Learning and recovering 3D surface deformations

Recovering the 3D deformations of a non-rigid surface from a single viewpoint has applications in many domains such as sports, entertainment, and medical imaging. Unfortunately, without any knowledge of the possible deformations that the object of interest can undergo, it is severely under-constrained, and extremely different shapes can have very similar appearances when reprojected onto an image plane. In this thesis, we first exhibit the ambiguities of the reconstruction problem when relying on correspondences between a reference image for which we know the shape and an input image. We then propose several approaches to overcoming these ambiguities. The core idea is that some a priori knowledge about how a surface can deform must be introduced to solve them. We therefore present different ways to formulate that knowledge that range from very generic constraints to models specifically designed for a particular object or material. First, we propose generally applicable constraints formulated as motion models. Such models simply link the deformations of the surface from one image to the next in a video sequence. The obvious advantage is that they can be used independently of the physical properties of the object of interest. However, to be effective, they require the presence of texture over the whole surface, and, additionally, do not prevent error accumulation from frame to frame. To overcome these weaknesses, we propose to introduce statistical learning techniques that let us build a model from a large set of training examples, that is, in our case, known 3D deformations. The resulting model then essentially performs linear or non-linear interpolation between the training examples. Following this approach, we first propose a linear global representation that models the behavior of the whole surface. As is the case with all statistical learning techniques, the applicability of this representation is limited by the fact that acquiring training data is far from trivial. A large surface can undergo many subtle deformations, and thus a large amount of training data must be available to build an accurate model. We therefore propose an automatic way of generating such training examples in the case of inextensible surfaces. Furthermore, we show that the resulting linear global models can be incorporated into a closed-form solution to the shape recovery problem. This lets us not only track deformations from frame to frame, but also reconstruct surfaces from individual images. The major drawback of global representations is that they can only model the behavior of a specific surface, which forces us to re-train a new model for every new shape, even though it is made of a material observed before. To overcome this issue, and simultaneously reduce the amount of required training data, we propose local deformation models. Such models describe the behavior of small portions of a surface, and can be combined to form arbitrary global shapes. For this purpose, we study both linear and non-linear statistical learning methods, and show that, whereas the latter are better suited for traking deformations from frame to frame, the former can also be used for reconstruction from a single image.

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