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Decomposing an image into interpretable and useful components is a fundamental problem in computer vision. In particular, decompositions involving shape are especially useful, since accurate estimated shape can often be used in downstream tasks in areas such as computer graphics, augmented reality, and robotics. While many previous approaches rely on large datasets in order to learn how to do this decomposition, they often fail to use all the information that is internal to the image itself. This dissertation explores methods for shape and appearance decomposition that advance the state of the art on three separate tasks while being internal, relying only on information found inside a single image or a small collection of images.The first method involves modeling a single image as a piecewise-smooth signal, with regions of similar color being grouped together, and separated by a sparse set of boundaries. The shape of the boundaries is modeled using a 2D field of overlapping generalized junctions, such that these boundaries separate regions of uniform appearance. The model is formulated as an optimization problem which encourages each junction to explain the local appearance of its underlying patch, and at the same time agree with its neighboring junctions. This model is shown to successfully recover the edges, corners, junctions, and uniform regions of an image, all under significant amounts of image noise. Despite not using an external dataset and having only a handful of tunable hyperparameters, the method is also shown to outperform large convolutional neural networks that are trained for the same tasks.In the second example, a single image of a textured object is decomposed into 2.5D shape (a field of surface normals), and a stochastic appearance texture process. Optimization is formulated as a three-player game, which upon convergence yields accurate shape and enables stochastically generating arbitrarily large flat texture samples. The three-player game enables accurate recovery of shape and texture appearance from a larger variety of textures compared with previous approaches. We also characterize the conditions under which this decomposition is unique, and under which conditions there may be an additional valid decompositions of the image into shape and texture.In the third and final example, a set of images of a single scene is decomposed into 3D scene shape and appearance using a light field encoded in the weights of a neural network. We show that explicitly modeling reflections and encouraging surface-like geometry significantly improves the estimated shape, and that it results in a significant boost to the accuracy of its novel view synthesis capabilities, especially for glossy objects. After convergence, this explicit parameterization also enables editing material properties.

We present a sufficient condition for recovering unique texture and viewpoints from unknown orthographic projections of a flat texture process. We show that four observations are sufficient in general, and we characterize the ambiguous cases. The results are applicable to shape from texture and texture-based structure from motion.

We introduce a bottom-up model for simultaneously finding many boundary elements in an image, including contours, corners and junctions. The model explains boundary shape in each small patch using a 'generalized M-junction' comprising M angles and a freely-moving vertex. Images are analyzed using non-convex optimization to cooperatively find M+2 junction values at every location, with spatial consistency being enforced by a novel regularizer that reduces curvature while preserving corners and junctions. The resulting 'field of junctions' is simultaneously a contour detector, corner/junction detector, and boundary-aware smoothing of regional appearance. Notably, its unified analysis of contours, corners, junctions and uniform regions allows it to succeed at high noise levels, where other methods for segmentation and boundary detection fail.

We introduce a probabilistic splat-based radiance field framework that retains the fast rasterization and test-time efficiency of 3D Gaussian Splatting (3DGS) while replacing heuristic primitive manipulation with gradient-based optimization of a volumetric probability density. Rather than relocating, splitting, or culling Gaussians via hand-tuned densification (e.g., ADC), we treat primitive locations as samples drawn from a persistent, learnable density. We instantiate this density using a novel, memory-efficient multi-scale hierarchical grid that enables end-to-end gradient-based optimization. To stabilize the optimization, we derive an unbiased gradient estimator with control variates that markedly reduces variance. By allowing probability mass to flow to where the loss demands, our framework eliminates brittle priors and naturally explores the volume, achieving state-of-the-art reconstruction quality on mip-NeRF 360 while preserving 3DGS-level rendering speed.

Radiance field methods (e.g. 3D Gaussian Splatting) have emerged as a powerful paradigm for novel view synthesis, yet their appearance modeling often relies on Spherical Harmonics (SH), which impose fundamental limitations. SH struggle with high-frequency signals, exhibit Gibbs ringing artifacts, and fail to capture specular reflections - a key component of realistic rendering. Although alternatives like spherical Gaussians offer improvements, they add significant optimization complexity. We propose Spherical Voronoi (SV) as a unified framework for appearance representation in 3D Gaussian Splatting. SV partitions the directional domain into learnable regions with smooth boundaries, providing an intuitive and stable parameterization for view-dependent effects. For diffuse appearance, SV achieves competitive results while keeping optimization simpler than existing alternatives. For reflections - where SH fail - we leverage SV as learnable reflection probes, taking reflected directions as input following principles from classical graphics. This formulation attains state-of-the-art results on synthetic and real-world datasets, demonstrating that SV offers a principled, efficient, and general solution for appearance modeling in explicit 3D representations. Project page: https://sphericalvoronoi.github.io/

We present an improved neural field architecture for solving partial differential equations (PDEs). Current physics-informed neural networks (PINNs) provide a flexible framework for solving PDEs, but they struggle to achieve highly accurate solutions and require computation that scales poorly with parameter count. Our model, which we call beignet (Bandlimited Embedding with Interpolated Grid Network), replaces the random Fourier feature embedding used by existing PINN models with a trainable multi-resolution Fourier feature pyramid. To query beignet at a continuous coordinate, we use Fourier interpolation at each level of the pyramid to return features at the input coordinate, and then decode this vector with a fully-connected neural network trunk. Our model provides multiple benefits: 1) Spatial derivatives can be computed efficiently by using the chain rule to compose derivatives of the neural network computed with automatic differentiation with derivatives of the feature grid computed spectrally by the Fast Fourier transform (FFT). 2) beignet can achieve higher accuracy in a compute-efficient manner by scaling the parameter count of this Fourier feature pyramid, instead of the less-efficient strategy of scaling the neural network architecture. 3) beignet can directly control the representation bandlimit, resulting in more stable optimization for difficult PDEs. We demonstrate that beignet finds significantly more accurate solutions on PDE benchmarks using fewer parameters than state-of-the-art PINN methods. We further evaluate beignet on the self-similar inviscid Burgers blowup problem and show that it can minimize residuals to near machine precision using Adam, an accuracy regime previously attained only by using computationally expensive higher-order optimizers.

We introduce a differentiable 3D representation that unifies the ray tracing capabilities of foam-based ray tracing with the efficiency of modern rasterization pipelines. While prior foam representations enable constant-time ray traversal through an explicit volumetric partition of space, their potentially unbounded cells hinder efficient tile-based rasterization. We address this limitation by generalizing Voronoi foams to bounded power diagrams with controllable cell extents, enabling spatially bounded primitives without requiring expensive Delaunay triangulations during training. We further introduce an oriented surface formulation that explicitly models interfaces between interior and exterior regions, and decouple geometry from appearance by embedding differentiable texture directly on these surfaces. Together, these contributions yield a representation that preserves state-of-the-art ray tracing efficiency while achieving rasterization performance competitive with current generation 3DGS, providing a practical path toward unified real-time differentiable rendering.

We present a method for relighting 3D reconstructions of large room-scale environments. Existing solutions for 3D scene relighting often require solving under-determined or ill-conditioned inverse rendering problems, and are as such unable to produce high-quality results on complex real-world scenes. Though recent progress in using generative image and video diffusion models for relighting has been promising, these techniques are either limited to 2D image and video relighting or 3D relighting of individual objects. Our approach enables controllable 3D relighting of room-scale scenes by distilling the outputs of a video-to-video relighting diffusion model into a 3D reconstruction. This side-steps the need to solve a difficult inverse rendering problem, and results in a flexible system that can relight 3D reconstructions of complex real-world scenes. We validate our approach on both synthetic and real-world datasets to show that it can faithfully render novel views of scenes under new lighting conditions.

Radiance field methods (e.g. 3D Gaussian Splatting) have emerged as a powerful paradigm for novel view synthesis, yet their appearance modeling often relies on Spherical Harmonics (SH), which impose fundamental limitations. SH struggle with high-frequency signals, exhibit Gibbs ringing artifacts, and fail to capture specular reflections - a key component of realistic rendering. Although alternatives like spherical Gaussians offer improvements, they add significant optimization complexity. We propose Spherical Voronoi (SV) as a unified framework for appearance representation in 3D Gaussian Splatting. SV partitions the directional domain into learnable regions with smooth boundaries, providing an intuitive and stable parameterization for view-dependent effects. For diffuse appearance, SV achieves competitive results while keeping optimization simpler than existing alternatives. For reflections - where SH fail - we leverage SV as learnable reflection probes, taking reflected directions as input following principles from classical graphics. This formulation attains state-of-the-art results on synthetic and real-world datasets, demonstrating that SV offers a principled, efficient, and general solution for appearance modeling in explicit 3D representations.

We present Neural Microfacet Fields, a method for recovering materials, geometry, and environment illumination from images of a scene. Our method uses a microfacet reflectance model within a volumetric setting by treating each sample along the ray as a (potentially non-opaque) surface. Using surface-based Monte Carlo rendering in a volumetric setting enables our method to perform inverse rendering efficiently by combining decades of research in surface-based light transport with recent advances in volume rendering for view synthesis. Our approach outperforms prior work in inverse rendering, capturing high fidelity geometry and high frequency illumination details; its novel view synthesis results are on par with state-of-the-art methods that do not recover illumination or materials.