顺序无关半透明:Weighted Blended

 
 
  所有基于排序的 OIT 算法实际上都需要大量的计算资源,不管是基于图元的排序还是基于像素的排序。
 

Blend 中的 OVER 操作

  Blend 计算的基本思想是:
C_f=C_1+(1-\alpha_1)C_0  这里,与上一篇博客 顺序无关半透明:Depth Peeling 中的定义一样,C_0 表示 premultiplied-alpha color,预先乘上了 \alpha,即 color.rgb * color.a。
  那么,当有 n 个半透明物体覆盖在背景上时,有
C_f=[C_n+(1-\alpha_n) {\cdot}{\cdot}{\cdot}[C_2+(1-\alpha_2)[C_1+(1-\alpha_1)C_0]]{\cdot}{\cdot}{\cdot}]  所以,背景 C_0 的贡献为:
C_0\prod_{i=1}^n(1-\alpha_i)  同理,C_i 的贡献为:
C_i\prod_{j=i+1}^n(1-\alpha_j)  由于上述累乘刨除掉 C_i 后实际上是顺序无关的,所以如何将 C_i 通过解析式估计出来,也就成了问题的关键。最终的 Weighted Blended 将上述式子融合为一个估计的解析式,它有一定偏差,但是在非极端情况下已经非常够用。

Weighted Blended OIT

  在不使用特殊缓冲区和排序的情况下,可以用 Weighted Blended OIT实现依赖现有数据和BlendFunction的OIT。 下面的介绍中对推导过程进行了省略,因为里边有大量的经验公式。

Meshkin的方法

C_f=(\sum_{i=1}^nC_i)+C_0(1-\sum_{i-1}^n\alpha_i)  这里,每次执行像素着色器之后,将 premultiplied 得到的 C_i 以及 \alpha_i 各自累加起来,最后通过单独的着色器将两者合并即可。
  这时,当 \alpha 的值很小并且颜色相似时可以得到较好的效果(类似于排序OIT),但随着 \alpha 的值增加,差距会越来越明显。

Bavoil和Myer的方法

C_f={\sum_{i=1}^nC_i\over \sum_{i=1}^n\alpha_i}\cdot(1-[1-{1\over n}\sum_{i=1}^n\alpha_i]^n)+C_0[1-{1\over n}\sum_{i=1}^n\alpha_i]^n  新的方法在 Meshkin 方法中引入了“加权平均”算子。[1-{1\over n}\sum_{i=1}^n\alpha_i]^n 被称为总透明度。根据计算的总透明度,然后通过 C_i\alpha_i 的加权估计最终的颜色。

新的 Blended OIT

C_f={\sum_{i=1}^nC_i\over \sum_{i=1}^n\alpha_i}(1-\prod_{i=1}^n(1-\alpha_i))+C_0\prod_{i=1}^n(1-\alpha_i)  在其它方法中,\alpha=0 以及 C_i=0 的情况下才有助于估计真实的半透明颜色。
  新的方法中,\prod_{i=1}^n(1-\alpha_i) 被估计为总透明度。对于半透明颜色,使用上述方法中的 \alpha_i 的加权平均值计算,计算最终的半透明颜色。

增加深度权重

C_f={\sum_{i=1}^nC_i\cdot w(z_i, \alpha_i)\over \sum_{i=1}^n\alpha_i\cdot w(z_i, \alpha_i)}(1-\prod_{i=1}^n(1-\alpha_i))+C_0\prod_{i=1}^n(1-\alpha_i)  由于上一个 Blend 公式中获得的半透明物体颜色是简单 \alpha 的加权平均值。所以不管 \alpha 权值大的像素上面还有多少层半透明像素,它都会是非常明显的一部分。
  所以,这里不仅使用简单的 \alpha 加权平均,将深度也作为权重标准之一,即相同 \alpha 的片段,距离相机的距离越近,则权重越大。计算出的贡献增大到视点越近。
  权重函数的选取要求单调递减,这里无法使用解析的公式推导,归根结底,只是经验公式罢了。以下四种权重函数可以提供参考:
\begin{aligned} w(z,\alpha)&=\alpha\cdot max[10^{-2},min[3\times10^{3},{10\over 10^{-5}+(\vert z\vert /5)^2+(\vert z\vert /200)^6}]]\\ w(z,\alpha)&=\alpha\cdot max[10^{-2},min[3\times10^{3},{10\over 10^{-5}+(\vert z\vert /10)^3+(\vert z\vert /200)^6}]]\\ w(z,\alpha)&=\alpha\cdot max[10^{-2},min[3\times10^{3},{10\over 10^{-5}+(\vert z\vert /200)^4}]]\\ w(z,\alpha)&=\alpha\cdot max[10^{-2},3\times10^3\cdot(1-d(z))^3] \end{aligned}  上述第四个式子中的 z 表示 OpenGL 中 gl_FragCoord.z,在相机空间为负值。有
d(z)={(z_{near}z_{far})/z-z_{far}\over z_{near}-z_{far}}

分析

  Weighted Blended OIT 是一种有效的手段,因为它可以节省半透明排序的时间,并且可以避免排序带来的 GPU 负载不均衡的问题。在笔记本 950M 中可以跑 300+ fps,相比 Depth Peeling 的 100 fps 要太多了。
  当然,估计肯定会带来误差。Weighted Blended OIT 还是有加权平均的传统矛盾,当 \alpha 值比较大偏向于不透明的时候,误差会越来越大,并不能完全达到基于排序 OIT 的效果。
  

结果

oit weighted

参考:
Weighted Blended Order-Independent Transparency

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