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An Isotropic 3 3 Image Gradient Operator

Presentation at Stanford A.I. Project 1968 02/2014;
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Available from: Irwin Sobel, Jun 14, 2015
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    ABSTRACT: In this paper we present a set of efficient algorithms for detection and identification of discontinuities in high dimensional space. The method is based on extension of polynomial annihilation for edge detection in low dimensions. Compared to the earlier work, the present method poses significant improvements for high dimensional problems. The core of the algorithms relies on adaptive refinement of sparse grids. It is demonstrated that in the commonly encountered cases where a discontinuity resides on a small subset of the dimensions, the present method becomes optimal , in the sense that the total number of points required for function evaluations depends linearly on the dimensionality of the space. The details of the algorithms will be presented and various numerical examples are utilized to demonstrate the efficacy of the method.
    Full-text · Article · May 2011 · Journal of Computational Physics
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    • "We pause to note two important distinctions between the polynomial annihilation edge detection method (13), and more traditional edge detection methods [5] [17]. The polynomial annihilation edge detection method is a high order reconstruction of the jump function, ½f ŠðyÞ, defined in (11). "
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    ABSTRACT: Edge detection has traditionally been associated with detecting physical space jump discontinuities in one dimension, e.g. seismic signals, and two dimensions, e.g. digital images. Hence most of the research on edge detection algorithms is restricted to these contexts. High dimension edge detection can be of significant importance, however. For instance, stochastic variants of classical differential equations not only have variables in space/time dimensions, but additional dimensions are often introduced to the problem by the nature of the random inputs. The stochastic solutions to such problems sometimes contain discontinuities in the corresponding random space and a prior knowledge of jump locations can be very helpful in increasing the accuracy of the final solution. Traditional edge detection methods typically require uniform grid point distribution. They also often involve the computation of gradients and/or Laplacians, which can become very complicated to compute as the number of dimensions increases. The polynomial annihilation edge detection method, on the other hand, is more flexible in terms of its geometric specifications and is furthermore relatively easy to apply. This paper discusses the numerical implementation of the polynomial annihilation edge detection method to high dimensional functions that arise when solving stochastic partial differential equations.
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    • "For flux computation, a surface is defined that passes halfway through the voxels located along the boundary of the region of interest. To avoid surface erosion, 2x1 derivative templates are used instead of the 3x3 Sobel templates [1], [7] typically used in CDI. Data shifting due to asymmetry of the template is corrected using linear interpolation to re-grid the data appropriately onto the surface. "
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