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  "contentMarkdown": "# Possion重建有关的一些概念\n\n> 创建时间：2020/5/23 8:28\n\n## 泛函\n\n简单的说， 泛函就是定义域是一个函数集，而值域是实数集或者实数集的一个子集，推广开来， 泛函就是从任意的向量空间到标量的映射。也就是说，它是从函数空间到数域的映射。\n设{y}是给定的函数集，如果对于这个函数集中任一函数y(x) 恒有某个确定的数与之对应，记为П(y(x))，则П(y(x))是定义于集合{y(x)}上的一个泛函。\n泛函定义域内的函数为可取函数或容许函数， y(x) 称为泛函П的变量函数。\n泛函П(y(x))与可取函数y(x)有明确的对应关系。泛函的值是由一条可取曲线的整体性质决定的。\n泛函也是一种“函数”，它的独立变量一般不是通常函数的“自变量”，而是通常函数本身。泛函是函数的函数。由于函数的值是由自变量的选取而确定的，而泛函的值是由自变量函数确定的，故也可以将其理解为函数的函数\n泛函的自变量是函数，泛函的自变量称为宗量。\n简言之，泛函就是函数的函数。 [2]\n\n## 梯度\n\n![Alt text](assets/a.png)\n\n## 二范数\n\n除了矩阵之外，向量和函数均有范数，其中：\n矩阵范数：矩阵A的2范数就是 A的转置乘以A矩阵特征根 最大值的开根号；\n向量范数：向量x的2范数是x中各个元素平方之和再开根号；\n函数范数：函数f(x)的2范数是x在区间（a,b）上f(x)的平方的积分再开根号。\n\n## 散度\n\n散度（divergence）可用于表征空间各点矢量场发散的强弱程度，物理上，散度的意义是场的有源性。当div F>0 ，表示该点有散发通量的正源（发散源）；当div F<0 表示该点有吸收通量的负源（洞或汇）；当div F=0，表示该点无源。\n\n![Alt text](assets/b.png)\n\n## 拉普拉斯算子\n\n拉普拉斯算子是n维欧几里德空间中的一个二阶微分算子，定义为梯度（▽f）的散度（▽·f）。因此如果f是二阶可微的实函数，则f的拉普拉斯算子定义为：\n\n![Alt text](assets/c.png)\n\n## 泊松方程\n\n![Alt text](assets/d.png)\n\n## Possion重建\n\n![Alt text](assets/a_1.png)\n\n指示函数(indicator function )\n重建的关键在于是找到一个指示函数。若一个元素属于这个集合则为1，否则为0。所以才有了论文中定义的在表面外为1，在表面内为0.如果这个时候我们对整个空间有效的指示函数进行梯度计算，我们会发现只有在接近物体表面的附近才有会有梯度向量，其余位置均为0向量。此时所得到的表面附近梯度就正好等于了论文中所阐述的inner surface normal，也就是朝向表面内的法向量\n\n输入数据为采样点的位置信息，还有采样点的法向量。\n\n通过求解这个指示函数找到函数值相同点，并提取出一个等值面，进而实现了重建。\n\n![Alt text](assets/5beb9622c7c3c.png)\n\n指示函数的梯度等于结合表面法线场计算得到的向量场。这也是高斯散度理论。\n\n![Alt text](assets/5beb964708830.png)\n\n指示函数不知道，梯度不知道。\n\n只能计算出向量场。所以应用散度算子这一媒介。转化成了一个泊松方程。\n\n![Alt text](assets/5beb965c4538f.png)\n\n正三角，就是梯度的散度(也是拉普拉斯算子)\n也就是\n\n![Alt text](assets/QQ%E6%88%AA%E5%9B%BE20200523090628.png)\n\n梯度的散度等于向量场的散度\n\n![Alt text](assets/QQ%E6%88%AA%E5%9B%BE20200523084715.png)\n\n解这个泊松方程就能找到指示函数。方程的解采用拉普拉斯矩阵迭代求出 。\n\n而后利用指示函数来找到等值面然后通过marching cube方法提取出等值面\n"
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