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摘要: 模糊认知图(Fuzzy cognitive map, FCM)是建立在认知图和模糊集理论上的一类代表性的软计算理论, 兼具神经网络和模糊决策两者的优势, 已成功地应用于复杂系统建模和时间序列分析等众多领域. 学习权重矩阵是基于模糊认知图建模的首要任务, 是模糊认知图研究领域的焦点. 针对这一核心问题, 首先, 全面综述模糊认知图的基本理论框架, 系统地总结近年来模糊认知图的拓展模型. 其次, 归纳、总结和分析模糊认知图学习算法的最新研究进展, 对学习算法进行重新定义和划分, 深度阐述各类学习算法的时间复杂度和优缺点. 然后, 对比分析各类学习算法在不同科学领域的应用特点以及现有的模糊认知图建模软件工具. 最后, 讨论学习算法未来潜在的研究方向和发展趋势.Abstract: Fuzzy cognitive maps (FCM) are a representative soft computing theory based on cognitive maps and fuzzy set theory. They combine the advantages of both neural networks and fuzzy decision-making and have been successfully applied in many fields, including complex system modeling and time series analysis. Learning the weight matrix is the primary task of modeling based on fuzzy cognitive maps and is the focus of research in this field. To address this core issue, we first comprehensively review the basic theoretical framework of fuzzy cognitive maps and systematically summarize the extended models developed in recent years. Next, the most recent advancements in fuzzy cognitive map learning algorithms are reviewed, analyzed, and summarized. The algorithms are redefined and categorized, with a detailed exploration of their time complexity, strengths, and weaknesses. Additionally, the application properties of various learning algorithms in various scientific domains are also compared and analyzed in this research, along with the software tools that are now available for creating fuzzy cognitive maps. Finally, potential research directions and development trends for learning algorithms are discussed.
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随着人工智能和传感器技术的发展, 人体姿态估计(Human pose estimation, HPE)逐渐应用于各种不同的领域, 如人机交互、运动捕捉[1-2]、虚拟替身、康复训练[3]、自动驾驶、视频监控和运动表现分析等[4-6]. 然而, 受视觉遮挡等因素影响[7], 这将增加视觉人体姿态估计中腕、肘等人体部件误识别的风险, 从而导致量测不确定性的存在, 而多视觉融合方法是处理视觉遮挡下HPE的主流方法之一[8-11].
针对多视觉融合估计问题, 文献[8]提出一种面向人体关节点位置信息的可靠性判别方法, 通过调节加权观测融合中的量测融合权重, 以提高HPE的鲁棒性. 而在文献[12]中, 将多视觉下的融合估计问题转换为优化问题, 利用骨骼长度作为约束条件, 并基于关节点位置信息的可靠性, 来调整优化过程中的权重大小, 从而减小视觉遮挡时的人体骨架抖动. 然而, 在求目标函数的过程中, 该方法易受初始数据的影响. 针对基于多视角融合的HPE问题, 文献[9]首次提出信息加权一致性滤波器(Information weighted consensus filter, IWCF), 通过平均一致性(Average consensus)[13]来获得邻近节点的信息. 同时, 使用Metropolis权重来提高IWCF的收敛速度, 实验证明融合后的人体姿态信息可获得更高的动作识别精度. 之后, 针对多视觉HPE中各传感器节点估计误差引起的关节点波动问题, 文献[10]将IWCF与交互式多模型(Interacting multiple model-based, IMM)相结合, 获得混合恒定速度(Constant velocity, CV)、恒定加速度(Constant acceleration, CA)和Singer运动等多模型下的姿态估计, 从而减小视觉遮挡的影响以及提高估计的精度.
另一方面, 针对人体姿态量测存在的噪声问题, 卡尔曼滤波(Kalman filtering, KF)[14]是一种有效的去噪方法. 其不仅在目标跟踪领域应用广泛[15-17], 而且在人体姿态估计领域也发挥重要的作用[18-20]. 例如, 文献[19]利用卡尔曼滤波器提高人体姿态估计的准确性. 针对人体姿态量测噪声统计特性的难以精确描述问题, 文献[20]提出一种基于鲁棒卡尔曼滤波的HPE方法, 利用假设检验对视觉遮挡下的复杂噪声进行识别, 并引入自适应因子来对量测噪声协方差进行调整, 从而减小量测不确定性对滤波器性能的影响. 此外, 针对量测信息缺失的目标跟踪问题, 文献[21]同样利用假设检验对量测信息进行有效筛选, 并利用渐进滤波方法来处理量测信息缺失造成的误差增大问题, 从而提高滤波器的鲁棒性. 针对渐进滤波对量测不确定性补偿的问题, 文献[22-23]提出带自适应量测更新的渐进高斯滤波方法, 给出渐进量测更新的终止条件. 这不仅有利于计算效率, 而且提高了对量测不确定性的自适应能力. 然而, 针对视觉遮挡造成量测噪声的复杂性, 现有估计方法并未充分考虑到局部量测不确定性的差异. 同时, 基于假设检验的方法局限于单一维度对量测进行筛选, 没有充分考虑到先验信息和局部量测以及不同局部量测之间的相容性问题.
为此, 本文构建分布式的渐进贝叶斯滤波融合框架, 提出基于渐进高斯滤波融合的人体姿态估计方法. 针对量测信息包含的复杂噪声, 设计分层性能评估方法, 从空间维度到时间维度对量测进行分类处理. 为解决量测不确定性下的融合估计问题, 设计一种分层分类的融合估计方法. 特别地, 针对量测统计特性变化问题, 引入渐进滤波方法, 利用局部估计间的交互信息来引导渐进量测更新, 从而隐式地补偿量测不确定性. 最后, 仿真与实验结果表明, 相比于现有的方法, 提高了人体姿态估计的准确性和鲁棒性.
1. 问题描述
如图1所示, 考虑一类多视觉融合环境下的人体姿态估计系统, 其中, 视觉传感器为深度相机, 用于采集人体目标的深度信息. 本文将人体目标视为由头、躯干、臂、手、腿、足等部件相互连接构成的多刚体系统. 这样, 人体姿态估计问题可看作人体各关节点位置估计问题. 首先, 利用卷积神经网络(Convolutional neural network, CNN) 的方法[24]从图像中识别出人体各部件, 并计算出人体各关节点在各个相机坐标系下的3D位置; 其次, 通过棋盘格标定法可获得相机坐标系相对于世界坐标系(即, 棋盘格)的旋转矩阵$ {R^{{c_i}2w}} $和平移向量$ {\boldsymbol{t}}^{{c_i}2w} $, 从而将在不同相机坐标系下检测的3D关节点统一到世界坐标系. 同时, 对人体运动建模如下:
$$ {{\boldsymbol{x}}_k} = {F_k}{{\boldsymbol{x}}_{k - 1}} + {{\boldsymbol{w}}_k} $$ (1) 其中, $k=1, 2,\cdots$ 表示离散时间序列, ${{\boldsymbol{x}}_k} = [{{( {{{\boldsymbol{x}}_{k,1}}} )}^\text{T}} \;\; \cdots\;\; {{{( {{{\boldsymbol{x}}_{k,L}}} )}{}^{\rm{T}}}} ]{}^{\rm{T}}$表示$ {k} $时刻人体姿态的状态, $ {{\boldsymbol{x}}_{k,l}} $表示关节点$ l $状态, $ l = 1, \cdots , L $, $ {L} $为选取的人体关节数量; $ {F_k} = {\rm{diag}}\{ {{F_{k,1}}}\;\; \cdots \;\; {{F_{k,L}}}\} $表示状态转移矩阵; 过程噪声${{\boldsymbol{w}}_k} = {[ {{{( {{{\boldsymbol{w}}_{k,1}}} )}^\text{T}}}\;\; \cdots \;\; {{{( {{{\boldsymbol{w}}_{k,L}}} )}{}^\text{T}}} ]{}^\text{T}}$服从零均值高斯分布, 其方差为$ {\mathop{\rm{cov}}} ( {{{\boldsymbol{w}}_k}} ) = {Q_k} $. 最后, 在此基础上, 将融合运动模型和单视觉量测信息形成人体姿态的局部估计, 进而融合各局部估计形成人体姿态的全局估计. 注意到视觉遮挡程度的不同, 将给人体关节点的检测与测量带来不同程度的影响, 从而导致复杂的量测噪声.
因此, 对人体姿态量测建模如下:
$$ {\boldsymbol{z}}_k^i = H_k^i{{\boldsymbol{x}}_k} + {\boldsymbol{v}}_k^i + {\boldsymbol{\xi}}_k^i $$ (2) 其中, ${\boldsymbol{z}}_k^i = {[ {{{( {{\boldsymbol{z}}_{k,1}^i} )}{}^\text{T}} \;\cdots\; {{( {{\boldsymbol{z}}_{k,L}^i} )}{}^\text{T}}} ]^\text{T}}$表示传感器$ i $的量测值, $ i = 1, \cdots , N $, $ {N} $为传感器总数, ${\boldsymbol{z}}_{k,l}^i = [ {z_{x,l}^i} \;\;{z_{y,l}^i}\;\;{z_{z,l}^i} ]^\text{T}$ 表示关节点的位置量测信息, $z_{x,l}^i, z_{y,l}^i, z_{z,l}^i$分别为关节点$ l $在$ {x} $、$ {y} $和$ {z} $轴上的量测值. $H_k^i = [ {{{( {H_{k,1}^i} )}^\text{T}}} \;\; \cdots \;\;{{{( {H_{k,L}^i} )}{}^\text{T}}} ]{}^\text{T}$为量测矩阵; 量测噪声${\boldsymbol{v}}_k^i = {[ {{{( {{\boldsymbol{v}}_{k,1}^i} )}^\text{T}}}\;\; \cdots \;\;{{{( {{\boldsymbol{v}}_{k,L}^i} )}{}^\text{T}}} ]{}^\text{T}}$服从零均值高斯分布, 且其协方差为$ {\mathop{\rm{cov}}} ( {{\boldsymbol{v}}_k^i} ) = R_k^i $. $ {\boldsymbol{\xi}}_k^i = U_k^i {\boldsymbol{\alpha}}_k^i + b{\boldsymbol{\beta}}_k^i $用来描述不同遮挡程度影响下引起的量测噪声. 其中, $ U_k^i = \text{diag}\{{{\boldsymbol{u}}_{k,1}^i}\;\; \cdots \;\;{{\boldsymbol{u}}_{k,L}^i} \} $, $ {\boldsymbol{u}}_{k,l}^i $服从零均值且协方差为$ R_{k, + }^i $的高斯分布; $ b $为幅值较大的数值, $ {\boldsymbol{\alpha}}_k^i $和$ {\boldsymbol{\beta}}_k^i $为随机变量且分别服从参数为$ {y_1}\;( {0 < {y_1} < 1} ) $和$ {y_2}\;( {0 < {y_2} < 1} ) $的伯努利分布.
相应地, 针对量测信息包含的复杂噪声, 将对量测进行检测和分类处理, 从而剔除高程度视觉遮挡下的量测野值, 同时通过渐进滤波隐式地补偿低程度视觉遮挡下的量测.
注1. 针对视觉遮挡程度的不同, 本文将量测主要分为两类. 即: 1)低程度视觉遮挡下的量测, 例如, 人体双臂交叉引起的腕、肘等关节小面积视觉遮挡, 用$ U_k^i {\boldsymbol{\alpha}}_k^i $来描述该情形下的量测不确定性; 2)高程度视觉遮挡下的量测, 例如, 人体侧身时腕、肘等关节受背部大面积视觉遮挡, 用$ b{\boldsymbol{\beta}}_k^i $来描述这种情况下的量测野值.
2. 自适应渐进高斯滤波融合
不同程度的视觉遮挡将造成量测统计特性变化, 进而导致局部滤波器性能下降并最终影响融合结果. 因此, 分两步从空间维度和时间维度上分别对量测进行相容性检测来实现量测筛选和分类处理.
考虑多视觉传感器的坐标位置和感知范围不同, 可能导致量测具有不同的噪声特性与误差模型. 如图2所示, 首先, 在空间维度上检测不同量测间马氏距离的平方, 即
$$ \begin{split} \gamma \left( {{\boldsymbol{z}}_k^i, {\boldsymbol{z}}_k^j} \right) = {\left( {{\boldsymbol{z}}_k^i - {\boldsymbol{z}}_k^j} \right)^\text{T}}\Sigma _{zz}^{ - 1}\left( {{\boldsymbol{z}}_k^i - {\boldsymbol{z}}_k^j} \right) \end{split} $$ (3) 其中, $ \Sigma _{zz}^{ - 1} $表示$( {\boldsymbol{z}}_k^i - {\boldsymbol{z}}_k^j ) $的协方差矩阵. 若$ \gamma ({\boldsymbol{z}}_k^i,{\boldsymbol{z}}_k^j) $落在置信区间内, 即$ \gamma ({\boldsymbol{z}}_k^i,{\boldsymbol{z}}_k^j) < {\chi _n} $, 则表示量测相容, 即视为正常量测, 否则认为其中可能存在异常量测, 需进一步在时间维度上分析相容性, 即检测预测值与量测的马氏距离平方:
$$ \begin{split} \gamma ({\boldsymbol{z}}_k^j, H_k^f\hat {{\boldsymbol{x}}}_{k|k - 1}^f) = \, & {\left( {{\boldsymbol{z}}_k^j - H_k^f\hat {{\boldsymbol{x}}}_{k|k - 1}^f} \right)^\text{T}}\Sigma _{zx}^{ - 1} \;\times \\ &\left( {{\boldsymbol{z}}_k^j - H_k^f\hat {{\boldsymbol{x}}}_{k|k - 1}^f} \right) \end{split} $$ (4) 其中, $ \Sigma _{zx}^{ - 1} $表示$ ( {{\boldsymbol{z}}_k^j - H_k^f\hat {{\boldsymbol{x}}}_{k|k - 1}^f} ) $的协方差矩阵. 若$ \gamma ({\boldsymbol{z}}_k^j, H_k^f\hat {{\boldsymbol{x}}}_{k|k - 1}^f) $ 落在置信区间内, 即
$$\gamma ({\boldsymbol{z}}_k^j, H_k^f\hat {{\boldsymbol{x}}}_{k|k - 1}^f) < {\chi _a} $$ 则表示量测中存在额外干扰, 否则视为野值.
根据量测相容性的检测结果, 将量测${Z_k} = \{ {\boldsymbol{z}}_k^1, \cdots , {\boldsymbol{z}}_k^N \}$分为$ G_k^n $, $ G_k^a $, $ G_k^d $等3组. 正常量测集合表示为
$$ \begin{split} G_k^n = &\left\{{{\boldsymbol{z}}_k^j|\gamma ({\boldsymbol{z}}_k^j,{\boldsymbol{z}}_k^i) < {\chi _n},{\boldsymbol{z}}_k^j,{\boldsymbol{z}}_k^i \in {Z_k},}\right.\\ & \left.{{\boldsymbol{z}}_k^j \ne {\boldsymbol{z}}_k^i,j = 1,2, \cdots}\right\} \end{split} $$ (5) 低程度视觉遮挡下的量测集合表示为
$$ \begin{split} G_k^a =\, & \left\{ {{\boldsymbol{z}}_k^j|\gamma ({\boldsymbol{z}}_k^j,H_k^f\hat {{\boldsymbol{x}}}_{k|k - 1}^f) < {\chi _a},\gamma \left( {{\boldsymbol{z}}_k^j,{\boldsymbol{z}}_k^i} \right) \ge } \right. \\ &\left.{{\chi _n},{\boldsymbol{z}}_k^j,{\boldsymbol{z}}_k^i \in {Z_k},{\boldsymbol{z}}_k^j \ne {\boldsymbol{z}}_k^i,j = 1,2, \cdots} \right\} \end{split} $$ (6) 集合$ G_k^d = {Z_k} - G_k^n - G_k^a $表示高程度视觉遮挡下的量测野值. $ {\chi_n} $, $ {\chi_a} $为置信区间, $ \hat {{\boldsymbol{x}}}_{k|k - 1}^f $为全局状态预测. 量测分组后, 得到不同视觉遮挡下的量测$ {\boldsymbol{z}}_k^{{n_j}} $, $ {\boldsymbol{z}}_k^{{a_j}} $, $ {\boldsymbol{z}}_k^{{d_j}} $, 其中${\boldsymbol{z}}_k^{{n_j}} \in G_k^n,{\boldsymbol{z}}_k^{{a_j}} \in G_k^a,{\boldsymbol{z}}_k^{{d_j}} \in G_k^d$.
本文方法框图如图3所示, 首先, 通过分层性能评估对量测进行分层和分类; 其次, 在局部估计中, 将拒绝量测野值$ {\boldsymbol{z}}_k^{{d_j}} $, 以避免量测野值对系统滤波性能造成较大的负面影响. 特别地, 在量测$ {\boldsymbol{z}}_k^{{a_j}} $更新过程中, 将渐进地引入量测信息对当前局部状态进行补偿, 即通过多次量测迭代得到相应补偿下的后验状态, 并通过局部估计间的交互信息来引导渐进量测更新. 最后, 融合人体姿态的各局部估计形成全局估计. 为此, 构建分布式渐进贝叶斯滤波融合框架如下.
1)人体姿态全局估计
$$ p({{\boldsymbol{x}}_k}|{Z_{1:k - 1}}) = \int {p({{\boldsymbol{x}}_k}|{{\boldsymbol{x}}_{k - 1}})p({{\boldsymbol{x}}_{k - 1}}|{Z_{1:k - 1}})\text{d}{{\boldsymbol{x}}_{k - 1}}} $$ (7) $$ \begin{split} & p({{\boldsymbol{x}}_k}|{Z_{1:k}}) = p({{\boldsymbol{x}}_k}|{Z_{1:k - 1}})\;\times\\ & \quad \frac{{\prod\limits_{{\boldsymbol{z}}_k^{{n_j}} \in G_k^n} {p\left( {{{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k}^{{n_j}}} \right)} \prod\limits_{{\boldsymbol{z}}_k^{{a_j}} \in G_k^a} {p\left( {{{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k}^{{a_j}}} \right)} }}{{\prod\limits_{{\boldsymbol{z}}_k^{{n_j}} \in G_k^n} {p\left( {{{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k - 1}^{{n_j}}} \right)} \prod\limits_{{\boldsymbol{z}}_k^{{a_j}} \in G_k^a} {p\left( {{{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k - 1}^{{a_j}}} \right)} }} \end{split}$$ (8) 2)人体姿态局部估计
$$ p({{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k - 1}^i) = \int {p({{\boldsymbol{x}}_k}|{{\boldsymbol{x}}_{k - 1}})p({{\boldsymbol{x}}_{k - 1}}|{\boldsymbol{z}}_{1:k - 1}^i)\text{d}{{\boldsymbol{x}}_{k - 1}}} $$ (9) $$ p({{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k}^i) = \frac{{p({{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k - 1}^i)p({\boldsymbol{z}}_k^i|{{\boldsymbol{x}}_k})}}{{p({\boldsymbol{z}}_k^i|{\boldsymbol{z}}_{1:k - 1}^i)}} $$ (10) 对于量测$ {\boldsymbol{z}}_k^{{n_j}} \in G_k^n $, 采用卡尔曼滤波方法得到人体姿态局部估计; 而对$ {\boldsymbol{z}}_k^{{a_j}} \in G_k^a $, 则采用渐进高斯滤波(Progressive Gaussian filtering, PGF)方法对量测不确定性进行隐式补偿. 可将量测分解为多个伪量测的集成, 即
$$ \begin{align} &R_k^{{a_j}} = {\left[ {\sum\limits_{m = 1}^M {{{\left( {R_{k, {\lambda _m}}^{{a_j}}} \right)}^{ - 1}}} } \right]^{ - 1}} \end{align} $$ (11) $$ \begin{align} &{\boldsymbol{z}}_k^{{a_j}} = {\left[ {\sum\limits_{m = 1}^M {{{\left( {R_{k, {\lambda _m}}^{{a_j}}} \right)}^{ - 1}}} } \right]^{ - 1}}\sum\limits_{m = 1}^M {\left[ {{{\left( {R_{k, {\lambda _m}}^{{a_j}}} \right)}^{ - 1}}{\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}}} \right]} \end{align} $$ (12) 利用量测迭代更新, 渐进地引入量测信息. 其中$ {\lambda _m} $表示伪时间, 且满足
$$\left\{ \begin{aligned} &{\Delta _m} = {\lambda _m} - {\lambda _{m - 1}}\\ &{\Delta _m} > 0 \\ & \sum\limits_{m = 1}^M {{\Delta _m} = 1} \end{aligned}\right. $$ (13) 其中, $ {\lambda _0} = 0 $, $ m = 1, \cdots , M $, $ M $为总渐进步数, $ {\Delta _m} $表示渐进步长, ${\boldsymbol{z}}_{k, {\lambda _1}:{\lambda _M}}^{{a_j}} = \{ {{\boldsymbol{z}}_{k, {\lambda _1}}^{{a_j}}, \cdots, {\boldsymbol{z}}_{k, {\lambda _M}}^{{a_j}}} \}$表示整个渐进过程中的伪量测, $ {\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}} $表示第$ m $步的伪量测. 在不考虑视觉遮挡所引起的量测不确定性时, $ p\left( {{\boldsymbol{z}}_k^{{a_j}}|{{\boldsymbol{x}}_k}} \right) $可以表示为
$$ \begin{split} & p\left( {{\boldsymbol{z}}_k^{{a_j}}|{{\boldsymbol{x}}_k}} \right) =\\ &\;\;\; \frac{1}{{\sqrt {2\pi \left| {R_k^{{a_j}}} \right|} }}\exp \Bigg[ { - \frac{1}{2}{{\left( {{\boldsymbol{z}}_k^{{a_j}} - H_k^{{a_j}}{{\boldsymbol{x}}_k}} \right)}^\text{T}}} \;\times \\ &\;\;\; {{{\left( {R_k^{{a_j}}} \right)}^{ - 1}}\left( {{\boldsymbol{z}}_k^{{a_j}} - H_k^{{a_j}}{{\boldsymbol{x}}_k}} \right)} \Bigg]= \\ &\;\;\; \frac{1}{{\sqrt {2\pi \left| {R_k^{{a_j}}} \right|} }}\prod\limits_{m = 1}^M {\frac{1}{{{{\left( {\sqrt {2\pi \left| {\frac{{R_k^{{a_j}}}}{{{\Delta _m}}}} \right|} } \right)}^{ - 1}}}}\frac{1}{{\sqrt {2\pi \left| {\frac{{R_k^{{a_j}}}}{{{\Delta _m}}}} \right|} }}} \;\times \\ &\;\;\; \exp \Bigg[ - \frac{1}{2}{\left( {{\boldsymbol{z}}_k^{{a_j}} - H_k^{{a_j}}{{\boldsymbol{x}}_k}} \right)^\text{T}}\times\Bigg. \\ &\;\;\; \Bigg.{\left( {\frac{{R_k^{{a_j}}}}{{{\Delta _m}}}} \right)^{ - 1}}\left( {{\boldsymbol{z}}_k^{{a_j}} - H_k^{{a_j}}{{\boldsymbol{x}}_k}} \right)\Bigg]\\[-15pt] \end{split} $$ (14) 因此, $p( {{\boldsymbol{z}}_k^{{a_j}}|{{\boldsymbol{x}}_k}} ) = c_k^{{a_j}}\prod\nolimits_{m = 1}^M {p( {{\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}}|{{\boldsymbol{x}}_k}} )},$ 其中,
$$ \begin{split} &p({\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}}|{{\boldsymbol{x}}_k}) = {\left(\sqrt {2\pi \left| {R_{k, {\lambda _m}}^{{a_j}}} \right|} \right)^{ - 1}}\;\times\\ & \qquad\exp \Bigg[ - \frac{1}{2}{({\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}} - H_k^{{a_j}}{{\boldsymbol{x}}_k})^\text{T}}\;\times \Bigg. \\ & \qquad \Bigg.{(R_{k, {\lambda _m}}^{{a_j}})^{ - 1}}({\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}} - H_k^{{a_j}}{{\boldsymbol{x}}_k})\Bigg] \end{split} $$ (15) $R_{k, {\lambda _m}}^{{a_j}} = \frac{{R_k^{{a_j}}}}{{{\Delta _m}}}$, $ {\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}} = {\boldsymbol{z}}_k^{{a_j}} $, 归一化因子为
$$ \begin{split} c_k^{{a_j}} = {\left( {\sqrt {2\pi \left| {R_k^{{a_j}}} \right|} } \right)^{ - 1}}\prod\limits_{m = 1}^M {\sqrt {2\pi \left| {\frac{{R_k^{{a_j}}}}{{{\Delta _m}}}} \right|} } \end{split} $$ (16) 对于$ G_k^a $中的量测, 其局部后验分布可进一步描述为
$$ \begin{split} p({{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k}^{{a_j}}) = \frac{{p({{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k - 1}^{{a_j}})\prod\limits_{m = 1}^M {p\left( {{\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}}|{{\boldsymbol{x}}_k}} \right)} }}{{{{\left( {c_k^{{a_j}}} \right)}^{ - 1}}p({\boldsymbol{z}}_k^{{a_j}}|{\boldsymbol{z}}_{1:k - 1}^{{a_j}})}} \end{split} $$ (17) 同时, 在其渐进量测更新过程中, 引入$ G_k^n $中量测作为参照量来引导其渐进迭代, 令
$$ \begin{split} {\varphi _{{\lambda _m}}} =\, & \gamma \left( {{\boldsymbol{z}}_k^{{n_j}},H_k^{{a_j}}\hat {{\boldsymbol{x}}}_{k|k, {\lambda _m}}^{{a_j}}} \right)-\\ &\gamma \left( {{\boldsymbol{z}}_k^{{n_j}},H_k^{{a_j}}\hat {{\boldsymbol{x}}}_{k|k, {\lambda _{m - 1}}}^{{a_j}}} \right) \end{split} $$ (18) $ {\varphi _{{\lambda _m}}} $表示在渐进量测更新前后的估计值与参照量间马氏距离的差值, 当$ {\varphi _{{\lambda _m}}} \ge 0 $时停止$ p({{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k}^{{a_j}}) $中的渐进量测更新, 从而对量测不确定性隐式地补偿, 即通过$ {\varphi _{{\lambda _m}}} $值来判断是否继续引入伪量测$ {\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}} $来渐进迭代和逐步修正状态估计, 而无需显式地将不确定性(如协方差矩阵)作为输入. 结合式(17)和式(18), 量测渐进更新过程中的后验概率密度函数(Probability density function, PDF)可以表示为
$$ \begin{split} &p\left({{\boldsymbol{x}}_k}, {\lambda _m}|{\boldsymbol{z}}_{1:k - 1}^{{a_j}}, {\boldsymbol{z}}_{k, {\lambda _1}:{\lambda _m}}^{{a_j}}\right) = \\ & \qquad {\eta _{k, {\lambda _m}}}p\left({{\boldsymbol{x}}_k}, {\lambda _1}|{\boldsymbol{z}}_{1:k - 1}^{{a_j}}, {\boldsymbol{z}}_{k, {\lambda _1}}^{{a_j}}\right)\times \\ & \qquad\prod\limits_{m = 2}^{{\varphi _{{\lambda _m}}} < 0} {p\left({\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}}|{{\boldsymbol{x}}_k}\right)} \end{split} $$ (19) 其中, $ {\eta _{k, {\lambda _m}}} $表示归一化因子.
注2. 在量测渐进更新过程中, $ {\varphi _{{\lambda _m}}} < 0 $表示估计值与参照量之间马氏距离的趋势减小, 即所修正的状态估计有效; 由式(14)可知, 渐进量测集成的等效协方差$ \bar R_k^{{a_j}} = {( {\Delta _1}+{\sum\nolimits_{m = 2}^{{\varphi _{{\lambda _m}}} < 0} {{\Delta _m}} } )^{ - 1}}R_k^{{a_j}} $, $ m = 2 $表示至少渐进一步(将简化为$ {\sum\nolimits_{m = 1}^{{\varphi _{{\lambda _m}}} < 0}} $). 通过控制量测渐进更新的步长从而自适应调整其协方差, 将量测不确定性的补偿问题转换为对量测渐进更新的步长控制问题.
令$ k-1 $时刻的局部估计和全局估计均为高斯分布, 即$p({{\boldsymbol{x}}_{k - 1}}|{\boldsymbol{z}}_{1:k - 1}^i) = \text{N}( {{{\boldsymbol{x}}_{k - 1}};\hat {{\boldsymbol{x}}}_{k - 1|k - 1}^i, P_{k-1|k-1}^i} ),$ $ p({{\boldsymbol{x}}_{k - 1}}|{Z_{1:k - 1}}) = \text{N}( {{{\boldsymbol{x}}_{k - 1}};\hat {{\boldsymbol{x}}}_{k - 1|k - 1}^f, P_{k-1|k-1}^f} ) $, 由状态方程(1)可得$ p({{\boldsymbol{x}}_k}|{{\boldsymbol{x}}_{k - 1}}) $ = $\text{N}( {{{\boldsymbol{x}}_k};{F_k}{{\boldsymbol{x}}_{k - 1}}, {Q_k}} ),$ 则由$ k $时刻的状态预测分布易知
$$ \begin{split} p\left( {{{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k - 1}^i} \right) = \text{N}\left( {{{\boldsymbol{x}}_k};\hat {{\boldsymbol{x}}}_{k|k - 1}^i, P_{k|k - 1}^i} \right) \end{split} $$ (20) $$ \begin{split} p\left( {{{\boldsymbol{x}}_k}|{Z_{1:k - 1}}} \right) = \text{N}\left( {{{\boldsymbol{x}}_k};\hat {{\boldsymbol{x}}}_{k|k - 1}^f, P_{k|k - 1}^f} \right) \end{split} $$ (21) 其中,
$$ \begin{split} \hat {{\boldsymbol{x}}}_{k|k - 1}^i = {F_k}\hat {{\boldsymbol{x}}}_{k - 1|k - 1}^i \end{split} $$ (22) $$ \begin{split} P_{k|k - 1}^i = {F_k}P_{k - 1|k - 1}^iF_k^\text{T} + {Q_k} \end{split} $$ (23) $$ \begin{split} \hat {{\boldsymbol{x}}}_{k|k - 1}^f = {F_k}\hat {{\boldsymbol{x}}}_{k - 1|k - 1}^f \end{split} $$ (24) $$ \begin{split} P_{k|k - 1}^f = {F_k}P_{k - 1|k - 1}^fF_k^\text{T} + {Q_k} \end{split} $$ (25) 定理1. 考虑系统(1)和(2)中, 当$ G_k^n \ne \emptyset $, $ G_k^a \ne \emptyset $, 若先验概率密度函数给出如式(20)和式(21), 则可得到全局状态滤波融合估计, 即
$$ \begin{split} &{\left( {P_{k|k}^f} \right)^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k}^f = {\left( {P_{k|k - 1}^f} \right)^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k - 1}^f\; + \\ &\;\;\sum\limits_{{\boldsymbol{z}}_k^{{n_j}} \in G_k^n} {\left[ {{{\left( {P_{k|k}^{{n_j}}} \right)}^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k}^{{n_j}} - {{\left( {P_{k|k - 1}^{{n_j}}} \right)}^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k - 1}^{{n_j}}} \right]}\;+\\ &\;\;\sum\limits_{{\boldsymbol{z}}_k^{{a_j}} \in G_k^a} {\left[ {{{\left( {P_{k|k, {\lambda _m}}^{{a_j}}} \right)}^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k, {\lambda _m}}^{{a_j}} - {{\left( {P_{k|k - 1}^{{a_j}}} \right)}^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k - 1}^{{a_j}}} \right]} \end{split} $$ (26) $$ \begin{split} {\left( {P_{k|k}^f} \right)^{ - 1}} =\;& {\left( {P_{k|k - 1}^f} \right)^{ - 1}}\;+\\ & \sum\limits_{{\boldsymbol{z}}_k^{{n_j}} \in G_k^n} {\left[ {{{\left( {P_{k|k}^{{n_j}}} \right)}^{ - 1}} - {{\left( {P_{k|k - 1}^{{n_j}}} \right)}^{ - 1}}} \right]}\;+\\ & \sum\limits_{{\boldsymbol{z}}_k^{{a_j}} \in G_k^a} {\left[ {{{\left( {P_{k|k, {\lambda _m}}^{{a_j}}} \right)}^{ - 1}} - {{\left( {P_{k|k - 1}^{{a_j}}} \right)}^{ - 1}}} \right]} \end{split} $$ (27) 其中,
$$ \begin{split} \hat {{\boldsymbol{x}}}_{k|k}^{{n_j}} = \hat {{\boldsymbol{x}}}_{k|k - 1}^{{n_j}} + K_k^{{n_j}}\left( {{\boldsymbol{z}}_k^{{n_j}} - H_k^{{n_j}}\hat {{\boldsymbol{x}}}_{k|k - 1}^{{n_j}}} \right) \end{split} $$ (28) $$ \begin{split} P_{k|k}^{{n_j}} = \left( {{{I}} - K_k^{{n_j}}H_k^{{n_j}}} \right)P_{k|k - 1}^{{n_j}} \end{split} $$ (29) $$ \begin{split} K_k^{{n_j}} = P_{k|k}^{{n_j}}{\left( {H_k^{{n_j}}} \right)^\text{T}}{\left( {R_k^{{n_j}}} \right)^{ - 1}} \end{split} $$ (30) $$ \begin{split} &{\left( {P_{k|k,{\lambda _m}}^{{a_j}}} \right)^{ - 1}}\hat {\boldsymbol{x}}_{k|k,{\lambda _m}}^{{a_j}} = {\left( {P_{k|k - 1}^{{a_j}}} \right)^{ - 1}}\hat {\boldsymbol{x}}_{k|k - 1}^{{a_j}} \;+\\ &\qquad\;\;\sum\limits_{m = 1}^{{\varphi _{{\lambda _m}}} < 0} {\left[ {{{\left( {H_k^{{a_j}}} \right)}^\text{T}}{{\left( {R_{k,{\lambda _m}}^{{a_j}}} \right)}^{ - 1}}{\boldsymbol{z}}_{k,{\lambda _m}}^{{a_j}}} \right]} \end{split} $$ (31) $$ \begin{split} &{\left( {P_{k|k, {\lambda _m}}^{{a_j}}} \right)^{ - 1}} = {\left( {P_{k|k - 1}^{{a_j}}} \right)^{ - 1}}\;+\\ &\qquad \;\;\sum\limits_{m = 1}^{{\varphi _{{\lambda _m}}} < 0} {\left[ {{{\left( {H_k^{{a_j}}} \right)}^\text{T}}{{\left( {R_{k, {\lambda _m}}^{{a_j}}} \right)}^{ - 1}}H_k^{{a_j}}} \right]} \end{split} $$ (32) 式中, ${{I}}$表示单位矩阵.
证明. 见附录A.
最后, 带量测分类处理的渐进高斯滤波融合算法(Progressive Gaussian filtering fusion with classification, PGFFwC)给出如下:
算法1. PGFFwC算法
1) 初始化;
2) while
3) 由式(21)得$\hat {\boldsymbol{x}}_{k{{|}}k - 1}^f, P_{k|k - 1}^f$;
4) for $i = 1:N$ do
5) 基于式(3)和式(4), 对量测$ {\boldsymbol{z}}_k^{i}$分层分类处理得$ {\boldsymbol{z}}_k^{j}$;
6) if $ {\boldsymbol{z}}_k^{j} \in G_k^n$
7) 由式(28)和式(29)得到局部估计$\hat {\boldsymbol{x}}_{k|k}^{{n_j}}, P_{k|k}^{{n_j}}\,;$
8) end if
9) if $ {\boldsymbol{z}}_k^{j} \in G_k^a$
10) 渐进量测更新;
11) while${\varphi _{{\lambda _m}}} < 0$ and $m<M$
12) 由式(31)和式(32), 渐进量测更新得$\hat {\boldsymbol{x}}_{k|k, {\lambda _m}}^{{a_j}},$ $ P_{k|k, {\lambda _m}}^{{a_j}} $;
13) end while
14) end if
15) if $ {\boldsymbol{z}}_k^{j} \in G_k^d$
16) 剔除该量测野值;
17) end if
18) end for
19) 由式(26)和式(27)状态融合, 得到$\hat {\boldsymbol{x}}_{k|k}^f, P_{k|k}^f\,;$
20) end while
如定理1所示, 人体姿态估计性能改善表现在两方面: 1)通过量测分层性能评估, 对量测进行分类处理; 2)利用局部估计间的交互信息来引导渐进量测更新, 从而隐式地补偿量测不确定性. 此外, 当渐进滤波中截止条件尚未触发时, 定理1将等价于集中式融合. 特别地, 当量测信息均为同一种情形下时, 则具有如下的等价形式:
推论1. 当$ G_k^n = \emptyset $, $ G_k^a \ne \emptyset $时, 式(26)和式(27)可以表示为
$$ \begin{split} &{\left( {P_{k|k}^f} \right)^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k}^f = {\left( {P_{k|k - 1}^f} \right)^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k - 1}^f\;{\rm{ + }}\\ &\qquad\sum\limits_{i = 1}^N {\sum\limits_{m = 1}^{{\varphi _{{\lambda _m}}} < 0} {\left[ {{{\left( {H_k^i} \right)}^\text{T}}{{\left( {R_{k, {\lambda _m}}^i} \right)}^{ - 1}} {\boldsymbol{z}}_{k, {\lambda _m}}^i} \right]} } \end{split} $$ (33) $$ \begin{split} &{\left( {P_{k|k}^f} \right)^{ - 1}} = {\left( {P_{k|k - 1}^f} \right)^{ - 1}}\;+\\ &\qquad \sum\limits_{i = 1}^N {\sum\limits_{m = 1}^{{\varphi _{{\lambda _m}}} < 0} {\left[ {{{\left( {H_k^i} \right)}^\text{T}}{{\left( {R_{k, {\lambda _m}}^i} \right)}^{ - 1}}H_k^i} \right]} } \end{split} $$ (34) 证明从略.
3. 仿真与实验
3.1 仿真验证
为验证本文方法的合理性与有效性, 设计由多个视觉传感器组成环境下的人体姿态估计仿真. 考虑存在不同程度视觉遮挡等因素, 采用式(2)的观测模型, 并假设人体关节目标的运动学模型如式(1), 状态转移矩阵参照文献[9, 19], 设置为${F_k} = {\rm{diag}}\left\{ {{F_0}}\;\;\cdots \;\;{{F_0}}\right\} ,$ 其中, ${F_0} = {\rm{diag}}\left\{ {{F_b}}\;\;{{F_b}}\;\;{{F_b}}\right\}$, ${F_b} = \left[ {\begin{aligned} & 1\;\;{\Delta t}\\& 0\;\;\;1 \end{aligned}} \right] ,$ 量测矩阵$ H_k^i = {\rm{diag}}\left\{ {H_0}\;\cdots \;{H_0}\right\} , $ 其中
$$ \begin{split} {H_0} = \left[ {\begin{array}{*{20}{l}} 1&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&1&0 \end{array}} \right] \end{split} $$ (35) $ {{\boldsymbol{x}}_{k,l}}={\left[{{x_{x,l}}}\;\;{{{\dot x}_{x,l}}}\;\;{{x_{y,l}}}\;\;{{{\dot x}_{y,l}}}\;\;{{x_{z,l}}}\;\;{{{\dot x}_{z,l}}} \right]^\text{T}} $, $ {x_{x,l}} $, $ {x_{y,l}} $, $ {x_{z,l}} $和$ {\dot {x}_{x,l}} $, $ {\dot {x}_{y,l}} $, $ {\dot {x}_{z,l}} $分别为人体关节点在$ x $, $ y $和$ z $轴上的位置和速度, 人体关节的总数量取为$ L $ = 17, 系统的采样时间$\Delta t =$ 0.03 s, 过程噪声$ {{\boldsymbol{w}}_k} $的协方差为$ {Q_k} = {\rm{diag}}\left\{{{Q_0}}\;\; \cdots\;\;{{Q_0}} \right\}$, 其中
$$ \begin{split} Q_0=\; &{\rm{diag}} \{0.09\;{\rm{cm}}^2, \;0.005\;{\rm{cm}}^2/{\rm{s}}^2, 0.09\;{\rm{cm}}^2,\;\\& 0.005 \;{\rm{cm}}^2/{\rm{s}}^2,\; 0.09\;{\rm{cm}}^2,\; 0.005 \;{\rm{cm}}^2/{\rm{s}}^2\} \end{split} $$ 人体骨架量测噪声$ {\boldsymbol{v}}_k^i $的协方差矩阵为$R_k^i =$${\rm{diag}}\left\{ {R_0} \cdots {R_0} \right\}$, 其中$R_0 =$ $ 5.0{I_{3 \times 3}} $$ {\rm{c}}{{\rm{m}}^{\rm{2}}} $, $ {I_{3 \times 3}} $表示3 × 3的单位阵. 设置不确定噪声$ {\boldsymbol{\xi}}_k^i $中低程度视觉遮挡干扰的协方差矩阵$ R_{k, + }^i $ = 5.0$ {I_{3 \times 3}} $$ {\rm{c}}{{\rm{m}}^{\rm{2}}} $, 出现的概率为$ {y_1} $ = 0.4; 高程度视觉遮挡下的野值设为幅值大小为40 cm的噪声, 即$ b= $40 cm, 出现的概率为$ {y_2}= $0.05. 以人体右臂腕关节点为例进行分析, 假设初始真实状态向量$ {{\boldsymbol{x}}_0} =$ [ 0 cm, 2.4 cm/s, 0 cm, 2.4 cm/s, 0 cm, 2.4 cm/s ]T, 关节点初始状态估计误差协方差为${P_0}= {I_{3 \times 3}}$, 状态估计初始值$ {\hat {{\boldsymbol{x}}}_{0|0}} $由高斯分布$ \text{N}\left( {{{\boldsymbol{x}}_0}, {P_0}} \right) $随机生成. PGF中渐进过程的总步数$ M $设为10步, 渐进步长$ \Delta _m $= 0.1, 量测评估机制中$ {\chi _n}= $ 15 cm, $ {\chi _a}= $ 30 cm.
便于仿真结果分析与比较, 定义位置误差指标为均方根误差(Root mean square error, RMSE), 其计算式为
$$ \begin{split}F_ {\rm{RMSE}} = \sqrt {\frac{1}{S}\sum\limits_{s = 1}^S {{{\left( {{H_k}{{\boldsymbol{x}}_k} - {H_k}{{\hat {{\boldsymbol{x}}}}_{k|k}}} \right)}^2}} } \end{split} $$ (36) 其中, $F_{\rm{RMSE}} $表示均方根误差, $ s = 1, \cdots , S $为仿真实验的序号, $ S $为蒙特卡罗仿真总次数, $ {\hat {{\boldsymbol{x}}}_{k|k}} $表示$ k $时刻的状态估计值, $ {{\boldsymbol{x}}_k} $表示$ k $时刻的状态真实值. 在局部滤波中采用带量测分类处理的渐进高斯滤波(PGF with classification, PGFwC) (即, PGFFwC中局部的滤波结果)、卡尔曼滤波、粒子滤波(Particle filtering, PF)、 鲁棒卡尔曼滤波(Robust Kalman filtering, RKF)[20]. 同时为进一步验证量测分层分类处理的作用, 加入不带量测分类处理的渐进高斯滤波方法(PGF without classification, PGFwoC) (即, 采用PGF方法无差别地处理量测数据)进行对比. 另外, 为验证PGFFwC的性能, 在融合算法中对比了集中式融合(Centralized fusion, CF)、协方差交叉(Covariance intersection, CI)融合、基于观测融合的自适应卡尔曼滤波(Adaptive measurement fusion-based Kalman filter, AMFKF)[25], 以及IWCF[9]的方法, 蒙特卡罗仿真结果如图4所示. 通过仿真结果可知, 无论在局部滤波还是在全局状态融合中, 本文所提方法的性能都更好. 同时, 通过图4可知, 带有量测分类处理的方法(PGFwC, PGFFwC)比未带量测分类处理的方法(PGFwoC, PGFFwoC)误差更小. 特别地, 包含量测分类处理的分布式状态融合方法(PGFFwC)提升的精度明显高于其他方法, 说明通过对量测进行分类处理后, 滤波器对量测不确定性的描述更准确, 从而在状态融合的过程中获得更高的精度.
3.2 实验验证
为进一步验证所提方法的有效性, 设计多视觉人体姿态估计实验, 实验平台如图5所示, 由两台微软公司的Azure Kinect DK相机[26-27], 一台Windows10操作系统的电脑和一个人体姿态估计对象组成. Azure Kinect DK视觉传感器包括彩色摄像头和深度摄像头, 采集到的彩色图像分辨率为1 920$ \times $1 080像素, 深度图像分辨率为512$ \times $512像素, 拍摄速度为30帧/s, 使用同步线缆硬件触发对两台相机进行同步数据采集, 并通过张正友相机标定法, 计算出从相机到主相机的旋转矩阵与平移向量, 以主相机坐标系作为世界坐标系. 在计算机上, 编写基于Visual Studio 2017的开发环境, 利用CNN的方法得到在深度相机空间下人体骨骼关节点的空间位置信息.
实验场景设置如下: 实验环境位于室内, 两台Azure Kinect DK呈约$ {45^ \circ } $角摆放, 人体目标位于两台相机前方1.5 m左右的位置进行挥臂运动, 用Azure Kinect DK来完成对人体关节点的捕捉, 整个过程会引入自遮挡以及由手持物遮挡造成的误识别. 这里需要补充说明的是, 人体关节点对应的实际人体位置并不明确, 即人体关节点的物理意义是不明确的. 故以高精度动作捕捉系统OptiTrack System[28] (精度0.5 mm) 来获取人体关节点的真实轨迹, 如图5所示, 该定位系统由12个Prime 13相机组成, 能够实时捕捉运动目标的位姿, 以追踪到的光学标记点的位置为真值, 即视为真实人体关节点位置进行对比.
在实验中, 采用的对比方法与仿真一致, 局部滤波分别采用PF, KF, RKF, PGFwC和PGFwoC的方法对比, 全局融合分别采用CF, CI, AMFKF, IWCF, PGFFwC和PGFFwoC的方法对比. 捕捉对象为人体右臂, 其中包括肩关节、肘关节和腕关节. 以人体右臂腕关节点为例分析, 滤波参数与仿真设置的一致, 图6表示该关节点在运动过程中, 不同方法处理下的累积位置误差分析图. 进一步, 表1所示为腕关节点、肘关节点以及肩关节点的位置误差均值, 从中可看出, PGFFwC方法下得到的误差更低. 由此说明该方法能有效提高人体姿态估计的精度和鲁棒性. 另外, 从3组关节点误差均值的整体对比中, 可看出腕关节点的误差相对更大, 肩关节点的误差相对更小, 表明机动性更强的关节点存在的误差也更大.
图 6 不同滤波融合方法下的累积位置误差Fig. 6 Cumulative position error under different filtering fusion methods表 1 累积误差均值统计(mm)Table 1 Cumulative error mean statistics (mm)实验方法 腕关节 肘关节 肩关节 观测融合 166.44 124.44 96.56 CF 157.55 118.00 95.00 AMFKF 147.81 113.85 93.08 CI 127.63 117.85 99.62 IWCF 153.12 113.21 92.53 PGFFwoC 151.77 114.12 92.83 PGFFwC 119.47 108.98 84.11 4. 结束语
为处理视觉遮挡下人体姿态估计性能下降问题, 提出基于渐进高斯滤波融合的姿态估计方法. 首先, 采用CNN的方法从深度图像中识别并得出人体各关节点在相机坐标系下的3D位置, 并将其转换到世界坐标系下; 其次, 在多视觉骨架数据融合中, 构建分布式的渐进贝叶斯滤波融合框架并提出基于渐进高斯滤波融合的人体姿态估计方法. 针对量测信息中包含的复杂噪声, 分别从空间、时间维度对量测进行相容性分析与分类处理. 同时, 引入渐进量测更新与引导机制, 隐式地补偿量测不确定性.
附录 A. 定理1的证明
证明. 因为$ p({\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}}|{{\boldsymbol{x}}_k}) $, $ p({{\boldsymbol{x}}_k}|{\boldsymbol{z}}_{1:k{\rm{ - }}1}^{{a_j}}) $为高斯分布, 易知$ p({{\boldsymbol{x}}_k}, {\lambda _m}|{\boldsymbol{z}}_{1:k - 1}^{{a_j}}, {\boldsymbol{z}}_{k, {\lambda _1}:{\lambda _m}}^{{a_j}}) $, $ p({{\boldsymbol{x}}_k}|{Z_{1:k}}) $也为高斯分布. 令
$$ \begin{split} &o({\boldsymbol{x}}_k) = \\ &\quad-\frac{1}{2}{\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k - 1}^f} \right)^\text{T}}{\left( {P_{k|k - 1}^f} \right)^{ - 1}}\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k - 1}^f} \right)-\\ &\quad\frac{1}{2}\sum\limits_{{\boldsymbol{z}}_k^{{n_j}} \in G_k^n} {\left[ {{{\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k}^{{n_j}}} \right)}^\text{T}}{{\left( {P_{k|k}^{{n_j}}} \right)}^{ - 1}}\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k}^{{n_j}}} \right)} \right]}\;- \\ &\quad\frac{1}{2}\sum\limits_{{\boldsymbol{z}}_k^{{a_j}} \in G_k^a} {\left[ {{{\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k,{\lambda _m}}^{{a_j}}} \right)}^\text{T}}{{\left( {P_{k|k,{\lambda _m}}^{{a_j}}} \right)}^{ - 1}}\left( {{{\boldsymbol{x}}_k} \;- } \right.} \right.} \\ &\;\;\left. {\left. {\hat {{\boldsymbol{x}}}_{k|k,{\lambda _m}}^{{a_j}}} \right)} \right] + \frac{1}{2}\sum\limits_{{\boldsymbol{z}}_k^{{n_j}} \in G_k^n} {\left[ {{{\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k - 1}^{{n_j}}} \right)}^\text{T}}{{\left( {P_{k|k}^{{n_j}}} \right)}^{ - 1}}} \right.} \times \\ \end{split} $$ $$\begin{split}&\quad\left. {\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k - 1}^{{n_j}}} \right)} \right] + \frac{1}{2}\sum\limits_{{\boldsymbol{z}}_k^{{a_j}} \in G_k^a} {\left[ {{{\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k - 1}^{{a_j}}} \right)}^\text{T}}} \right.} \times\\&\quad \left. {{{\left( {P_{k|k,{\lambda _m}}^{{a_j}}} \right)}^{ - 1}}\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k - 1}^{{a_j}}} \right)} \right]\end{split} \tag{A1} $$ 根据后验概率密度函数(8), 可得最大后验估计为
$$ \begin{split} \hat {{\boldsymbol{x}}}_{k|k}^f = &\arg \max p({{\boldsymbol{x}}_k}|{Z_{1:k}}) = \\ &\arg \mathop {\max }\limits_{{{\boldsymbol{x}}_k}} o\left( {{{\boldsymbol{x}}_k}} \right) \end{split} \tag{A2}$$ 求解$ \frac{{\partial o\left( {{{\boldsymbol{x}}_k}} \right)}}{{\partial {{\boldsymbol{x}}_k}}} = 0 $, 得全局状态估计为
$$ \begin{split} &{\left( {P_{k|k}^f} \right)^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k}^f = {\left( {P_{k|k - 1}^f} \right)^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k - 1}^f\;+ \\ &\qquad\sum\limits_{{\boldsymbol{z}}_k^{{n_j}} \in G_k^n} {\left[ {{{\left( {P_{k|k}^{{n_j}}} \right)}^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k}^{{n_j}} - {{\left( {P_{k|k - 1}^{{n_j}}} \right)}^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k - 1}^{{n_j}}} \right]} \;+ \\ &\qquad\sum\limits_{{\boldsymbol{z}}_k^{{a_j}} \in G_k^a} {\left[ {{{\left( {P_{k|k,{\lambda _m}}^{{a_j}}} \right)}^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k,{\lambda _m}}^{{a_j}}} \right.}\; - \\ &\qquad\left. { {{\left( {P_{k|k - 1}^{{a_j}}} \right)}^{ - 1}}\hat {{\boldsymbol{x}}}_{k|k - 1}^{{a_j}}} \right] \end{split}\tag{A3} $$ $$ \begin{split} &{\left( {P_{k|k}^f} \right)^{ - 1}} = {\left( {P_{k|k - 1}^f} \right)^{ - 1}}\; +\\ & \qquad\sum\limits_{{\boldsymbol{z}}_k^{{n_j}} \in G_k^n} {\left[ {{{\left( {P_{k|k}^{{n_j}}} \right)}^{ - 1}}} \right.} \left. { - \;{{\left( {P_{k|k - 1}^{{n_j}}} \right)}^{ - 1}}} \right] +\\ &\qquad\sum\limits_{{\boldsymbol{z}}_k^{{a_j}} \in G_k^a} {\left[ {{{\left( {P_{k|k, {\lambda _m}}^{{a_j}}} \right)}^{ - 1}} - {{\left( {P_{k|k - 1}^{{a_j}}} \right)}^{ - 1}}} \right]} \end{split} \tag{A4}$$ 其中, 由于量测$ {\boldsymbol{z}}_k^{{n_j}} \in G_k^n $中不含额外噪声干扰与野值, 局部状态估计$ \hat {{\boldsymbol{x}}}_{k|k}^{{n_j}} $即可用卡尔曼滤波得到
$$ {\hat{\boldsymbol{x}}_{k|k}^{{n_j}}} = \hat {{\boldsymbol{x}}}_{k|k - 1}^{{n_j}} + K_k^{{n_j}}\left( {{\boldsymbol{z}}_k^{{n_j}} - H_k^{{n_j}}\hat {{\boldsymbol{x}}}_{k|k - 1}^{{n_j}}} \right) \tag{A5}$$ $$ P_{k|k}^{{n_j}} = \left( {I - K_k^{{n_j}}H_k^{{n_j}}} \right)P_{k|k - 1}^{{n_j}} \tag{A6}$$ $$ K_k^{{n_j}} = P_{k|k}^{{n_j}}{\left( {H_k^{{n_j}}} \right)^\text{T}}{\left( {R_k^{{n_j}}} \right)^{ - 1}}\tag{A7} $$ 对于${\boldsymbol{z}}_k^{{a_j}} \in G_k^a\, ,$ 在其局部状态更新中, 通过PDF式(19), 可求得最大后验状态估计
$$ \begin{split} \hat {{\boldsymbol{x}}}_{k|k}^{{a_j}} = \arg \max p({{\boldsymbol{x}}_k}, {\lambda _m}|{\boldsymbol{z}}_{1:k - 1}^{{a_j}}, {\boldsymbol{z}}_{k, {\lambda _1}:{\lambda _m}}^{{a_j}}) \end{split} $$ 根据高斯分布的连乘性质, $p({{\boldsymbol{x}}_k}, {\lambda _m}|{\boldsymbol{z}}_{1:k - 1}^{{a_j}}, {\boldsymbol{z}}_{k, {\lambda _1}:{\lambda _m}}^{{a_j}} )$也为高斯分布, 求得
$$ \begin{split} &\hat {{\boldsymbol{x}}}_{k|k, {\lambda _m}}^{{a_j}} = \arg \mathop { \max }\limits_{{{\boldsymbol{x}}_k}} p({{\boldsymbol{x}}_k}, {\lambda _m}|{\boldsymbol{z}}_{1:k - 1}^{{a_j}}, {\boldsymbol{z}}_{k, {\lambda _1}:{\lambda _m}}^{{a_j}}) = \\ &\quad\arg \mathop {\max }\limits_{{{\boldsymbol{x}}_k}} \exp \left\{ - \frac{1}{2}{\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k - 1}^{{a_j}}} \right)^\text{T}}{\left( {P_{k|k - 1}^{{a_j}}} \right)^{ - 1}}\;\times\right. \end{split} $$ $$ \begin{split} & \quad\left( {{{\boldsymbol{x}}_k} - \hat {{\boldsymbol{x}}}_{k|k - 1}^{{a_j}}} \right) - \frac{1}{2}\sum\limits_{m = 1}^{{\varphi _{{\lambda _m}}} < 0} \left[{{\left( {{\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}} - H_k^{{a_j}}{{\boldsymbol{x}}_k}} \right)}^\text{T}} \;\times \right.\\ &\left.\left.\quad{\left( {R_{k, {\lambda _m}}^{{a_j}}} \right)^{ - 1}}\left( {{\boldsymbol{z}}_{k, {\lambda _m}}^{{a_j}} - H_k^{{a_j}}{{\boldsymbol{x}}_k}} \right)\right]\right\} \end{split}\tag{A8} $$ 解法与式(A1) ~ (A4)类似, 得
$$ \begin{split} &{\left( {P_{k|k,{\lambda _m}}^{{a_j}}} \right)^{ - 1}}\hat {\boldsymbol{x}}_{k|k,{\lambda _m}}^{{a_j}} = {\left( {P_{k|k - 1}^{{a_j}}} \right)^{ - 1}}\hat {\boldsymbol{x}}_{k|k - 1}^{{a_j}} \;+\\ & \qquad\sum\limits_{m = 1}^{{\varphi _{{\lambda _m}}} < 0} {\left[ {{{\left( {H_k^{{a_j}}} \right)}^\text{T}}{{\left( {R_{k,{\lambda _m}}^{{a_j}}} \right)}^{ - 1}}{\boldsymbol{z}}_{k,{\lambda _m}}^{{a_j}}} \right]} \end{split}\tag{A9} $$ $$ \begin{split} &{\left( {P_{k|k, {\lambda _m}}^{{a_j}}} \right)^{ - 1}} = {\left( {P_{k|k - 1}^{{a_j}}} \right)^{ - 1}}\;+\\ &\qquad\sum\limits_{m = 1}^{{\varphi _{{\lambda _m}}} < 0} {\left[ {{{\left( {H_k^{{a_j}}} \right)}^\text{T}}{{\left( {R_{k, {\lambda _m}}^{{a_j}}} \right)}^{ - 1}}H_k^{{a_j}}} \right]} \end{split}\tag{A10}$$ □ -
图 4 专家知识驱动的学习算法的基本流程
Fig. 4 The basic process of expert knowledge-driven learning algorithms
表 1 拓展认知图模型对比
Table 1 Comparison of extension cognitive map models
类别 名称 特点 优点 缺点 应用领域 基于不同
模糊理论
的拓展
认知图RBFCM[28] 引入模糊进位累加器计算因果权重 涵盖多种概念关系并具有
多功能性和简单性建模要求高 决策支持 FGCM[29] 引入灰色数衡量因果强度 建模概念之间的不确定信息 推理复杂 可靠性工程 IFCM[30] 利用直觉模糊集建模因果关系 衡量了因果关系中的不确定性 推理复杂 时间序列预测 IVFCM[31] 利用区间值描述因果关系的强度 考虑了非结构化环境相关的不确定性 依赖专家知识 决策支持, 时间序列
预测模型ECM[32] 在因果推理中融入了证据理论 既能表示不确定性又能进行知识融合 依赖专家知识,
推理复杂决策支持 RCN[33] 利用粗糙集表示因果关系 解决了不确定情况下的决策问题 依赖专家知识 决策支持软件,
可靠性评估FRCN[34] 利用模糊粗糙结构构建神经网络 建模了因果关系的不确定性 推理复杂 决策支持 zT2FSs-FCM[35] 引入二型模糊集建模节点间的
因果权重捕获了概念间的不确定性关系 推理复杂 系统评估 面向动态
系统建模
的拓展
认知图DCN[37] 考虑了因果关系的时变性 结构上具有更高的可扩展性和灵活性 依赖拉普拉斯框架,
建模复杂度高决策支持 DRFCM[39] 推理过程中引入非线性动态函数 能够捕获动态因果关系,
具有自适应性建模要求高 风险评估, 决策支持 FTCM[40] 考虑了因果关系强度和时间滞后性 能够随时间推移分析系统的动态行为 建模复杂 时间序列预测 E-FCM[41] 采用检查机制模拟动态因果关系 能够自我进化适应不断发展的行为 计算耗时 动态场景建模 HFCM[42] 考虑了复杂系统建模过程中的
多阶动态性准确地描述了系统行为 随着阶数增加,
计算复杂度增加时间序列预测 TAFCM[43] 引入了定时自动机理论建模
系统的时间粒度推理过程具有动态性和自适应性 建模要求高 人类情绪建模 DFCM[44] 嵌入在深度神经网络的框架中 构建可解释预测器, 挖掘隐藏的
因果关系训练耗时, 容易面临
“数据饥饿”问题时间序列预测 AFCM[45] 构建基于趋势的信息粒引入
自适应更新机制自适应权重长期预测 计算耗时 时间序列预测 
下载: 导出CSV
表 2 基于学习范式的模糊认知图学习算法分类
Table 2 Classification of fuzzy cognitive map learning algorithms based on the learning paradigm
类别 学习方法 时间复杂度 优点 缺点 作者 发表年份 专家知识
驱动的方法DHL[47] ${\rm{O}}(N^2)$ 简单, 易操作 只考虑了当前的一对概念 Dickerson等 1994 BDA[48] ${\rm{O}}(N^2)$ 考虑多个概念的影响 只适用于二进制计算 Huerga 2002 AHL[49] ${\rm{O}}(N^2)$ 考虑了所有概念的影响 训练耗时 Papageorgiou等 2004 NHL[50] ${\rm{O}}(N^2)$ 保留了原始的图结构, 具有合理的物理解释性 依赖专家标准 Papageorgiou等 2003 INHL[51] ${\rm{O}}(N^2)$ 避免陷入局部最小值 需要先验知识 Li等 2004 DDNHL[52] ${\rm{O}}(N^2)$ 数据驱动 依赖专家知识 Stach 等 2008 带终端约束的
NHL算法[53]${\rm{O}}(N^2)$ 提高结果的可行性 需要先验知识 陈宁等 2016 FBN[54] ${\rm{O}}(N^2)$ 利用模糊因果规则推理 性能受激活参数的影响 Carvalho等 2007 基于bagging增强的NHL算法[55] ${\rm{O}}(N^2)$ 泛化性能较好 依赖专家知识 Papageorgiou等 2012 自动学习
算法GA[56] ${\rm{O}}(N^2)$ 数据驱动 受限于二进制编码 Mateou等 2005 RCGA[57] ${\rm{O}}(N^2)$ 数据驱动, 实数编码 参数寻优耗时 Stach 等 2005 PSO[58−59] ${\rm{O}}(N!)$ 数据驱动, 元启发式算法 依赖专家知识 Parsopoulos 等Oikonomou 等 2003
2013SOMA[60] ${\rm{O}}(N^2)$ 数据驱动 计算耗时 Vaščák 2010 ACO[61] ${\rm{O}}(N^2)$ 概率型算法鲁棒性强 计算耗时, 容易早熟收敛 Chen等 2012 ABC[62] ${\rm{O}}(N^2)$ 数据驱动 参数寻优耗时 Yesil等 2013 ICA[63] ${\rm{O}}(N^3)$ 数据驱动 计算复杂, 耗时 Ahmadi 等 2015 DE[64] ${\rm{O}}(N^2)$ 容易理解, 计算简单 易局部收敛 Juszczuk等 2009 SA[65−66] ${\rm{O}}(N^2)$ 计算简单 参数寻优耗时 Ghazanfari 等
Alizadeh等2007
2009BB-BC[67] ${\rm{O}}(N^2)$ 算法简单, 泛化能力较好 不适用于解决高维问题 Yesil等 2010 CA[68] ${\rm{O}}(N^2)$ 全局搜索与局部搜索结合 参数寻优耗时, 对问题的依赖性强 Ahmadi等 2014 基于互信息的
模因算法[70]${\rm{O}}(N^2)$ 适用于大规模图学习 无法在搜索过程中
关注图的稀疏性Zou等 2018 MARO[71] ${\rm{O}}(N^2)$ 只需调用一次目标
函数, 无需设置参数计算复杂, 易陷入局部最优 Salmeron等 2019 分解RCGA[72−73] ${\rm{O}}(N^2)$ 分解并行计算 计算复杂 Chen等, Stach等 2015, 2010 D&C RCGA[74] ${\rm{O}}(N^2)$ 可并行计算并具有可扩展性 随着图的大小和处理器数量增加, 算法性能下降 Stach等 2007 dMAGA[75] ${\rm{O}}(N^2)$ 适用于大规模图学习
具有鲁棒性受 FCM 节点的取值范围限制, 需在算法执行前进行数据归一化 Liu等 2015 MA-NN[76] ${\rm{O}}(N^2)$ 分布式计算框架适用于
大规模网络重建受FCM节点的取值范围限制, 需在算法执行前进行数据归一化 Chi等 2019 MOEA[77, 79−80] ${\rm{O}}(N^2)$ 多目标进化考虑了图的稀疏性 不适用于大规模图学习 Liu等, Poczeta 等,
Chi等2019, 2018, 2016 IMFPSO[78] ${\rm{O}}(N!)$ 优化过程考虑了知识迁移 算法易早熟, 过早收敛 Liang等 2022 SRCGA[15] ${\rm{O}}(N^2)$ 考虑了图的稀疏性 不适用于处理大规模数据 Stach等 2012 MMMA[17] ${\rm{O}}(N^2)$ 多图优化知识转移 有可能发生负信息迁移, 导致收敛速度缓慢 Shen等 2020 CS[81] ${\rm{O}}(N^3)$ 适用于大规模稀疏图学习 参数寻优耗时 Wu等 2017 内点法[82] ${\rm{O}}(N^4)$ 精度高, 可扩展性好 对初值敏感, 难以处理
不等式约束问题Lu等 2020 约束优化[83] ${\rm{O}}(N^3)$ 考虑了矩阵分布具有抗噪能力 仅适用于有监督学习 Feng等 2021 近似梯度下降[84] ${\rm{O}}(N^3)$ 适用于解决大规模数据问题 对初始点敏感, 可能
陷入局部最优Ding等 2021 Moore-Penrose逆[85] ${\rm{O}}(N^3)$ 参数较少 计算复杂 Vanhoenshoven等 2020 Lasso回归[86] ${\rm{O}}(N^3)$ 考虑了图的稀疏性, 适用于
大规模图学习可能出现过拟合 Wu等 2016 岭回归[87] ${\rm{O}}(N^3)$ 泛化性能较好, 适用于
大规模图学习对特征的缩放敏感 Yang等 2018 弹性网络回归[88] ${\rm{O}}(N^3)$ 增加了L1 和L2 正则化, 适用于大规模图学习 参数调节困难 Shen等 2020 支持向量回归[89] ${\rm{O}}(N^4)$ 适用于高维非线性数据 对缺失数据敏感 Gao等 2020 贝叶斯岭回归[90] ${\rm{O}}(N^4)$ 简单、模型适应性较强 对模型的假设较多依赖先验分布 Liu等 2020 FTRL[91] ${\rm{O}}(N^3)$ 在线学习 计算、推理过程复杂 Wu 等 2021 半自动学习算法 DE+NHL[92] ${\rm{O}}(N^2)$ 进化过程中保留了图的物理意义 依赖专家知识 Papageorgiou等 2005 RCGA+NHL[93] ${\rm{O}}(N^2)$ 利用了遗传算法的全局优化能力 受限于专家经验 Zhu等 2008 PSO+NHL[94] ${\rm{O}}(N!)$ 避免人为因素产生的训练误差 受限于专家经验 Yazdi等 2008 EGDA+NHL[95] ${\rm{O}}(N^2)$ 全局搜索, 参数少 受限于专家经验 Ren 2012 DDNHL+GA[96] ${\rm{O}}(N^3)$ 数据驱动分类推理能力强 受限于专家经验 Natarajan等 2016 RCGA+DE+
梯度下降[97]${\rm{O}}(N^2)$ 全局搜索 参数寻优耗时 Madeiro 等 2012 注: 时间复杂度为该算法更新一次FCM权重矩阵所需时间开销, 未考虑数据量大小及最大迭代次数. N表示节点个数.
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表 3 大规模模糊认知图学习算法分类
Table 3 Large-scale fuzzy cognitive map learning algorithm classification
类别 方法 转换函数 最大FCM规模 发表年份 基于暴力求解的方法 D&C RCGA[73] sigmoid 40 2010 并行RCGA[74] sigmoid 80 2007 dMAGA[75] sigmoid 200 2015 MA-NN[76] sigmoid 100 2019 MOEA[80] sigmoid 40 2015 SRCGA[15] sigmoid 40 2012 基于维度缩减的方法 MIMA[70] sigmoid 500 2018 文献[100] sigmoid/tanh 25 2015 文献[101] sigmoid 10 2018 基于分解的方法 CS[81] sigmoid 1 000 2017 MMMA[17] sigmoid/tanh 600 2020 内点法[82] sigmoid/tanh 200 2020 约束优化[83] sigmoid/tanh 200 2021 近似梯度下降[84] sigmoid 200 2021 Lasso回归[86] sigmoid 500 2016 弹性网络回归[88] sigmoid 200 2020 HTMA-DRA[99] sigmoid 200 2022 dMAGA-FCM$_D$[102] sigmoid 500 2017 NMMMAGA[103] sigmoid 200 2019 Parallel FCM[104] sigmoid 1 000 2023
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表 4 模糊认知图学习算法的应用文献总结
Table 4 Literature review on the application of fuzzy cognitive map learning algorithms
类别 应用领域 文献 专家知识驱动的方法 模式分类 [55] 前列腺癌诊断 [105] 公司信用风险评估 [106] 自闭症预测 [107] 结构损伤检测 [108] 帕金森病预测 [110] 事故成因预测 [111] 裂纹严重程度分级 [112] 乳腺癌风险评估 [113] 自动学习算法 基因调控网络重建 [17, 75–77, 82, 84, 86] 多变量时间序列预测 [44−45, 78, 85, 116–119] 单变量时间序列预测 [83, 87, 89−91, 120−127, 140–143] 情景意识评估 [114] 病情趋势预测 [134] 前列腺癌预测 [135] 日需水量预测 [136] 电器能耗预测 [137] RFID物流操作评估 [138] 分类 [5, 128–133, 140] 半自动学习算法 医学诊断 [139] 甘蔗产量预测 [96] 决策支持 [92] 化学控制 [94] 太阳能发电 [97]
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表 5 模糊认知图建模工具对比
Table 5 Comparison of fuzzy cognitive map modeling tools
工具名称 受众定位 适用场景 应用形式 学习算法数量 图形页面 年份 FCM Modeler[144] 学术研究 静态建模, 群体决策 Java Applet 1 √ 1997 FCMappers.net[145] 学术研究 网络分析, 系统建模 网站 — — 2009 FCM Tool[146] 商业产品, 学术研究 决策支持, 系统建模 软件 1 √ 2011 FCM Designer[147] 学术研究 系统建模 Java Applet — √ 2010 FCM Designer Version 2.0[148] 学术研究 医学诊断, 推荐系统建模 Java Applet — √ 2016 Mental Modeler[149] 商业产品, 学术研究 群体决策, 系统建模 Web 页面 — √ 2013 JFCM[150] 教学工具, 学术研究 系统建模 Java开源库 — — 2014 ISEMK[152] 商业产品, 学术研究 决策支持, 时间序列预测 — 6 √ 2015 FCM Expert[154] 学术研究 决策支持, 系统建模 Java软件 4 √ 2017 FCMpy[158] 学术研究 系统建模 开源Python模块 5 √ 2022 注: “—”表示“无”或者未查询到.
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