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Seeded segmentation methods have gained a lot of attention due to their good performance in fragmenting complex images, easy usability and synergism with graph-based representations. These methods usually rely on sophisticated computational tools whose performance strongly depends on how good the training data reflect a sought image pattern. Moreover, poor adherence to the image contours, lack of unique solution, and high computational cost are other common issues present in most seeded segmentation methods. In this work we introduce Laplacian Coordinates, a quadratic energy minimization framework that tackles the issues above in an effective and mathematically sound manner. The proposed formulation builds upon graph Laplacian operators, quadratic energy functions, and fast minimization schemes to produce highly accurate segmentations. Moreover, the presented energy functions are not prone to local minima, i.e., the solution is guaranteed to be globally optimal, a trait not present in most image segmentation methods. Another key property is that the minimization procedure leads to a constrained sparse linear system of equations, enabling the segmentation of high-resolution images at interactive rates. The effectiveness of Laplacian Coordinates is attested by a comprehensive set of comparisons involving nine state-of-the-art methods and several benchmarks extensively used in the image segmentation literature. 

Most 3D laser scanners are based on 3D optical triangulation algorithms, where the location of each 3D point is estimated as the intersection of a camera ray and a plane of light projected by a laser line generator. Since a physical laser line generator projects a sheet of light of finite thickness, inaccurate measurement and errors result from assuming that the plane of light is infinitesimally thin. We propose a new mathematical formulation for 3D optical triangulation based on interval arithmetic, where 3D points are only determined within certain bounds along the camera rays, and multiple measurements are used to tighten these bounds. We propose the Line Segment Cloud as an alternative surface representation to visualize the measurement errors within the proposed framework. We introduce the Iterative Line Segment Tightening algorithm to convert line segment clouds to point clouds, as a preprocessing step prior to surface reconstruction. We describe how to construct a low cost laser line 3D scanner, where the camera is fixed with respect to the object and the laser line generator is mounted on a high resolution motion platform. We describe a GPU-based implementation where the large number of captured images are processed in real time. Finally, we present some experimental results. 

3D surface reconstruction usually begins with a point cloud and aims to build a representation of the object producing that point cloud. There are several algorithms to solve this problem, each with different priors over the point cloud, such as the type of object represented, or the method by which it was obtained. In this work, we focus on an algorithm called Non-Convex Hull (NCH), which reconstructs surfaces through a concept similar to the Medial Axis Transform. A new algorithm called Shrinking Planes is proposed to compute the NCH, based on the Shrinking Ball method with a few improvements. We prove that the new method can approximate surfaces to arbitrarily small error, and evaluate its performance on the surface reconstruction task. The new method maintains the same reconstruction quality as the Naïve Non-Convex Hull method, while achieving a large performance improvement. 

Carving is a subtractive process where we get the shape by removing materials. While most people can get roughly the right intended shape, it is usually challenging not to over-cut the model. We propose a method that helps an unskilled user to carve a rough physical replica of a 3D model using the minimum number of cuts while only using manual cutting tools. The method starts by analyzing the input 3D model and generates the minimum set of cutting steps that remove most of the material. Then using a projector, we project the instructions sequentially onto a block of material to guide the user in performing them. We use the projector-camera setup to 3D scan the object after cutting and automatically detect the changes to reflect them on the digital model. We demonstrate a complete system to support this operation and show several examples of manually carved 3D models while using the system. 

We propose a method that helps an unskilled user to carve a physical replica of a 3D CAD model while only using manual cutting tools. The method starts by analyzing the input CAD model and generates a set of carving instructions. Then using a projector, we project the instructions sequentially one at a time to a block of material to guide the user in performing each of them. After each cutting step, we use the projector-camera setup to 3D scan the object after cutting. And automatically align the scanned point cloud to the CAD model, to prepare the position for the next instruction. We demonstrate a complete system to support this operation and show several examples manually carved while using the system.