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Left: the level sets of the solution to the Poisson equation for the silhouette of an elephant, right: the level sets of the distance transform for the same silhouette.
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We present a novel approach that allows us to reliably compute many useful properties of a silhouette. Our approach assigns, for every internal point of the silhouette, a value reflecting the mean time required for a random walk beginning at the point to hit the boundaries. This function can be computed by solving Poisson's equation, with the silho...
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... ( x, y ) S , subject to Dirichlet boundary conditions U ( x, y ) = 0 at the bounding contour ∂S . Fig. 2 shows the solution to the Poisson equation obtained for the silhouettes in Fig. 1. High values of U are attained in the central part of the shape, whereas the external protrusions (the limbs, head, and tail) disappear at relatively low values of U . The level sets of U represent smoother versions of the bounding contour. This is different from the distance transform, which smoothes the shape near concavities while introducing discontinuities near convex sections of the contour (see Fig. 3). Also unlike the distance transform in which every value is determined by a single contour point (the nearest), the values assigned by the Poisson equation take into account many points on the boundaries and so they reflect more global properties of the silhouette. Below we exploit these properties of the Poisson solution to characterize a silhouette using measures constructed with derivatives of the solution. In addition we will explain (in Section V) that the solution to the Poisson equation can be calculated much faster than the distance transform. Poisson’s equation arises in gravitation and electrostatics. Here we only mention a few of its relevant properties. The solution to the Poisson equation exists and is unique for any closed region with boundary conditions given by any integrable function. Uniqueness is shown by noticing that the solution to the related homogeneous equation ∆ U = 0 (called the Laplace equation ) with zero boundary conditions is identically zero. More generally, the values of U along any closed curve within S determine the values of U inside the region bounded by this curve, but they are insufficient to uniquely determine the values of U outside the curve. For silhouettes described by conics the Poisson equation takes a particularly simple form. Consider a silhouette composed of the points ( x, y ) ...
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... Background. Many prior works have observed the central relationship between silhouettes and 3D shape [Ikeuchi and Horn 1981;Cipolla et al. 1998;DeCarlo et al. 2003], which allows geometric quantities, like depth, to be modeled in image space as the solution to a Poisson equation, with boundary conditions imposed along silhouettes [Ikeuchi and Horn 1981;Gorelick et al. 2006]. Inspired by this relationship, Baran and Lehtinen [2009] propose an artistic curve inflation model where height is modeled as a solution to the BVP Equation (3) with = 0, f = 4, and g = 0. ...
We introduce a Monte Carlo method for evaluating derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions). Derivatives can be evaluated at arbitrary points without performing a global solve, or constructing a volumetric grid or mesh. The method is hence well-suited to inverse problems with complex geometry, such as PDE-constrained shape optimization. Like other walk on spheres (WoS) algorithms, our method is trivial to parallelize, and is agnostic to boundary representation (meshes, splines, implicit surfaces etc.), supporting large topological changes. We focus in particular on screened Poisson equations, which model diverse problems from scientific and geometric computing. As in differentiable rendering, we jointly estimate derivatives with respect to all parameters -- hence, cost does not grow significantly with parameter count. In practice, even noisy derivative estimates exhibit fast, stable convergence for stochastic gradient-based optimization -- as we show via examples from thermal design, shape from diffusion, and computer graphics.
... The proposed distances include the Hausdorff and Frechet distances [7]. Other techniques are based on the Poisson equation [8], integral invariants [9], and an elastic shape distance on the energy required to elastically deform one boundary contour to the other [10,11]. ...
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... Although stochastic fluctuation is a classical topic, it is still one of the popular subjects to be applied into various areas even in the present years [11,12]. Random walks are also applied into various areas, including decision support systems [10,[13][14][15][16], computer visions [17][18][19][20][21], social network analysis [22] and knowledge discovery [23]. The Brownian motion process is a Wiener stochastic process which is the random motion of a particle suspended in a medium [24][25][26]. ...
This article analyzes the behavior of a Brownian fluctuation process under a mixed strategic game setup. A variant of a compound Brownian motion has been newly proposed, which is called the Shifted Brownian Fluctuation Process to predict the turning points of a stochastic process. This compound process evolves until it reaches one step prior to the turning point. The Shifted Brownian Fluctuation Game has been constructed based on this new process to find the optimal moment of actions. Analytically tractable results are obtained by using the fluctuation theory and the mixed strategy game theory. The joint functional of the Shifted Brownian Fluctuation Process is targeted for transformation of the first passage time and its index. These results enable us to predict the moment of a turning point and the moment of actions to obtain the optimal payoffs of a game. This research adapts the theoretical framework to implement an autonomous trader for value assets including stocks and cybercurrencies.
... To generate spatial mapping of the lung masks, we first use the concept of Poisson distance map (PDM), introduced in [23], to encode the shape of individual lung masks V . PDM is commonly used for characterizing the silhouette of an object via continuous labeling of voxel positions with scalar field values U 3d in the range of [0, 1]. ...
... The solution for U proposed in [23] is guaranteed to be smooth according to [24]. It has the advantage of generating distance values that are sensitive to global shape characteristics, unlike other distance metrics (e.g. ...
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... Some other characteristics of the PGVF merit further study. For example, in [13] the solution of the Poisson PDE u = −1 with Dirichlet initial condition was used for object decomposition to parts. In this paper, special functions were designed to generate GVFs whose SPs are located at different body parts. ...
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... The distance transform acts as a generator function for the medial axis, such that points x ∈ MA if and only if they satisfy some constraint involving their distance to the boundary. However, other authors have proposed alternative generator functions in their pruning strategies [21,4]. ...
... For example, [21] and [4] introduce what they call Poisson skeletons by approximating D Ω (x) as the solution of the Poisson equation with constant source function. ...
We present the CPMA, a new method for medial axis pruning with noise robustness and equivariance to isometric transformations. Our method leverages the discrete cosine transform to create smooth versions of a shape $\Omega$. We use the smooth shapes to compute a score function $\scorefunction$ that filters out spurious branches from the medial axis. We extensively compare the CPMA with state-of-the-art pruning methods and highlight our method's noise robustness and isometric equivariance. We found that our pruning approach achieves competitive results and yields stable medial axes even in scenarios with significant contour perturbations.
... Hence, random walks are also effective in link prediction and recommendation system [20], [21], [22], [23], [24], [25], [26], [27]. Random walks can also be applied in computer vision [7], [28], [29], [30], [31], [32], [33], [34], [35], [36], semi-supervised learning [37], [38], [39], [40], [41], network embedding [42], [43], and complex social network analysis [44]. There are also some literature illustrating the applications of random walks on graphs [45], [46], text analysis [47], science of science [48], and knowledge discovery [49]. ...
... Gorelick et al. [29] characterize the shape of a picture using random walks. For each internal pixel, they calculate the value reflecting the mean time required for a random walker beginning at the pixel to reach the boundary. ...
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