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The Hilbert transform is useful for image processing because it can select which edges of an input image are enhanced and to what degree the edge enhancement occurs. However, the transform operation is one dimensional and is not applicable for arbitrarily shaped two-dimensional objects. We introduce a radially symmetric Hilbert transform that permits two-dimensional edge enhancement. We implement one-dimensional, two-dimensional, and radial Hilbert transforms with a programmable phase-only liquid-crystal spatial light modulator. Experimental results are presented.

We show how to two dimensionally encode the polarization state of an incident light beam using a parallel-aligned liquid-crystal spatial light modulator (LCSLM). Each pixel of the LCSLM acts as a voltage-controlled wave plate and can be programmed over a 2pi phase range at a wavelength of 514.5 nm. Techniques are reviewed for either rotating the major axis of elliptically polarized light or for converting an input linearly polarized beam into an arbitrary elliptically polarized beam. Experimental results are demonstrated in which we generate various two-dimensional spatial patterns of polarized light. Several potential applications are suggested. We also report an unexpected edge-enhancement effect that might be useful in image processing applications.

The community structure of sedentary organisms is largely controlled by the outcome of direct competition for space. Understanding factors defining competitive outcomes among neighbors is thus critical for predicting large-scale changes, such as transitions to alternate states within coral reefs. Using a spatially explicit model, we explored the importance of variation in two spatial properties in benthic dynamics on coral reefs: (1) patterns of herbivory are spatially distinct between fishes and sea urchins and (2) there is wide variation in the areal extent into which different coral species can expand. We reveal that the size-specific, competitive asymmetry of corals versus fleshy algae highlights the significance of spatial patterning of herbivory and of coral growth. Spatial dynamics that alter the demographic importance of coral recruitment and maturation have profound effects on the emergent structure of the reef benthic community. Spatially constrained herbivory (as by sea urchins) is more effective than spatially unconstrained herbivory (as by many fish) at opening space for the time needed for corals to settle and to recruit to the adult population. Further, spatially unconstrained coral growth (as by many branching coral species) reduces the number of recruitment events needed to fill a habitat with coral relative to more spatially constrained growth (as by many massive species). Our model predicts that widespread mortality of branching corals (e.g., Acropora spp) and herbivorous sea urchins (particularly Diadema antillarum) in the Caribbean has greatly reduced the potential for restoration across the region.

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