Analytical framework for computer-generated holography of screen space primitives with point-source integration in the Fourier domain

Computer-generated holography enables precise optical wavefront modulation and supports applications ranging from 3D displays to industrial manufacturing. However, simulating wavefronts is computationally expensive, particularly for high-pixel-count digital holograms. We present a unified analytical framework for directly computing digital holograms from fundamental geometric primitives. The framework models each primitive as a continuous distribution of infinitesimal point sources and evaluates its response by integration done in the 2D discrete Fourier domain. We derive novel closed-form expressions in their direct computational form for voxels, 3D Gaussians, quadratic splines, rings, circles, and spheres, along with new simplified expressions for triangles. Numerical reconstructions obtained from these formulations are validated against densely sampled point clouds representing the same geometries. We further quantify the numerical accuracy of the proposed method and identify the factors that govern the error. We also introduce a nearby-reference-point formulation that improves numerical accuracy and enables tiled 2D-DFT evaluation of very large holograms. Benchmarks on compound objects with polygonal and voxel primitives demonstrate improved numerical reconstruction fidelity by capturing the object geometry more faithfully than sparse point-cloud representations do, while requiring only approximately 4 times the processing time of a single point-source primitive.