projective TSDF/TSDF/flipped TSDF三种截断符号距离函数比较的个人理解

先直接上图：

说明：projective TSDF只找在摄像机视线上最靠近的表面点，这样虽然加速了运算，但是有严重的视角依赖性。TSDF选择计算在所有可观测表面上任一点中最近的点的距离，但在沿着遮挡（-dmax）边界的空白空间部分(+dmax)，有很强的梯度。flipped TSDF则通过变换使得在表面呈现最强的梯度（可以参考下图中颜色变化）。

最近在学习普林斯顿大学的一篇三维场景理解的论文中，出现了TSDF数据编码方式。论文中三种方式的对比图如下：

于是找了一些资料进行学习。
（主要参考11th International Conference, ICIAR 2014《Image Analysis and Recognition》中《Truncated Signed Distance Function: Experiments on Voxel Size》，Diana Werner et al.）

图为二维TSDF例子。Solid object (green), camera with field of view, optical axis and ray (blue), and TSDF grid (unseen voxels are white, for others see color bar). The signed distance value of voxel x is determined by the depth of the corresponding surface point p and the voxel’s camera distance camz(x).

pic(x) is the projection of the voxel center x onto the depth image. So depthi(pic(x)) is the measured depth in between the camera and the nearest object surface point p on the viewing ray crossing x. Accordingly, camz(x) is the distance in between the voxel and the camera along the optical axis. Consequently, sdfi(x) is a distance along the optical axis as well.

The SDF has been truncated at ±t. This is beneficial, because large distances are not relevant for surface reconstruction and a restriction of the value range can be utilized to memory footprint. The truncated variant of sdfi(x) is denoted by tsdfi(x).

In Fig tsdfi(x) of the voxel grid is encoded by color.

The TSDF representation requires to select several parameters：
①Grid volume size determines the dimensions of the TSDF grid.
②Voxel size v is a crucial parameter as it influences memory requirements and surface reconstruction accuracy. If dimensions of a 3D grid are fixed, doubling the voxel size means to reduce the number of voxels to one-eighths. This is associated with the same reduction in memory footprint. Further, it reduces computational cost for updating the TSDF and for ray tracing. The other way around, an increased voxel size facilitates to increase the scene volume without needing more memory or increasing computational cost. However, an increase in voxel size comes along with a decrease in the level of representable details resp. with lowered reconstruction accuracy. So it is worth thinking about the optimal voxel size for a particular application.
③Distance representation and truncation distance t i. e. the coding of distance values TSDFi(x) is crucial for the reconstruction accuracy. The selection of t influences reconstruction accuracy. t should be larger than length of voxel diagonal(√d) · v (voxel size)and the level of noise.

---