利用Ritz方法和Tent函數由投影中求三維立體之重建

Abstract

從投影中重建三度空間立體(Shape from Shading)的技術是電腦視覺科技 中一項重要的研究題目．利用此技術，我們可以由物體在空間的二維投影 影像去推算出物體的三度空間結構．由於只給定此單張二維投影影像，光 源角度以及由三度空間物體投影到二維影像的數學模型 . 所以如何發展 出有效且正確的演算法，是目前研究的重要課題．我們以能量函數最小化 的觀點將問題正規化(Regularization)，這可使問題的解唯一．由於無法 直接求得表面高度，所以必須利用高度的斜率來間接解出表面高度．首先 ，所求的變數以一階的三角波形(Tent function)為基底函數來展開，使 問題離散化(Discretization)，以便於用數值方法求得解答．此外，在能 量函數中另外加上高階變化率的限制，可使得變數間的關係更加密切而且 使變數更有規律地變化．接著我們用擾動的方法求出使得能量趨近於最小 的擾動量，即完成整個演算法． A robust approach to recover the shape of an object from shading information is presented. The scheme is derived based on the perturbation method that minimizes a cost function for computing a surface height from a single image given a imaging model and light direction. To describe the desired solution variables as a linear combination of a set of basis functions, the Ritz method is applied in our approach to discretize the cost function associated with the regularized shape from shading problem. This kind of discretization is simple and the concept of signal representation by a set of weighted basis functions can be clearly stated. To enhance the robustness of iterative scheme, two new constraints are added into the cost function. They constrain the behavior of high-order change rate between the variables, and bound the variables more tightly. Thus, the changes of variables will be more regular. Hence, new constraints strengthen the relations between the input image and the reconstructed surface height. After finding the weighting coefficients using perturbation method, the solution of the SFS problem will finally be decided.