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统计学
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抽样
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推断
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概率论
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计数
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</a>
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概率概念
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分布
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</a>
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贝叶斯
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</span>
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</a>
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信息论
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</span>
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<span class="md-ellipsis">
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机器学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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机器学习
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经典机器学习
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</span>
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梯度机器学习
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</span>
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</a>
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深度学习
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</span>
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</a>
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强化学习
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</span>
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</a>
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分布式深度学习
|
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</span>
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|
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|
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|
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</a>
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</nav>
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计算语言学
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</span>
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<span class="md-nav__icon md-icon"></span>
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计算语言学
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语言学基础
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</a>
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</span>
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</a>
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嵌入与序列模型
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</span>
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</a>
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<span class="md-ellipsis">
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Transformer 与语言模型
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</span>
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|
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</a>
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<span class="md-ellipsis">
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高级文本生成
|
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|
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|
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|
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</span>
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|
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|
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|
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</a>
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<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
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计算机视觉
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图像基础
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</span>
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</a>
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卷积网络
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|
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</a>
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ViT 与生成模型
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说话人与音频分析
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</span>
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</a>
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源分离与降噪
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</span>
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多模态学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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多模态学习
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多模态表征
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</span>
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</a>
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视觉语言模型
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</span>
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</a>
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图像与视频 Token 化
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</span>
|
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|
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|
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|
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</a>
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跨模态生成
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</span>
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|
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|
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|
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</a>
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</li>
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<span class="md-ellipsis">
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统一多模态架构
|
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|
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|
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|
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</span>
|
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|
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|
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自主系统
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感知
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</span>
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</a>
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机器人学习
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</span>
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</a>
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视觉-语言-动作模型
|
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|
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</span>
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|
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|
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|
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</a>
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</li>
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自动驾驶
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</span>
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</a>
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<a href="../../chapter%2011%3A%20autonomous%20systems/05.%20space%20and%20extreme%20robotics/" class="md-nav__link">
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<span class="md-ellipsis">
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太空与极端机器人
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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</nav>
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图神经网络
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</span>
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<span class="md-nav__icon md-icon"></span>
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图神经网络
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</label>
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几何深度学习
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<span class="md-ellipsis">
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|
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硬件基础
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</a>
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ARM 与 NEON
|
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x86 与 AVX
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Triton、TPU 与 Pallas
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量化
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边缘推理
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ML 系统设计
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应用 AI
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前沿 AI
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<h1 id="3d">视频与3D视觉<a class="headerlink" href="#3d" title="Permanent link">¶</a></h1>
|
||
<p><em>视频与3D视觉将图像理解扩展到时间域和空间域。本文涵盖光流、视频分类（3D卷积网络、TimeSformer）、目标跟踪（SORT、DeepSORT）、动作识别、深度估计（单目与立体）、点云、神经辐射场（NeRF）和3D高斯泼溅。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>文件01-04将图像视为孤立快照。但视觉世界是连续的：物体在运动，场景在变化，深度真实存在。本文将计算机视觉扩展到时间域（视频）和空间域（3D），涵盖模型如何理解运动、跟踪目标、估计深度和重建场景。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>视频</strong>是一系列随时间捕获的图像（帧）。以30帧/秒计算，一段10秒的片段包含300帧。关键挑战在于建模<strong>时间维度</strong>：物体如何运动，场景如何演变，以及如何跨帧关联信息。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>光流</strong>估计两帧连续图像之间像素的表观运动。对于帧<span class="arithmatex">\(t\)</span>中的每个像素，光流产生一个二维位移向量<span class="arithmatex">\((u, v)\)</span>，指向该像素在帧<span class="arithmatex">\(t+1\)</span>中的位置。结果是一个与图像大小相同的稠密运动场。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="两帧连续视频帧及其之间的光流场，以彩色箭头可视化显示像素运动方向和大小" src="../../images/optical_flow.svg" /></p>
|
||
<ul>
|
||
<li>光流在<strong>亮度恒常性假设</strong>下计算：像素的强度在其移动时不变。如果帧<span class="arithmatex">\(t\)</span>中位置<span class="arithmatex">\((x, y)\)</span>处的像素强度为<span class="arithmatex">\(I(x, y, t)\)</span>，并在小时间间隔<span class="arithmatex">\(\delta t\)</span>内移动了<span class="arithmatex">\((u, v)\)</span>：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[I(x + u\delta t, \, y + v\delta t, \, t + \delta t) = I(x, y, t)\]</div>
|
||
<ul>
|
||
<li>进行一阶泰勒展开（见第03章）并除以<span class="arithmatex">\(\delta t\)</span>：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[I_x u + I_y v + I_t = 0\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>其中<span class="arithmatex">\(I_x, I_y\)</span>是空间梯度（Sobel算子，见文件01），<span class="arithmatex">\(I_t\)</span>是时间梯度（相邻帧的差值）。这就是<strong>光流约束方程</strong>。一个方程，两个未知数<span class="arithmatex">\((u, v)\)</span>：我们需要额外的约束条件。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>Lucas-Kanade</strong>假设光流在一个小窗口内（例如5x5像素）是恒定的。这给出了一个超定系统（25个方程，2个未知数），通过最小二乘法求解（第06章的正规方程）：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[
|
||
\begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} \sum I_x^2 & \sum I_x I_y \\ \sum I_x I_y & \sum I_y^2 \end{bmatrix}^{-1} \begin{bmatrix} -\sum I_x I_t \\ -\sum I_y I_t \end{bmatrix}
|
||
\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>这个2x2矩阵就是文件01中的结构张量（与Harris角点检测中使用的矩阵相同）。Lucas-Kanade适用于小运动，但当物体在帧间移动超过几个像素时会失效。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>Farneback方法</strong>对每个像素邻域进行多项式展开，并估计最能解释帧间变化的位移场。它产生稠密光流（每个像素一个向量），能处理比Lucas-Kanade更大的运动。</p>
|
||
</li>
|
||
<li>
|
||
<p>现代<strong>深度学习光流</strong>方法（FlowNet、RAFT）学习从帧对端到端预测光流。<strong>RAFT</strong>（Recurrent All-Pairs Field Transforms，Teed和Deng，2020）计算两帧中所有像素对之间的4D相关体，并使用基于GRU的更新算子迭代优化光流估计。RAFT达到了最先进的精度，并已成为标准的光流骨干网络。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>双流网络</strong>（Simonyan和Zisserman，2014）是视频理解的早期方法。一个流处理单帧RGB图像（外观），另一个流处理光流帧的堆叠（运动）。两个流在末端融合（通过平均或拼接）。这种架构明确区分了"事物看起来像什么"与"它们如何运动"。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>3D卷积网络</strong>将2D卷积扩展到时间维度。3D卷积使用大小为<span class="arithmatex">\(k \times k \times k_t\)</span>的滤波器，同时跨越空间和时间维度，直接学习时空特征。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>C3D</strong>（Tran等人，2015）堆叠了3x3x3滤波器的3D卷积，展示了时间卷积可以在没有显式光流的情况下学习运动特征。代价是高昂的：3D卷积的参数和计算量是其2D对应物的<span class="arithmatex">\(k_t\)</span>倍。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>I3D</strong>（Inflated 3D，Carreira和Zisserman，2017）采用了一种更实用的方法：从预训练的2D CNN（如Inception或ResNet）开始，将所有2D滤波器沿时间维度"膨胀"为3D，重复权重并除以<span class="arithmatex">\(k_t\)</span>。这将ImageNet预训练迁移到视频，同时增加了时间建模能力。一个2D的<span class="arithmatex">\(k \times k\)</span>滤波器变为<span class="arithmatex">\(k \times k \times k_t\)</span>的滤波器，初始化为<span class="arithmatex">\(W_{\text{3D}}[:,:,j] = W_{\text{2D}} / k_t\)</span>，对所有时间位置<span class="arithmatex">\(j\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>SlowFast网络</strong>（Feichtenhofer等人，2019）使用两条并行的路径，以不同的时间分辨率运行：</p>
|
||
<ul>
|
||
<li><strong>Slow路径</strong>以低帧率（例如每16帧）处理帧，具有高空间分辨率和更多通道，捕获精细的空间细节。</li>
|
||
<li><strong>Fast路径</strong>以高帧率（每2帧）处理帧，空间分辨率降低且通道数较少（通常为Slow路径的<span class="arithmatex">\(1/8\)</span>），捕获快速的时间变化。</li>
|
||
<li>侧向连接通过步长卷积将信息从Fast融合到Slow。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>其核心洞见是：空间和时间信息具有不同的带宽需求——物体外观变化缓慢，但运动可以很迅速。SlowFast通过设计匹配这种非对称性。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>TimeSformer</strong>（Bertasius等人，2021）将Vision Transformer应用于视频。它将完整的时空注意力（代价过高：<span class="arithmatex">\(O((T \times N)^2)\)</span>，其中<span class="arithmatex">\(T\)</span>为帧数，<span class="arithmatex">\(N\)</span>为每帧的块数）分解为<strong>分块注意力</strong>：每个块在时间注意（每个块在相同空间位置跨时间进行注意力）和空间注意（每个块在同一帧内跨空间进行注意力）之间交替。这使代价从<span class="arithmatex">\(O(T^2 N^2)\)</span>降低到<span class="arithmatex">\(O(T^2 + N^2)\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>VideoMAE</strong>（Tong等人，2022）将掩码自编码器思想（见文件04）扩展到视频。使用极高的掩码比例（90-95%），因为视频具有高度的时间冗余性：相邻帧看起来几乎相同，因此掩码大部分块后仍然留有足够的信息进行重建。VideoMAE在无标签视频上预训练ViT骨干网络，并迁移到下游任务。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>动作识别</strong>将视频片段分类为多种动作类别之一（例如"跑步"、"烹饪"、"弹吉他"）。它是图像分类的视频对应任务。标准基准数据集包括Kinetics-400（400个动作类别，约30万个片段）、Something-Something（174个需要时间推理的细粒度动作）和ActivityNet（200个类别，包含长时未裁剪视频）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>时间动作检测</strong>超越了分类：给定一个长段未裁剪的视频，找到每个动作的开始时间、结束时间和类别。这是目标检测的时间对应任务。ActionFormer等方法使用Transformer处理时间特征并预测动作边界。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>视频目标跟踪</strong>在第一帧识别出特定目标后，跨帧跟踪该目标。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>SORT</strong>（Simple Online and Realtime Tracking，Bewley等人，2016）将检测模型（独立检测每帧中的目标）与<strong>卡尔曼滤波器</strong>（用于运动预测）和<strong>匈牙利算法</strong>（用于分配）相结合。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>卡尔曼滤波器</strong>为每个跟踪的目标维护一个状态估计（位置、速度、大小），并使用线性运动模型预测它在下一帧中的位置。当新的检测结果到达时，卡尔曼滤波器通过结合预测值和观测值（按各自的不确定性加权）来更新其估计。这是贝叶斯更新（第05章）在跟踪中的应用。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>匈牙利算法</strong>解决双线性分配问题：给定<span class="arithmatex">\(M\)</span>个已跟踪目标和<span class="arithmatex">\(N\)</span>个新检测结果，找到使总代价最小化的最优一对一匹配（使用文件03中的IoU距离）。未匹配的检测结果开始新的轨迹；未匹配的轨迹在宽限期后被终止。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>DeepSORT</strong>通过添加<strong>深度外观特征</strong>扩展了SORT：每个检测到的目标经过一个小型CNN，产生一个外观嵌入（描述子向量）。匹配代价结合了IoU距离和嵌入空间中的余弦距离（第01章）。这处理了遮挡和重识别：即使一个目标在其他目标后消失数帧，其外观嵌入允许在重新出现时重新匹配。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>ByteTrack</strong>（Zhang等人，2022）通过使用所有检测结果（包括低置信度的）来改进跟踪。大多数跟踪器会丢弃低于置信度阈值的检测结果。ByteTrack首先将高置信度检测结果与现有轨迹匹配，然后将剩余的低置信度检测结果与未匹配的轨迹匹配。这恢复了暂时被遮挡或模糊（因此检测置信度低）的目标。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>3D视觉</strong>恢复在2D图像投影中丢失的第三个空间维度（见文件01）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>深度估计</strong>预测从相机到场景中每个点的距离。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>立体深度</strong>使用两个相距基线距离<span class="arithmatex">\(b\)</span>的相机。同一个点在左右图像中出现在不同的水平位置（这个偏移称为<strong>视差</strong><span class="arithmatex">\(d\)</span>）。深度与视差成反比：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[Z = \frac{f \cdot b}{d}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>其中<span class="arithmatex">\(f\)</span>是焦距，<span class="arithmatex">\(b\)</span>是基线距离。计算视差需要找到两个图像之间的对应点（立体匹配），这是沿水平扫描线的一维搜索（因为相机水平对齐，3D中同一高度的点投影到两幅图像的同一行）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>单目深度估计</strong>从单张图像预测深度，这本质上是病态问题（无限多个3D场景可以产生相同的2D图像）。然而，人类利用相对大小、纹理梯度、遮挡和大气雾霾等线索毫不费力地做到这一点。深度学习网络从训练数据中学习这些线索。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>MiDaS</strong>和<strong>Depth Anything</strong>等模型从单张图像预测相对深度图（排序哪些物体更近）。它们使用尺度不变损失在各种数据集上训练，尽管理论上存在歧义，但仍能产生非常准确的结果。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>点云</strong>是3D点<span class="arithmatex">\((x, y, z)\)</span>的集合，可选地带有颜色或其他属性，由LiDAR传感器或立体重建捕获。与图像不同，点云是无序且不规则间隔的。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>PointNet</strong>（Qi等人，2017）通过独立地对每个点应用共享MLP，然后使用最大池化聚合（这是置换不变的，解决了排序问题），直接处理点云。<strong>PointNet++</strong>增加了层次化分组，以捕获多尺度的局部结构。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>神经辐射场（NeRF）</strong>（Mildenhall等人，2020）将3D场景表示为一个连续函数，将3D位置<span class="arithmatex">\((x, y, z)\)</span>和视角方向<span class="arithmatex">\((\theta, \phi)\)</span>映射到颜色<span class="arithmatex">\((r, g, b)\)</span>和密度<span class="arithmatex">\(\sigma\)</span>。该函数由一个MLP参数化：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[F_\theta: (x, y, z, \theta, \phi) \to (r, g, b, \sigma)\]</div>
|
||
<ul>
|
||
<li>为了渲染一个像素，从相机穿过该像素向场景投射一条射线。沿射线采样点，MLP预测每个点的颜色和密度。像素颜色通过<strong>体渲染</strong>计算：沿射线按密度加权积分颜色：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[C(\mathbf{r}) = \int_{t_n}^{t_f} T(t) \cdot \sigma(\mathbf{r}(t)) \cdot \mathbf{c}(\mathbf{r}(t), \mathbf{d}) \, dt\]</div>
|
||
<ul>
|
||
<li>其中<span class="arithmatex">\(T(t) = \exp(-\int_{t_n}^{t} \sigma(\mathbf{r}(s)) \, ds)\)</span>是累积透射率（已吸收的光总量）。在实际中，该积分通过沿射线采样<span class="arithmatex">\(N\)</span>个点并求和来近似：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\hat{C} = \sum_{i=1}^{N} T_i \cdot (1 - \exp(-\sigma_i \delta_i)) \cdot c_i\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>NeRF通过最小化渲染像素与一组带位姿照片的真实像素之间的MSE来训练。训练完成后，NeRF可以从任何相机位置渲染逼真的新视角。其局限性在于速度：渲染需要对MLP进行数百万次评估（每个像素每个采样点一次），这使得实时渲染变得困难。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>3D高斯泼溅</strong>（Kerbl等人，2023）通过将场景表示为3D高斯原语的集合（而非连续的体积函数）来解决NeRF的速度限制。每个高斯原语有一个3D位置（均值）、一个3D协方差矩阵（控制形状和朝向）、不透明度及颜色（表示为球谐函数以实现视角相关效果）。</p>
|
||
</li>
|
||
<li>
|
||
<p>渲染将每个3D高斯投影到图像平面（产生一个2D高斯"泼溅"），按深度排序，并使用alpha混合从前往后合成。这是一个在GPU上实时运行的栅格化过程（100+ FPS），比NeRF的射线步进快几个数量级。高斯泼溅达到或超过NeRF的质量，同时实现实时渲染。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>SLAM</strong>（同时定位与地图构建）是在未知环境中构建地图同时跟踪相机自身位置的问题。这是机器人、自动驾驶和AR的基础。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>视觉里程计</strong>通过跨图像跟踪特征来估计相机从一帧到另一帧的运动。特征点（SIFT、ORB，见文件01）在连续帧之间匹配，并利用这些匹配关系通过<strong>本质矩阵</strong>（编码两视图之间的几何关系，由文件01的内参和外参推导）估计相机的旋转和平移。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>基于特征的SLAM</strong>通过维护持久地图来扩展视觉里程计。<strong>ORB-SLAM</strong>（Mur-Artal等人，2015）是使用最广泛的基于特征的SLAM系统。它有三个并行线程：</p>
|
||
<ol>
|
||
<li><strong>跟踪</strong>：将每帧中的ORB特征与地图匹配，使用PnP（Perspective-n-Point）和RANSAC估计相机位姿。</li>
|
||
<li><strong>局部建图</strong>：从匹配的特征三角化新的地图点，通过光束法平差（最小化所有观察到每个点的视图的重投影误差）优化其位置。</li>
|
||
<li><strong>闭环检测</strong>：检测相机何时重新访问先前建图的区域（使用视觉词袋），然后通过全局优化地图来校正累积漂移。</li>
|
||
</ol>
|
||
</li>
|
||
<li>
|
||
<p><strong>LiDAR SLAM</strong>使用来自LiDAR传感器的3D点云替代（或补充）相机图像。LiDAR提供直接的深度测量，使几何估计更鲁棒，但硬件成本更高。LOAM（LiDAR Odometry and Mapping）等方法使用迭代最近点（ICP）配准来对齐连续扫描之间的点云。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>视觉-惯性SLAM</strong>融合相机数据与IMU（加速度计+陀螺仪）的测量结果。IMU提供高频的旋转和加速度估计，弥补相机帧之间的间隙，并处理快速运动或临时视觉特征丢失的情况。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>VR/AR</strong>应用是计算机视觉最苛刻的消费者之一。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>姿态估计</strong>从图像中确定人体（或面部、手部）的位置和朝向。<strong>身体姿态</strong>通常表示为一组2D或3D关键点位置（关节点：肩膀、肘部、手腕、髋部、膝盖、脚踝）。<strong>OpenPose</strong>和<strong>MediaPipe</strong>等模型使用热图回归预测这些关键点：对于每个关节点，模型输出一个热图，其中峰值指示关节点的位置。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>自上而下</strong>的方法首先使用边界框检测器（见文件03）检测人物，然后在每个框内估计姿态。<strong>自下而上</strong>的方法首先检测图像中的所有关键点，然后使用部位亲和场（编码连接关节点之间关联的向量场）将它们分组为个体。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>场景重建</strong>从传感器数据构建环境的3D模型。在AR中，这使得可以将虚拟物体放置在真实表面上、遮挡真实物体后面的虚拟物体以及投射虚拟阴影。实时场景重建方法（如ARKit和ARCore中基于深度传感器的系统）构建环境的稀疏网格，并随着用户移动而更新。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>VR中的实时渲染</strong>约束极为苛刻：双眼需要独立渲染90+ FPS（以避免晕动症），从头部位移到显示更新的延迟需低于20毫秒。<strong>注视点渲染</strong>（仅渲染用户注视位置的高分辨率，使用眼动追踪）和<strong>重投影</strong>（基于新头部位姿扭曲上一帧以填补下一帧渲染间隙）等技术对于满足这些约束至关重要。</p>
|
||
</li>
|
||
<li>
|
||
<p>实时神经渲染（3D高斯泼溅）、鲁棒跟踪（视觉-惯性SLAM）和高效姿态估计的融合，正使逼真的交互式AR/VR体验变得越来越可行。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colabnotebook">编程任务（使用CoLab或Notebook）<a class="headerlink" href="#colabnotebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>从头实现Lucas-Kanade光流算法。计算一个方块向右移动的两帧合成图像之间的光流。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="k">def</span><span class="w"> </span><span class="nf">lucas_kanade</span><span class="p">(</span><span class="n">frame1</span><span class="p">,</span> <span class="n">frame2</span><span class="p">,</span> <span class="n">window_size</span><span class="o">=</span><span class="mi">5</span><span class="p">):</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="sd">"""Lucas-Kanade光流。"""</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="c1"># 计算梯度</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="n">Ix</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">frame1</span><span class="p">)</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="n">Iy</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">frame1</span><span class="p">)</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="n">It</span> <span class="o">=</span> <span class="n">frame2</span> <span class="o">-</span> <span class="n">frame1</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="c1"># Sobel风格梯度</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="n">Ix</span> <span class="o">=</span> <span class="n">Ix</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">set</span><span class="p">((</span><span class="n">frame1</span><span class="p">[</span><span class="mi">2</span><span class="p">:,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">frame1</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="p">:])</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="n">Iy</span> <span class="o">=</span> <span class="n">Iy</span><span class="o">.</span><span class="n">at</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">((</span><span class="n">frame1</span><span class="p">[:,</span> <span class="mi">2</span><span class="p">:]</span> <span class="o">-</span> <span class="n">frame1</span><span class="p">[:,</span> <span class="p">:</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="n">H</span><span class="p">,</span> <span class="n">W</span> <span class="o">=</span> <span class="n">frame1</span><span class="o">.</span><span class="n">shape</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a> <span class="n">half_w</span> <span class="o">=</span> <span class="n">window_size</span> <span class="o">//</span> <span class="mi">2</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a> <span class="n">u</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">frame1</span><span class="p">)</span>
|
||
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a> <span class="n">v</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">frame1</span><span class="p">)</span>
|
||
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>
|
||
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">half_w</span><span class="p">,</span> <span class="n">H</span> <span class="o">-</span> <span class="n">half_w</span><span class="p">):</span>
|
||
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">half_w</span><span class="p">,</span> <span class="n">W</span> <span class="o">-</span> <span class="n">half_w</span><span class="p">):</span>
|
||
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a> <span class="n">Ix_win</span> <span class="o">=</span> <span class="n">Ix</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="n">half_w</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="n">half_w</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="o">-</span><span class="n">half_w</span><span class="p">:</span><span class="n">j</span><span class="o">+</span><span class="n">half_w</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
|
||
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a> <span class="n">Iy_win</span> <span class="o">=</span> <span class="n">Iy</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="n">half_w</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="n">half_w</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="o">-</span><span class="n">half_w</span><span class="p">:</span><span class="n">j</span><span class="o">+</span><span class="n">half_w</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
|
||
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a> <span class="n">It_win</span> <span class="o">=</span> <span class="n">It</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="n">half_w</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="n">half_w</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="o">-</span><span class="n">half_w</span><span class="p">:</span><span class="n">j</span><span class="o">+</span><span class="n">half_w</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
|
||
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>
|
||
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a> <span class="n">A</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">Ix_win</span><span class="p">,</span> <span class="n">Iy_win</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a> <span class="n">ATA</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">A</span>
|
||
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a> <span class="n">ATb</span> <span class="o">=</span> <span class="o">-</span><span class="n">A</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">It_win</span>
|
||
<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a>
|
||
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a> <span class="c1"># 检查系统是否良态</span>
|
||
<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a> <span class="n">det</span> <span class="o">=</span> <span class="n">ATA</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">ATA</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">ATA</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">ATA</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
|
||
<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a> <span class="k">if</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">det</span><span class="p">)</span> <span class="o">></span> <span class="mf">1e-6</span><span class="p">:</span>
|
||
<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a> <span class="n">flow</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">ATA</span><span class="p">,</span> <span class="n">ATb</span><span class="p">)</span>
|
||
<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a> <span class="n">u</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">flow</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||
<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a> <span class="n">v</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">flow</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||
<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>
|
||
<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a> <span class="k">return</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span>
|
||
<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a>
|
||
<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a><span class="c1"># 创建两帧：一个向右移动的白色方块</span>
|
||
<a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a><span class="n">frame1</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">64</span><span class="p">,</span> <span class="mi">64</span><span class="p">))</span>
|
||
<a id="__codelineno-0-41" name="__codelineno-0-41" href="#__codelineno-0-41"></a><span class="n">frame1</span> <span class="o">=</span> <span class="n">frame1</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">20</span><span class="p">:</span><span class="mi">40</span><span class="p">,</span> <span class="mi">15</span><span class="p">:</span><span class="mi">35</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mf">1.0</span><span class="p">)</span>
|
||
<a id="__codelineno-0-42" name="__codelineno-0-42" href="#__codelineno-0-42"></a>
|
||
<a id="__codelineno-0-43" name="__codelineno-0-43" href="#__codelineno-0-43"></a><span class="n">frame2</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">64</span><span class="p">,</span> <span class="mi">64</span><span class="p">))</span>
|
||
<a id="__codelineno-0-44" name="__codelineno-0-44" href="#__codelineno-0-44"></a><span class="n">frame2</span> <span class="o">=</span> <span class="n">frame2</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">20</span><span class="p">:</span><span class="mi">40</span><span class="p">,</span> <span class="mi">20</span><span class="p">:</span><span class="mi">40</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mf">1.0</span><span class="p">)</span> <span class="c1"># 向右移动5个像素</span>
|
||
<a id="__codelineno-0-45" name="__codelineno-0-45" href="#__codelineno-0-45"></a>
|
||
<a id="__codelineno-0-46" name="__codelineno-0-46" href="#__codelineno-0-46"></a><span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">lucas_kanade</span><span class="p">(</span><span class="n">frame1</span><span class="p">,</span> <span class="n">frame2</span><span class="p">,</span> <span class="n">window_size</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
|
||
<a id="__codelineno-0-47" name="__codelineno-0-47" href="#__codelineno-0-47"></a>
|
||
<a id="__codelineno-0-48" name="__codelineno-0-48" href="#__codelineno-0-48"></a><span class="c1"># 可视化</span>
|
||
<a id="__codelineno-0-49" name="__codelineno-0-49" href="#__codelineno-0-49"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<a id="__codelineno-0-50" name="__codelineno-0-50" href="#__codelineno-0-50"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">frame1</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'gray'</span><span class="p">);</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'帧1'</span><span class="p">);</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'off'</span><span class="p">)</span>
|
||
<a id="__codelineno-0-51" name="__codelineno-0-51" href="#__codelineno-0-51"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">frame2</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'gray'</span><span class="p">);</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'帧2'</span><span class="p">);</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'off'</span><span class="p">)</span>
|
||
<a id="__codelineno-0-52" name="__codelineno-0-52" href="#__codelineno-0-52"></a>
|
||
<a id="__codelineno-0-53" name="__codelineno-0-53" href="#__codelineno-0-53"></a><span class="c1"># 光流的箭矢图（为清晰起见降采样）</span>
|
||
<a id="__codelineno-0-54" name="__codelineno-0-54" href="#__codelineno-0-54"></a><span class="n">step</span> <span class="o">=</span> <span class="mi">4</span>
|
||
<a id="__codelineno-0-55" name="__codelineno-0-55" href="#__codelineno-0-55"></a><span class="n">Y</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mgrid</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">64</span><span class="p">:</span><span class="n">step</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">64</span><span class="p">:</span><span class="n">step</span><span class="p">]</span>
|
||
<a id="__codelineno-0-56" name="__codelineno-0-56" href="#__codelineno-0-56"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">frame1</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'gray'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
|
||
<a id="__codelineno-0-57" name="__codelineno-0-57" href="#__codelineno-0-57"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">quiver</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">u</span><span class="p">[::</span><span class="n">step</span><span class="p">,</span> <span class="p">::</span><span class="n">step</span><span class="p">],</span> <span class="n">v</span><span class="p">[::</span><span class="n">step</span><span class="p">,</span> <span class="p">::</span><span class="n">step</span><span class="p">],</span>
|
||
<a id="__codelineno-0-58" name="__codelineno-0-58" href="#__codelineno-0-58"></a> <span class="n">color</span><span class="o">=</span><span class="s1">'#e74c3c'</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">width</span><span class="o">=</span><span class="mf">0.005</span><span class="p">)</span>
|
||
<a id="__codelineno-0-59" name="__codelineno-0-59" href="#__codelineno-0-59"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'光流'</span><span class="p">);</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'off'</span><span class="p">)</span>
|
||
<a id="__codelineno-0-60" name="__codelineno-0-60" href="#__codelineno-0-60"></a>
|
||
<a id="__codelineno-0-61" name="__codelineno-0-61" href="#__codelineno-0-61"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span> <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<a id="__codelineno-0-62" name="__codelineno-0-62" href="#__codelineno-0-62"></a>
|
||
<a id="__codelineno-0-63" name="__codelineno-0-63" href="#__codelineno-0-63"></a><span class="c1"># 检查运动区域的平均光流</span>
|
||
<a id="__codelineno-0-64" name="__codelineno-0-64" href="#__codelineno-0-64"></a><span class="n">region_u</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="mi">20</span><span class="p">:</span><span class="mi">40</span><span class="p">,</span> <span class="mi">15</span><span class="p">:</span><span class="mi">35</span><span class="p">]</span>
|
||
<a id="__codelineno-0-65" name="__codelineno-0-65" href="#__codelineno-0-65"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"物体区域的平均水平光流: </span><span class="si">{</span><span class="n">region_u</span><span class="p">[</span><span class="n">region_u</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2"> 像素"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>实现一个用于2D目标跟踪的简单卡尔曼滤波器。模拟一个带噪声的轨迹，并展示卡尔曼滤波器如何平滑估计。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="k">def</span><span class="w"> </span><span class="nf">kalman_predict</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Q</span><span class="p">):</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="sd">"""卡尔曼滤波器预测步骤。"""</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a> <span class="n">x_pred</span> <span class="o">=</span> <span class="n">F</span> <span class="o">@</span> <span class="n">x</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a> <span class="n">P_pred</span> <span class="o">=</span> <span class="n">F</span> <span class="o">@</span> <span class="n">P</span> <span class="o">@</span> <span class="n">F</span><span class="o">.</span><span class="n">T</span> <span class="o">+</span> <span class="n">Q</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a> <span class="k">return</span> <span class="n">x_pred</span><span class="p">,</span> <span class="n">P_pred</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="k">def</span><span class="w"> </span><span class="nf">kalman_update</span><span class="p">(</span><span class="n">x_pred</span><span class="p">,</span> <span class="n">P_pred</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="sd">"""卡尔曼滤波器更新步骤。"""</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a> <span class="n">y</span> <span class="o">=</span> <span class="n">z</span> <span class="o">-</span> <span class="n">H</span> <span class="o">@</span> <span class="n">x_pred</span> <span class="c1"># 创新</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a> <span class="n">S</span> <span class="o">=</span> <span class="n">H</span> <span class="o">@</span> <span class="n">P_pred</span> <span class="o">@</span> <span class="n">H</span><span class="o">.</span><span class="n">T</span> <span class="o">+</span> <span class="n">R</span> <span class="c1"># 创新协方差</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a> <span class="n">K</span> <span class="o">=</span> <span class="n">P_pred</span> <span class="o">@</span> <span class="n">H</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">S</span><span class="p">)</span> <span class="c1"># 卡尔曼增益</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a> <span class="n">x_updated</span> <span class="o">=</span> <span class="n">x_pred</span> <span class="o">+</span> <span class="n">K</span> <span class="o">@</span> <span class="n">y</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a> <span class="n">P_updated</span> <span class="o">=</span> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x_pred</span><span class="p">))</span> <span class="o">-</span> <span class="n">K</span> <span class="o">@</span> <span class="n">H</span><span class="p">)</span> <span class="o">@</span> <span class="n">P_pred</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a> <span class="k">return</span> <span class="n">x_updated</span><span class="p">,</span> <span class="n">P_updated</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a>
|
||
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="c1"># 状态: [x, y, vx, vy]</span>
|
||
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="n">dt</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="n">F</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="c1"># 状态转移</span>
|
||
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">dt</span><span class="p">],</span>
|
||
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
|
||
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
|
||
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="n">H</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="c1"># 观测：测量 x, y</span>
|
||
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
|
||
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="n">Q</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.01</span> <span class="c1"># 过程噪声</span>
|
||
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="n">R</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="mf">4.0</span> <span class="c1"># 测量噪声（有噪声的检测器）</span>
|
||
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a>
|
||
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="c1"># 模拟真实轨迹：圆周运动</span>
|
||
<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="n">n_steps</span> <span class="o">=</span> <span class="mi">50</span>
|
||
<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="n">t</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
|
||
<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="n">true_x</span> <span class="o">=</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">+</span> <span class="mi">20</span>
|
||
<a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="n">true_y</span> <span class="o">=</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">+</span> <span class="mi">20</span>
|
||
<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a>
|
||
<a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a><span class="c1"># 带噪声的观测</span>
|
||
<a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
<a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a><span class="n">noise</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="p">(</span><span class="n">n_steps</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="o">*</span> <span class="mf">2.0</span>
|
||
<a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a><span class="n">obs_x</span> <span class="o">=</span> <span class="n">true_x</span> <span class="o">+</span> <span class="n">noise</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
|
||
<a id="__codelineno-1-41" name="__codelineno-1-41" href="#__codelineno-1-41"></a><span class="n">obs_y</span> <span class="o">=</span> <span class="n">true_y</span> <span class="o">+</span> <span class="n">noise</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span>
|
||
<a id="__codelineno-1-42" name="__codelineno-1-42" href="#__codelineno-1-42"></a>
|
||
<a id="__codelineno-1-43" name="__codelineno-1-43" href="#__codelineno-1-43"></a><span class="c1"># 运行卡尔曼滤波器</span>
|
||
<a id="__codelineno-1-44" name="__codelineno-1-44" href="#__codelineno-1-44"></a><span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">obs_x</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">obs_y</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span> <span class="c1"># 初始状态</span>
|
||
<a id="__codelineno-1-45" name="__codelineno-1-45" href="#__codelineno-1-45"></a><span class="n">P</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span> <span class="o">*</span> <span class="mf">10.0</span> <span class="c1"># 初始不确定性</span>
|
||
<a id="__codelineno-1-46" name="__codelineno-1-46" href="#__codelineno-1-46"></a>
|
||
<a id="__codelineno-1-47" name="__codelineno-1-47" href="#__codelineno-1-47"></a><span class="n">kalman_x</span><span class="p">,</span> <span class="n">kalman_y</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
|
||
<a id="__codelineno-1-48" name="__codelineno-1-48" href="#__codelineno-1-48"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_steps</span><span class="p">):</span>
|
||
<a id="__codelineno-1-49" name="__codelineno-1-49" href="#__codelineno-1-49"></a> <span class="n">x</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">kalman_predict</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span>
|
||
<a id="__codelineno-1-50" name="__codelineno-1-50" href="#__codelineno-1-50"></a> <span class="n">z</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">obs_x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">obs_y</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span>
|
||
<a id="__codelineno-1-51" name="__codelineno-1-51" href="#__codelineno-1-51"></a> <span class="n">x</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">kalman_update</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
|
||
<a id="__codelineno-1-52" name="__codelineno-1-52" href="#__codelineno-1-52"></a> <span class="n">kalman_x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||
<a id="__codelineno-1-53" name="__codelineno-1-53" href="#__codelineno-1-53"></a> <span class="n">kalman_y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||
<a id="__codelineno-1-54" name="__codelineno-1-54" href="#__codelineno-1-54"></a>
|
||
<a id="__codelineno-1-55" name="__codelineno-1-55" href="#__codelineno-1-55"></a><span class="n">kalman_x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">kalman_x</span><span class="p">)</span>
|
||
<a id="__codelineno-1-56" name="__codelineno-1-56" href="#__codelineno-1-56"></a><span class="n">kalman_y</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">kalman_y</span><span class="p">)</span>
|
||
<a id="__codelineno-1-57" name="__codelineno-1-57" href="#__codelineno-1-57"></a>
|
||
<a id="__codelineno-1-58" name="__codelineno-1-58" href="#__codelineno-1-58"></a><span class="c1"># 可视化</span>
|
||
<a id="__codelineno-1-59" name="__codelineno-1-59" href="#__codelineno-1-59"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
|
||
<a id="__codelineno-1-60" name="__codelineno-1-60" href="#__codelineno-1-60"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">true_x</span><span class="p">,</span> <span class="n">true_y</span><span class="p">,</span> <span class="s1">'k-'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'真实轨迹'</span><span class="p">)</span>
|
||
<a id="__codelineno-1-61" name="__codelineno-1-61" href="#__codelineno-1-61"></a><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">obs_x</span><span class="p">,</span> <span class="n">obs_y</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'#e74c3c'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'带噪声的观测'</span><span class="p">)</span>
|
||
<a id="__codelineno-1-62" name="__codelineno-1-62" href="#__codelineno-1-62"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">kalman_x</span><span class="p">,</span> <span class="n">kalman_y</span><span class="p">,</span> <span class="s1">'#3498db'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'卡尔曼滤波'</span><span class="p">)</span>
|
||
<a id="__codelineno-1-63" name="__codelineno-1-63" href="#__codelineno-1-63"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span> <span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
|
||
<a id="__codelineno-1-64" name="__codelineno-1-64" href="#__codelineno-1-64"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'卡尔曼滤波跟踪'</span><span class="p">)</span>
|
||
<a id="__codelineno-1-65" name="__codelineno-1-65" href="#__codelineno-1-65"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'y'</span><span class="p">)</span>
|
||
<a id="__codelineno-1-66" name="__codelineno-1-66" href="#__codelineno-1-66"></a><span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<a id="__codelineno-1-67" name="__codelineno-1-67" href="#__codelineno-1-67"></a>
|
||
<a id="__codelineno-1-68" name="__codelineno-1-68" href="#__codelineno-1-68"></a><span class="n">obs_error</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">((</span><span class="n">obs_x</span> <span class="o">-</span> <span class="n">true_x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">obs_y</span> <span class="o">-</span> <span class="n">true_y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
|
||
<a id="__codelineno-1-69" name="__codelineno-1-69" href="#__codelineno-1-69"></a><span class="n">kalman_error</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">((</span><span class="n">kalman_x</span> <span class="o">-</span> <span class="n">true_x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">kalman_y</span> <span class="o">-</span> <span class="n">true_y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
|
||
<a id="__codelineno-1-70" name="__codelineno-1-70" href="#__codelineno-1-70"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"观测RMSE: </span><span class="si">{</span><span class="n">obs_error</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-71" name="__codelineno-1-71" href="#__codelineno-1-71"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"卡尔曼滤波RMSE: </span><span class="si">{</span><span class="n">kalman_error</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-72" name="__codelineno-1-72" href="#__codelineno-1-72"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"误差降低: </span><span class="si">{</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">kalman_error</span><span class="o">/</span><span class="n">obs_error</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">100</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2">%"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>实现一个简化的NeRF风格体渲染管线。通过一个简单的3D场景（已知颜色和密度的球体）投射射线，并沿每条射线积分来渲染图像。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="k">def</span><span class="w"> </span><span class="nf">render_ray</span><span class="p">(</span><span class="n">origin</span><span class="p">,</span> <span class="n">direction</span><span class="p">,</span> <span class="n">spheres</span><span class="p">,</span> <span class="n">n_samples</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span> <span class="n">t_near</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">t_far</span><span class="o">=</span><span class="mf">6.0</span><span class="p">):</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w"> </span><span class="sd">"""穿过球体场景对单条射线进行体渲染。"""</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a> <span class="n">t_vals</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">t_near</span><span class="p">,</span> <span class="n">t_far</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">)</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="n">deltas</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">jnp</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">t_vals</span><span class="p">),</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1e-3</span><span class="p">])])</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="n">colour</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="n">transmittance</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
|
||
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="n">point</span> <span class="o">=</span> <span class="n">origin</span> <span class="o">+</span> <span class="n">t_vals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">direction</span>
|
||
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a>
|
||
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a> <span class="c1"># 计算该点的密度和颜色</span>
|
||
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a> <span class="n">density</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a> <span class="n">point_colour</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
|
||
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a>
|
||
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a> <span class="k">for</span> <span class="n">center</span><span class="p">,</span> <span class="n">radius</span><span class="p">,</span> <span class="n">col</span><span class="p">,</span> <span class="n">sigma</span> <span class="ow">in</span> <span class="n">spheres</span><span class="p">:</span>
|
||
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="n">dist</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">point</span> <span class="o">-</span> <span class="n">center</span><span class="p">)</span>
|
||
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a> <span class="c1"># 软球体：密度随距表面的距离指数衰减</span>
|
||
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a> <span class="n">d</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">jnp</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dist</span> <span class="o">-</span> <span class="n">radius</span><span class="p">)</span> <span class="o">*</span> <span class="n">sigma</span><span class="p">)</span> <span class="o">*</span> <span class="n">sigma</span>
|
||
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a> <span class="n">density</span> <span class="o">+=</span> <span class="n">d</span>
|
||
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a> <span class="n">point_colour</span> <span class="o">+=</span> <span class="n">d</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">col</span><span class="p">)</span>
|
||
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a>
|
||
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a> <span class="c1"># 按总密度归一化颜色</span>
|
||
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a> <span class="n">point_colour</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">density</span> <span class="o">></span> <span class="mf">1e-6</span><span class="p">,</span> <span class="n">point_colour</span> <span class="o">/</span> <span class="n">density</span><span class="p">,</span> <span class="n">point_colour</span><span class="p">)</span>
|
||
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a>
|
||
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a> <span class="c1"># 体渲染方程</span>
|
||
<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a> <span class="n">alpha</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">-</span> <span class="n">jnp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">density</span> <span class="o">*</span> <span class="n">deltas</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
||
<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a> <span class="n">colour</span> <span class="o">+=</span> <span class="n">transmittance</span> <span class="o">*</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">point_colour</span>
|
||
<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a> <span class="n">transmittance</span> <span class="o">*=</span> <span class="p">(</span><span class="mf">1.0</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">)</span>
|
||
<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a>
|
||
<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a> <span class="k">return</span> <span class="n">colour</span>
|
||
<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a>
|
||
<a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="c1"># 场景：三个彩色球体</span>
|
||
<a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a><span class="n">spheres</span> <span class="o">=</span> <span class="p">[</span>
|
||
<a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">4.0</span><span class="p">]),</span> <span class="mf">0.8</span><span class="p">,</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span> <span class="mf">5.0</span><span class="p">),</span> <span class="c1"># 红色</span>
|
||
<a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">]),</span> <span class="mf">0.6</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span> <span class="mf">5.0</span><span class="p">),</span> <span class="c1"># 绿色</span>
|
||
<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">]),</span> <span class="mf">0.5</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">],</span> <span class="mf">5.0</span><span class="p">),</span> <span class="c1"># 蓝色</span>
|
||
<a id="__codelineno-2-42" name="__codelineno-2-42" href="#__codelineno-2-42"></a><span class="p">]</span>
|
||
<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a>
|
||
<a id="__codelineno-2-44" name="__codelineno-2-44" href="#__codelineno-2-44"></a><span class="c1"># 相机设置</span>
|
||
<a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a><span class="n">img_h</span><span class="p">,</span> <span class="n">img_w</span> <span class="o">=</span> <span class="mi">64</span><span class="p">,</span> <span class="mi">64</span>
|
||
<a id="__codelineno-2-46" name="__codelineno-2-46" href="#__codelineno-2-46"></a><span class="n">focal</span> <span class="o">=</span> <span class="mf">60.0</span>
|
||
<a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a><span class="n">origin</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
|
||
<a id="__codelineno-2-48" name="__codelineno-2-48" href="#__codelineno-2-48"></a>
|
||
<a id="__codelineno-2-49" name="__codelineno-2-49" href="#__codelineno-2-49"></a><span class="n">image</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">img_h</span><span class="p">,</span> <span class="n">img_w</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
|
||
<a id="__codelineno-2-50" name="__codelineno-2-50" href="#__codelineno-2-50"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">img_h</span><span class="p">):</span>
|
||
<a id="__codelineno-2-51" name="__codelineno-2-51" href="#__codelineno-2-51"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">img_w</span><span class="p">):</span>
|
||
<a id="__codelineno-2-52" name="__codelineno-2-52" href="#__codelineno-2-52"></a> <span class="c1"># 计算射线方向</span>
|
||
<a id="__codelineno-2-53" name="__codelineno-2-53" href="#__codelineno-2-53"></a> <span class="n">px</span> <span class="o">=</span> <span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">img_w</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">focal</span>
|
||
<a id="__codelineno-2-54" name="__codelineno-2-54" href="#__codelineno-2-54"></a> <span class="n">py</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">img_h</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">focal</span>
|
||
<a id="__codelineno-2-55" name="__codelineno-2-55" href="#__codelineno-2-55"></a> <span class="n">direction</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">px</span><span class="p">,</span> <span class="n">py</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
|
||
<a id="__codelineno-2-56" name="__codelineno-2-56" href="#__codelineno-2-56"></a> <span class="n">direction</span> <span class="o">=</span> <span class="n">direction</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">direction</span><span class="p">)</span>
|
||
<a id="__codelineno-2-57" name="__codelineno-2-57" href="#__codelineno-2-57"></a>
|
||
<a id="__codelineno-2-58" name="__codelineno-2-58" href="#__codelineno-2-58"></a> <span class="n">colour</span> <span class="o">=</span> <span class="n">render_ray</span><span class="p">(</span><span class="n">origin</span><span class="p">,</span> <span class="n">direction</span><span class="p">,</span> <span class="n">spheres</span><span class="p">)</span>
|
||
<a id="__codelineno-2-59" name="__codelineno-2-59" href="#__codelineno-2-59"></a> <span class="n">image</span> <span class="o">=</span> <span class="n">image</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="n">colour</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
|
||
<a id="__codelineno-2-60" name="__codelineno-2-60" href="#__codelineno-2-60"></a>
|
||
<a id="__codelineno-2-61" name="__codelineno-2-61" href="#__codelineno-2-61"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
|
||
<a id="__codelineno-2-62" name="__codelineno-2-62" href="#__codelineno-2-62"></a><span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image</span><span class="p">)</span>
|
||
<a id="__codelineno-2-63" name="__codelineno-2-63" href="#__codelineno-2-63"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'NeRF风格体渲染</span><span class="se">\n</span><span class="s1">(3个球体)'</span><span class="p">)</span>
|
||
<a id="__codelineno-2-64" name="__codelineno-2-64" href="#__codelineno-2-64"></a><span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'off'</span><span class="p">)</span>
|
||
<a id="__codelineno-2-65" name="__codelineno-2-65" href="#__codelineno-2-65"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span> <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<a id="__codelineno-2-66" name="__codelineno-2-66" href="#__codelineno-2-66"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"图像形状: </span><span class="si">{</span><span class="n">image</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-67" name="__codelineno-2-67" href="#__codelineno-2-67"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"渲染了 </span><span class="si">{</span><span class="n">img_h</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">img_w</span><span class="si">}</span><span class="s2"> 条射线，每条 </span><span class="si">{</span><span class="mi">64</span><span class="si">}</span><span class="s2"> 个采样点"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
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