Geometric optimization and neural models for advanced computer graphics applications

Recent developments in machine learning have provided additional opportunities for data-driven solutions to complex geometry problems that were previously intractable. Nonetheless, the classical optimization techniques are still crucial, particularly where data is scarce. And combining neural methods with classical optimization promises more efficient solutions to broader geometry processing challenges. In this thesis, we explore novel geometric optimization algorithms and neural-based solutions to address critical challenges in computer graphics, specifically in digital fabrication, shape representation, image processing, and artistic rendering. First, we design a new, robust, end-to-end computational milling pipeline that efficiently enables production of complex 3D shapes from commercially available material slabs using 3-axis CNC milling. It involves a novel decomposition algorithm that partitions objects into double height-field (DHF) slices adhering to fabrication constraints. This algorithm significantly reduces milling time and material waste by minimizing slice count while maximizing slice heights. Based on the strengths of DHF, we propose a novel hybrid neural implicit 3D shape representation — Compact Neural Double Height-Fields (CN-DHFs). It approximates a target surface via intersections of DHFs while a Multi-Layer Perceptron (MLP) encodes each DHF. This representation delivers high-quality reconstructions, reducing reconstruction error compared to prior art. Next, we present a hybrid algorithm that combines neural networks with geometric optimization to deblur low resolution anti-aliased raster clip-art images, aligning with user expectations. The approach first generates low-blur approximations through a learned network, then refines them into blur-free images consistent with human expectations using discrete partitioning. Our outputs feature a compact color palette comparable to the ground truth and are preferred by users over prior art. Finally, we introduce the first dedicated method to model and learn the perspective projection humans employ when creating line drawings. We algorithmically replicate the perspective deviations by learning a spatially continuous local perspective deviation function in a one-shot setup due to data paucity and heterogeneity of human drawing choices. The resulting learned perspective functions faithfully capture input sketch artist perspectives, remain consistent across views, and generalize to new similar shapes. Collectively, these contributions harness geometric optimization and machine learning to enhance precision, efficiency, and perceptual authenticity in computer graphics.