Variational approaches to 3D reconstruction from multiple depth images

This dissertation introduces a novel variational framework for reconstructing 3D surfaces from depth data acquired by commercial sensors. The work is organized into three main contributions. First, by leveraging variational methods and differential geometry, we derive explicit expressions for occluding boundaries—a key source of non-differentiability in 3D-2D matching tasks. Unlike prior approaches that rely on curvature measures or implicit representations, our work characterizes the local structure of occluding curves directly from the surface geometry. In particular, we demonstrate that the tangent of an occluding curve can be extracted from the surface’s second-order structure along the viewing direction, decomposing naturally into components corresponding to geodesic torsion and normal curvature. This result not only clarifies the role of occluding boundaries but also integrates seamlessly into our implicit reconstruction framework. The second contribution addresses the challenges of reconstructing surfaces from raw, noisy, and incomplete depth maps—a common limitation of commercial depth cameras. We propose a variational framework that enforces global data fidelity while incorporating flexible regularization strategies. A key innovation is the separation of foreground and background modeling, which minimizes the influence of erroneous background data. To further mitigate issues from missing measurements, we introduce a novel inpainting technique that works in concert with an area-penalty regularizer. In addition, a new initialization scheme for the implicit function is presented, accelerating convergence and enhancing reconstruction accuracy, particularly for smooth surfaces. Finally, we exploit the ideas from shape analysis, and develop a new regularizer for neural signed distance function (SDF) reconstruction. By directly addressing the instability introduced by the prevalent Eikonal loss, our regularizer not only improves convergence but also preserves finer geometric details compared to existing methods. Rigorous analysis and extensive experiments on public benchmarks validate its superior performance. In summary, this dissertation demonstrates that integrating variational methods and shape analysis can overcome significant data-quality challenges in 3D reconstruction, while also offering new insights that enrich learning-based approaches.

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