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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Horn–Schunck method ： ウィキペディア英語版 Horn–Schunck method The Horn–Schunck method of estimating optical flow is a global method which introduces a global constraint of ''smoothness'' to solve the ''aperture problem'' (see Optical Flow for further description). == Mathematical details == The Horn-Schunck algorithm assumes smoothness in the flow over the whole image. Thus, it tries to minimize distortions in flow and prefers solutions which show more smoothness. The flow is formulated as a global energy functional which is then sought to be minimized. This function is given for two-dimensional image streams as: :$E=\backslash iint\; \backslash left(+\; I\_yv\; +\; I\_t)^2\; +\; \backslash alpha^2(\backslash lVert\backslash nabla\; u\backslash rVert^2+\backslash lVert\backslash nabla\; v\backslash rVert^2)\backslash right)xy\}$ where $I\_x$, $I\_y$ and $I\_t$ are the derivatives of the image intensity values along the x, y and time dimensions respectively, $\backslash vec\; =\; ()^\backslash top$ is the optical flow vector, and the parameter $\backslash alpha$ is a regularization constant. Larger values of $\backslash alpha$ lead to a smoother flow. This functional can be minimized by solving the associated multi-dimensional Euler-Lagrange equations. These are :$\backslash frac\; -\; \backslash frac\backslash frac\; -\; \backslash frac\backslash frac\; =\; 0$ :$\backslash frac\; -\; \backslash frac\backslash frac\; -\; \backslash frac\backslash frac\; =\; 0$ where $L$ is the integrand of the energy expression, giving :$I\_x(I\_xu+I\_yv+I\_t)\; -\; \backslash alpha^2\; \backslash Delta\; u\; =\; 0$ :$I\_y(I\_xu+I\_yv+I\_t)\; -\; \backslash alpha^2\; \backslash Delta\; v\; =\; 0$ where subscripts again denote partial differentiation and $\backslash Delta\; =\; \backslash frac\; +\; \backslash frac$ denotes the Laplace operator. In practice the Laplacian is approximated numerically using finite differences, and may be written $\backslash Delta\; u(x,y)\; =\; \backslash overline(x,y)\; -\; u(x,y)$ where $\backslash overline(x,y)$ is a weighted average of $u$ calculated in a neighborhood around the pixel at location (x,y). Using this notation the above equation system may be written :$(I\_x^2\; +\; \backslash alpha^2)u\; +\; I\_xI\_yv\; =\; \backslash alpha^2\backslash overline-I\_xI\_t$ :$I\_xI\_yu\; +\; (I\_y^2\; +\; \backslash alpha^2)v\; =\; \backslash alpha^2\backslash overline-I\_yI\_t$ which is linear in $u$ and $v$ and may be solved for each pixel in the image. However, since the solution depends on the neighboring values of the flow field, it must be repeated once the neighbors have been updated. The following iterative scheme is derived: :$u^=\backslash overline^k\; -\; \backslash frac^k+I\_t)\}$ :$v^=\backslash overline^k\; -\; \backslash frac^k+I\_t)\}$ where the superscript ''k+1'' denotes the next iteration, which is to be calculated and ''k'' is the last calculated result. This is in essence the Jacobi method applied to the large, sparse system arising when solving for all pixels simultaneously. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Horn–Schunck method」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.