图1 本文对控制变量技术在渲染领域中应用的分类梳理

Fig. 1 A classification of the applications of control variate techniques in rendering

图2 在超过5 000个灯光和高遮挡的场景中，文献[11]的直接照明采样方法的不同组件的等时比较

Fig. 2 Equal-time comparison of different components of reference [11] direct illumination sampling method in a scene with over 5 000 lights and high occlusion ((a) Proportional; (b) Scalable; (c) ML (overfitting); (d) MAP for mean; (e) d + MAP for variance; (f) e + regression; (g) f + visibility model; (h) g + control variate)

图3 漫反射毛发状纤维渲染的比较((a)使用控制变量；(b)不使用控制变量)[15]

Fig. 3 Comparison of diffuse hair-like fiber rendering ((a) With control variates; (b) Without control variates)[15]

图4 毛皮生产渲染中使用球谐控制变量采样方法结果[15]

Fig. 4 Results of using spherical harmonic control variate sampling in fur production rendering[15]

图5 2种光源产生的出射辐射((a) 多边形光源; (b) 透过窗口限定进入的环境光) [20]

Fig. 5 Outgoing radiance produced by twotypes of light ((a) Polygon light; (b) Portal-masked environment)[20]

图6 经过训练的光选择技术(Trained)和不同防御性光选择技术组合的等样本比较[22]

Fig. 6 Equal-sample comparison of trained light selection (Trained) and different combinations of defensive light selection techniques ((a) Uniform; (b) NoMax)[22]

图7 在多光源选择应用中的等时渲染比较

Fig. 7 Equal-time comparison in many-light selection application ((a) Reference [23]; (b) Reference [11])

图8 基于路径跟踪的渐进式分层求解器的示意图[27]

Fig. 8 Illustration of a progressive hierarchical solver based on path tracing[27]

图9 辐射度估计的单颜色通道值重建结果((a) 存储在方向网格中; (b) B样条重建结果)[29]

Fig. 9 Reconstruction results of single-color-channel radiance estimates ((a) Stored in the directional grid; (b) Reconstructed using B-splines)[29]

图10 CRESPO等[32]方法在一维积分问题上的流程示意图((a) 原始被积函数和PDF；(b) 将被积函数映射到主空间；(c) 选择误差最大区域；(d) 细分与更新；(e) 最终分段控制变量；(f) 蒙特长洛采样残差)

Fig. 10 Flowchart of CRESPO et al.’s [32] method for a one-dimensional integration problem ((a) Original integrand and PDF; (b) Map integrand to primary space; (c) Select region with highest error; (d) Subdivide and update; (e) Final piecewise control variate; (f) Monte Carlo sample of the residual)

图11 SALAüN等[33]方法和MC方法在一维积分问题上的流程对比示意图((a) 原被积函数；(b) 采样；(c) 常数函数拟合；(d) 解析积分；(e) 多项式函数拟合；(f) 求和)

Fig. 11 Comparison flowchart of SALAüN et al.’s[33] method and the MC method for a one-dimensional integration problem ((a) Original integrand; (b) Sampling; (c) Constant function fitting; (d) Analytic integration; (e) Polynomial function fitting; (f) Summation)

图12 文献[35]示意图((a) 神经控制变量；(b) 残差神经拟合；(c) 启发式终止

Fig. 12 Figure from reference [35] ((a) Neural control variate; (b) Residual neural fitting; (c) Heuristic termination)

图13 光线进入介质后的4个过程((a) 吸收; (b) 向外散射; (c) 向内散射; (d) 自发光)

Fig. 13 Four processes after ray enters medium ((a) Absorption; (b) Out-scattering; (c) In-scattering; (d) Emission)

图14 虚拟介质粒子会改变密度,但不会改变辐射率和光线的方向[41]

Fig. 14 Virtual medium particles alter the density but do not change the radiance or the direction of rays[41]

图15 文献[42]示意图((a) 原始介质；(b) 均匀的控制分量；(c) 异质的残差分量)

Fig. 15 Figure from reference [42] ((a) Original medium;(b) Homogeneous control component; (c) Heterogeneous residual component)

图16 分解跟踪方法可以在异质介质中构建部分自由路径(黄色段)，而无需评估空间变化的碰撞参数[42]

Fig. 16 The decomposed tracking method constructs partially free paths (yellow segments) in heterogeneous media without evaluating spatially varying collision parameters[42]

图17 文献[44]应用控制变量示意图((a) 原消光系数; (b) 控制消光系数; (c) 残差消光系数)

Fig. 17 Illustration of the application of control variates in reference[44] ((a) Original extinction coefficient; (b) Control extinction coefficient; (c) Residual extinction coefficient)

图18 文献[47]通过截断幂级数进行透射率估计(此处以采样阶数N=3)

Fig. 18 Reference [47] estimate transmittance by truncating the power series (here with order N=3)

图19 不同方法在程序化生成的云场景下的性能对比

Fig. 19 Performance comparison of different methods in procedurally generated cloud scenes ((a) Reference [49]; (b) Reference [44]; (c) Reference [42]; (d) Reference [40])

图20 文献[51]设计的在线自适应采样框架(包含采样数估计、CV系数估计、估计历史更新3个模块)

Fig. 20 The online adaptive sampling framework designed by reference [51] (Including three modules: sample count estimation, control variate coefficient estimation, and estimation history update)

图21 递归控制变量在优化过程中动态权重变化的可视化效果[54]

Fig. 21 Visualization of dynamic weight changes of recursive control variates during the optimization process[54]