利用離散格林函數計算大型變場於臉部變形應用

Abstract

影像校準(Image registration)廣泛應用於電腦視覺與MRI影像中。對於我們的研究，我們主要著重於clamped-plate splines(CPS)與large deformation dffeomorphic metric mapping(LDDMM)的演算法並且將之應用於三維影像的臉部變形問題。其中，黎曼映射(Riemann mapping)被使用於將一個三維曲面S映射至一個二維的圓盤 D，而形成帶有保角因子 λ(x)（conformal factor）與 平均曲率 h(x) (mean curvature)的圓盤影像。為了藉由CPS與LDDMM在圓盤影像間建造一個對應關係，我們提出將圓盤影像視為一個影像強度(intensity)為 h(x) 並於整個原盤上帶有彈性係數 λ(x) 的保角影像(conformal image) I(λ,h)(x)。但格林函數(Green’s function)對於此帶有彈性係數的圓盤並無解析解，因此，我們利用有限元素法(finite element method)與HCT基底函數(HCT element)去獲得離散格林函數，並將之作為基底去計算LDDMM 與CPS的形變場。最後我們內插出不同時間點的 λ(x) 與h(x) 用於臉部曲面的重建

Image registration is widely used in computer vision and MRI image. In our work, we focus on clamped-plate splines(CPS) and large deformation diffeomorphic metric mapping(LDDMM) and apply the methods to 3D face morphing problem. Riemann mapping is used to map 3D surfaces to a 2D disk D, which form disk images with conformal factor λ(x) and mean curvature h(x). To build the correspondence between these disk images via CPS and LDDMM, we consider the disk as a conformal image I(λ,h)x)。 where h(x). is the image intensity and λ(x) represents the material factor of the disk. However, there is no analytic form of the Green's function for the elastic disk with variable material factor. Therefore, we use finite element method with HCT element to obtain the discrete Green's function which is used as basis to compute the deformation map of LDDMM and CPS. Finally, we interpolate λ(x) and h(x) to reconstruct the missing frames.