vela, and a straight cingular suture is present [2.38]. Toward the end of cribra formation, this structure finishes at valve surface level in Trigonium arcticum, and rotae within porelli of the pseudocelli are also being formed [2.143]. For Trigonium formosum, rimoportulae are centrally located and cribra are highly domed [2.143].
2.2.2 Centric Diatoms, Morphology, and Valve Formation
Centric diatom valve formation occurs from a central area to the valve margin in three general stages [2.126] and is illustrated in Figure 2.5. First, horizontal silica deposition occurs to form a basal layer. From an annulus in the central area at the site of valve formation initiation, radial rows of silica are deposited as areolae with cross extensions and connections so that a branching pattern of silica strands emerges. Gaps in this overall pattern are filled in with silica, and internal rimoportulae tubes commence [2.126]. Second, vertical silica deposition occurs so that areolae walls increase in height. Round pores are transformed to hexagonal or other shapes, and external rimoportulae are evident [2.126]. Third, areolar associated structures of cribra and cribella are completed during horizontal silica deposition so that the size and spacing of these structures occur as a response to the constraints imposed by the areolae [2.126]. Successive layers of silica are deposited, and the completion of the valve and its structures extends to the margin [2.116].
Figure 2.5 Three stages of valve formation (as defined in [2.125]) in Biddulphia. Row 1: First stage shows formation of the elongated annulus, branching virgae and vimines producing open holed areolae, and the internal tubes of rimoportulae just beginning. Row 2: Second stage shows the external emergence of rimoportulae, the filling in of areolae, and ridges are more hyaline with increasing silica deposition. Row 3: Third stage shows the beginning of pseudocelli formation, fusing of the valve margin, and the mature cribra. All photos by Mary Ann Tiffany.
Prior to valve formation, mitotic and cytokinetic processes occur [2.122]. Silica aggregation occurs within the silicalemma/SDV where new wall formation occurs [2.59, 2.116, 2.119, 2.126]. The general outline of the cell wall is molded via the plasmalemma [2.122].
As silica deposition occurs from center to the periphery, it is evident that 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