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3D scene modeling has long been a fundamental problem in computer graphics and computer vision. With the popularity of consumer-level RGB-D cameras, there is a growing interest in digitizing real-world indoor 3D scenes. However, modeling indoor
3D scenes remains a challenging problem because of the complex structure of interior objects and poor quality of RGB-D data acquired by consumer-level sensors. Various methods have been proposed to tackle these challenges. In this survey, we provide an overview of recent advances in indoor scene modeling techniques, as well as public datasets and code libraries which can facilitate experiments and evaluation.
An order 3 magic hexagon resembles the shape of a 19-cell honeycomb, arranged in a 3 4 5 4 3 manner. The requirement is to fill the numbers 1-19 in the grids so that each row (15 in total) adds up to 38.
Previously invented methods aimed at solving this problem and proving its uniqueness were either not rigorous enough or too intricate. So by analyzing its properties, I wanted to find a combinatorial solution to its construction, prove its uniqueness, and investigate whether its mathematical principles can be used in real-world applications. The difficulty depends on the viewpoint, so the first step was to label each grid in a convenient way. I chose to look at the magic hexagon as a network composed of a center and rings. Then the connections and restrictions of each number set could be found by formula
derivation. In a similar fashion, symmetrical properties were also found. The next step was to analyze possible distributions of odd and even numbers. Out of the 9 configurations, only 1 proved to be usable. The final step was construction. With all the properties known, the few impossibilities were easily eliminated, and only one solution remained, thus proving its uniqueness.
The procedures used on the order 3 magic hexagon may be extended to those of higher orders, providing more ease in their construction. The unique properties of magic hexagons may be used in some fields of application, such as in password systems, large-scale roof structure, composite material, national security systems and many other fields.
We introduce an interactive technique for manipulating simple 3D shapes based on extracting them from a single photograph. Such extraction requires understanding of the components of the shape, their projections, and relations. These simple cognitive tasks for humans are particularly difficult for automatic algorithms. Thus, our approach combines the cognitive abilities of humans with the computational accuracy of the machine to solve this problem. Our technique provides the user the means to quickly create editable 3D parts¡ª human assistance implicitly segments a complex object into its components, and positions them in space. In our interface, three strokes are used to generate a 3D component that snaps to the shape¡¯s outline in the photograph, where each stroke defines one dimension of the component. The computer reshapes the component to fit the image of the object in the photograph as well as to satisfy various inferred geometric constraints imposed by its global 3D structure. We show that with this intelligent interactive modeling tool, the daunting task of object extraction is made simple. Once the 3D object has been extracted, it can be quickly edited and placed back into photos or 3D scenes, permitting object-driven photo editing tasks which are impossible to perform in image-space.
It is well known that the nonlinear filter has important applications in military, engineering and commercial industries. In this paper, we propose efficient and accurate numerical algorithms for the realization of the Yau-Yau method for solving nonlinear filtering problems by using finite difference schemes. The Yau-Yau method reduces the nonlinear filtering problem to the initial-value problem of Kolmogorov equations. We first solve this problem by the implicit Euler method, which is stable in most cases, but costly. Then, we propose a quasi-implicit Euler method which is feasible for acceleration by fast Fourier transformations. Furthermore, we propose a superposition technique which enables us to deal with the nonlinear filtering problem in an off-time process and thus, save a large amount of computational cost. Next, we prove that the numerical solutions of Kolmogorov equations by our schemes are always nonnegative in each iteration. Consequently, our iterative process preserves the probability density functions. In addition, we prove convergence of our schemes under some mild conditions. Numerical results show that the proposed algorithms are efficient and promising.