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摘要: 图像的模糊问题影响人们对信息的感知、获取及图像的后续处理. 无参考模糊图像质量评价是该问题的主要研究方向之一. 本文分析了近20年来无参考模糊图像质量评价相关技术的发展. 首先, 本文结合主要数据集对图像模糊失真进行分类说明; 其次, 对主要的无参考模糊图像质量评价方法进行分类介绍与详细分析; 随后, 介绍了用来比较无参考模糊图像质量评价方法性能优劣的主要评价指标; 接着, 选择典型数据集及评价指标, 并采用常见的无参考模糊图像质量评价方法进行性能比较; 最后, 对无参考模糊图像质量评价的相关技术及发展趋势进行总结与展望.Abstract: The blurriness distortion of image affects information perception, acquisition and subsequent processing. No-reference blurred image quality assessment is one of main research directions for the problem. This paper analyzes the relevant technique development of no-reference blurred image quality assessment in recent 20 years. Firstly, combining with main databases, different types of blurriness distortions are described. Secondly, main methods for no-reference blurred image quality assessment are classified and analyzed in detail. Thirdly, performance measures for no-reference blurred image assessment are introduced. Then, the typical databases, performance measures and methods are introduced for performance comparisons. Finally, the relevant technologies and development trends of no-reference blurred image assessment are summarized and prospected.
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随着信息技术的高速发展, 各领域中所产生的数据维度正在以前所未有的速度增长, 例如社交网络数据、金融交易数据和城市交通流量数据等.
然而, 传统的数据表征方法无法适用于具有复杂关联特征的网络数据集. 所以, 图网络[1]——一种非规则域中用于表征关联数据的模型应运而生. 如何更好地分析这些基于图网络表征的数据集, 从而更加高效地挖掘数据集的深度信息成为当下研究的热点问题之一.
近年来, 随着图信号处理的兴起和发展, 图网络中的信号(数据)分析与处理引起了研究者们的广泛关注. 图信号处理是将传统的信号处理理论衍生至基于图网络表征的非规则域信号处理理论[2]. 目前,图信号处理的理论研究主要包括图滤波器(组)的设计[3]、图信号采样/恢复[4]、图信号压缩[5]和图拓扑学习[6]等. 相关的应用研究有传感网络中的异常数据检测[7]及修复[8], 基于图数据的机器学习等[9-10]. 然而, 目前该研究领域中仍然存在着许多亟待探索和解决的理论问题和应用瓶颈[11]. 例如, 图信号处理中尚未出现类似于奈奎斯特采样定理的统一采样理论[12]. 相关的挑战还包括图信号的大规模分布式计算[13]、异构网络中的图信号处理[14]、如何融合多尺度下的图信号特征而进行信号多分辨分析[15], 以及如何分析张量图网络中的多层图数据之间的关联性[16]等. 随着图信号处理的不断发展, 必将成为有效应对数据泛滥现象和降低数据冗余的重要工具, 并为网络数据的高效处理提供理论支撑.
由于存在图网络的拓扑结构复杂多变以及数据维度带来的计算消耗大的问题, 如何利用尽可能少的采样节点信号和网络拓扑信息更加高效和完备地表征未采样节点信号, 从而为网络数据的传输和处理提供高效的技术支撑是图信号处理中的核心问题[17]. 在图信号重构的相关研究中, 由于带限图信号重构问题可作为其他类型图信号重构问题的源问题进行相关推广; 如何设计高效的带限图信号重构算法是一个重要的研究课题, 它为设计平滑图信号重构算法和实际网络数据重构方法提供了理论基础.
基于Papoulis-Gerchberg信号重构算法[18], Narang等[19]提出一种基于空域迭代图滤波的信号重构方法(Iteration least square reconstruction, ILSR). 该方法通过将采样信号和每次迭代后产生的采样信号残差进行累加后, 再进行图谱域带限滤波处理, 从而达到重构目的. 在ILSR重构算法的基础上, Wang等[20]提出了基于迭代加权策略的信号重构算法(Iteration weighting reconstruction, IWR)和基于迭代传播策略的信号重构算法(Iteration propagating reconstruction, IPR), 两种算法优于ILSR算法的原因在于对采样节点进行了残差滤波处理. 在IWR算法中, Wang等[20]首先将采样信号的残差扩大相应的权重, 然后进行图滤波处理; 而在IPR算法中, 首先是基于预先划分好的局部集将采样节点的信号残差传递给相邻的未采样节点, 然后进行图滤波处理. 由于两种算法在每步迭代中加入了对于采样信号残差的处理, 增大了未采样信号在插值过程中的增量, 进而提高了重构的效率和精度. 为了进一步地提高对于残差信号的估计精度, Yang等[21]提出了基于扩散算子的迭代重构算法(Iteration graph reconstruction based diffusion operator, IGDR). IGDR算法修正了IWR和IPR算法中由于采样信号残差在局部集内均匀传递而导致的过平滑现象, 在每步迭代中基于局部扩散算子和全局扩散算子对信号采样进行了联合处理, 使得迭代滤波得到的未采样信号为图带限滤波信号和残差扩散信号的总和. 不同于IWR、IPR和IGDR算法聚焦于迭代残差信号的处理方法, Brugnoli等[22]同样在ILSR算法的基础上提出了基于最优参数的Papoulis-Gerchberg信号迭代重构算法(Optimal Papoulis-Gerchberg iterative reconstruction, O-PGIR), 该算法通过在每步迭代中设置松弛参数的最优解而达到较高的迭代效率.
不同于基于空域滤波的重构算法研究, 为了完善图信号谱域理论框架及提升图信号的谱域特征分析能力, 基于图傅里叶变换的图谱域重构算法同样是近年来的研究热点.
Tseng等先后提出基于压缩感知的硬阈值截断图谱域重构算法[23]和基于图傅里叶变换的图谱域重构算法[24]. 在硬阈值截断图谱域重构算法中,作者首先将图信号重构问题转化为图谱域中的稀疏优化问题, 然后采用经典压缩感知理论中的基追踪算法和正交匹配算法或迭代硬阈值截断法分别进行求解. 通过上述方法估计出未采样图信号在图谱域中的频率分量, 最后基于图傅里叶逆变换将估计的频率分量转换为空域图信号. 在正交匹配算法的基础上作者又提出了基于图傅里叶变换的信号重构算法; 在正交匹配算法中, 完整频率分量是通过逐步重构出每个图频率分量值而实现的. 而在基于图傅里叶变换的信号重构算法中, 作者通过重构出小于截止图频率内的频率分量值实现信号重构. 该算法实质上是将ILSR算法转化到图频域进行处理. 然而, 两种方法并没有针对低通带限图信号的谱域特性进行更深入的分析, 只是将空域重构算法转化到变换域进行.
本文首先基于图傅里叶变换的分块矩阵形式和图带限信号特性分析得出图带限分量的恒等不变性. 基于该特性, 本文将重构问题建模为一个最小二乘模型. 本文所提出的重构模型是根据图高频部分的恒等关系, 相比于基于图低频段相似性的ILSR重构模型, 更加能够准确地表征信号的图谱域带限特性, 提高了重构精度. 此外, 由于根据重构模型而设计的迭代算法采用拟牛顿法进行求解, 在避免海森矩阵求解的同时高效利用了模型的二阶梯度信息, 相比于ILSR和O-PGIR提高了迭代效率. 而在基于残差信号的重构算法中, 本文根据残差信号同样具备图带限分量的恒等不变性, 设计了一种基于残差谱移位的重构算法. 相比于IWR/IPR和IGDR算法, 本文算法具有较好的重构性能. 此外, 由于本文提出的图带限分量的恒等不变性不需要考虑带限频率所在的频段, 所以针对分段带限图信号的重构问题同样适用, 并且具有良好的重构性能.
1. 基于谱移位的重构算法
图信号是指定义在具有网络拓扑结构中的信号集合, 其拓扑结构采用图模型
$G = (V{\rm{, }}\;E{\rm{, }}\;{\boldsymbol{W}})$ 进行表征. 其中, 节点集为$V = \{ {v_1}, \cdot \cdot \cdot ,{v_N}\} $ .$E = \{ e(i,j)\} $ 是图模型中的边集合,$e(i,j)$ 表示节点${v_i}$ 和节点${v_j}$ 之间有边相连. 信号${\boldsymbol{f}} = \{ f(i)\} \in {{\mathbf{R}}^N},$ 其中$f(i)$ 为图模型${{G}}$ 中节点${v_i}$ 上的信号值. 邻接矩阵${\boldsymbol{W}} = $ $ \{ w(i,j)\} \in {{\mathbf{R}}^{N \times N}}$ 用于表征节点之间的相关性,${\boldsymbol{W}}$ 中的元素$w(i,j)$ 如式(1)所示.$$w(i,j) = \left\{ {\begin{aligned} &1,\;\;\;\;{e(i,j) \in {{E}}}\\ &0,\;\;\;\;{\text{其他}} \end{aligned}} \right.$$ (1) 由矩阵
${\boldsymbol{W}}$ 可得到图拉普拉斯矩阵${\boldsymbol{L}} = {\boldsymbol{D}} - {\boldsymbol{W}}$ 和归一化图拉普拉斯矩阵${{\boldsymbol{L}}_{{Nor}}} = {{\boldsymbol{D}}^{ - 1/2}}{\boldsymbol{L}}{{\boldsymbol{D}}^{ - 1/2}}$ , 其中的度矩阵定义为${\boldsymbol{D}} = {\rm{diag}}\{ {d_i}\}$ ,对角线元素${d_i}$ 为邻接矩阵中第$i$ 行元素之和. 通过对归一化图拉普拉斯矩阵${{\boldsymbol{L}}_{Nor}}$ 进行特征值分解, 得到特征向量矩阵${\boldsymbol{U}} = [{{\boldsymbol{u}}_1}\;\cdots\;{{\boldsymbol{u}}_N}]$ 和与其对应的特征值矩阵${\boldsymbol{\Lambda }} = $ $ {\rm{diag}} \{ {\lambda _1},\cdots,{\lambda _N}\}$ .在图信号处理理论中, 图傅里叶变换对建立了图信号在空间域和图谱域之间的联系, 从谱聚类的角度分析和处理图信号[25]. 其正变换和逆变换分别如式(2)和式(3)所示, 其中
${\boldsymbol{f}}$ 和${\tilde{\boldsymbol f}}$ 分别表示空域信号和图频率分量.$${\tilde{\boldsymbol f}} = {{\boldsymbol{U}}^{\rm{T}}}{\boldsymbol{f}}$$ (2) $${\boldsymbol{f}} = {\boldsymbol{U\tilde f}}$$ (3) 根据图傅里叶变换对的定义, 图带限信号(Band-limited graph signals)
${{\boldsymbol{f}}_{BLG}} \in P{W_\omega }$ 的定义为: 当${\lambda _i} > \omega $ 时,${{\tilde{\boldsymbol f}}_{BLG}}(i) = 0$ ; 其中$\omega $ 为带限图信号${{\boldsymbol{f}}_{BLG}}$ 的截止图频率. 如图1所示, 图1(a)为空间域中节点信号的分布图; 下层为图信号的拓扑结构, 上层为将各节点信号连接而成的平面图. 图1(b)表示的是图信号经过图傅里叶变换后得到的图谱域示意图; 在其图谱域示意图中, 其高频段的图频率分量为零. 由于在图信号重构问题中, 如何设计采样策略同样对于能否实现精确重构有着一定的影响. 在本文中将采用基于重构唯一性条件而设计采样策略. 在满足该条件的情况下, 任意的带限图信号均可实现精确重构. 图截止频率($\omega $ )的重构唯一性的条件为[26]: 当带限图信号${{\boldsymbol{f}}_{BLG}} \in P{W_\omega }$ 的截止图频率${\omega ^2} \leq \eta$ 时, 从任意的采样节点集合重构得到的带限图信号具有唯一性. 其中$\eta $ 是关于${\boldsymbol{L}}_{Nor}^ *$ 的最小特征值,${\boldsymbol{L}}_{Nor}^ *$ 是由${{\boldsymbol{L}}_{Nor}}^2$ 中对应于未采样节点集合的行和列而构成的子矩阵.本文所研究的是带限图信号重构问题,即是在已知图信号
${\boldsymbol{f}}$ 的先验信息 —图带限特性$({\boldsymbol{f}} \in $ $ P{W_\omega })$ 和采样信号${\boldsymbol{f}}(S)$ 的情况下, 如何重构得到未采样信号${\boldsymbol{f}}({S^c})$ .采样矩阵${{\boldsymbol{P}}_S} = {\rm{diag}}\{ {{\boldsymbol{1}}_S}\} \in {{\mathbf{R}}^{N \times N}}$ (对应于采样节点的主对角线元素为1,其余为0). 本文定义带限图信号的带宽为$B$ , 即共有$B$ 个${\lambda _i} \leq \omega$ , 采样节点个数为$M$ , 未采样节点个数为$N - M .$ 若将图信号
${\boldsymbol{f}}$ 中的采样信号${\boldsymbol{f}}(S)$ 和未采样信号${\boldsymbol{f}}({S^c})$ 进行适当的重新排序后可得图信号${\boldsymbol{f}} = $ $ {[{\boldsymbol{f}}^{\rm{T}}{(S)}{\boldsymbol{\Phi }}^{\rm{T}}{(S)}{\rm{ }}\quad\Phi^{\rm{T}} {({S^c})}{\boldsymbol{f}}^{\rm{T}}{({S^c})}]^{\rm{T}}}, $ 其中${\boldsymbol{\Phi }}(S) \in {{\mathbf{R}}^{M \times N}}$ 是由${{\boldsymbol{P}}_{\rm{S}}}$ 中$M$ 个非全零的行向量而构成,${\boldsymbol{\Phi }}({S^c}) \in $ $ {{\mathbf{R}}^{(N - M) \times N}}$ 是由$({\boldsymbol{I}} - {{\boldsymbol{P}}_{\rm{S}}})$ 中$N - M$ 个非全零行向量组成.图傅里叶变换对的分块矩阵表示形式如式(4)和式(5)所示.
${{\boldsymbol{U}}_L}(S) \in {{\mathbf{R}}^{M \times B}}$ 和${{\boldsymbol{U}}_L}({S^c}) \in $ $ {{\mathbf{R}}^{(N - M) \times B}}$ 分别由矩阵${\boldsymbol{U}}$ 的子矩阵$[{{\boldsymbol{u}}_1}\;\cdots\;{{\boldsymbol{u}}_B}]$ 中对应于采样节点和未采样节点的行向量所构成的子矩阵; 子矩阵${{\boldsymbol{U}}_H}(S)$ 和${{\boldsymbol{U}}_H}({S^c})$ 分别是由${\boldsymbol{U}}$ 的子矩阵$[{{\boldsymbol{u}}_{B + 1}}\;\cdots\;{{\boldsymbol{u}}_N}]$ 中对应于采样节点和未采样节点的行向量所构成的子矩阵.${{\tilde{\boldsymbol f}}_L} \in {{\mathbf{R}}^B}$ 和${{\tilde{\boldsymbol f}}_H} \in {{\mathbf{R}}^{(N - B)}}$ 分别表示对应于前$B$ 个图频率分量(图低频分量)和第$(B + 1)$ 至第$N$ 个图频率分量(图高频分量).$$\begin{split} \left[ \begin{array}{l} {{{\tilde{\boldsymbol f}}}_L} \\ {{{\tilde{\boldsymbol f}}}_H} \end{array} \right] = \left[ \begin{array}{l} {{\boldsymbol{U}}^{\rm{T}}_L}{(S)}{\rm{ }}\;\quad{{\boldsymbol{U}}^{\rm{T}}_L}{({S^c})} \\ {{\boldsymbol{U}}^{\rm{T}}_H}{(S)}{\rm{ }}\quad{{\boldsymbol{U}}^{\rm{T}}_H}{({S^c})} \end{array} \right]\left[ \begin{array}{l} \Phi (S){\boldsymbol{f}}(S) \\ \Phi ({S^c}){\boldsymbol{f}}({S^c}) \end{array} \right] \end{split}$$ (4) $$ \left[ \begin{array}{l} {\Phi(S)}{\boldsymbol{f}}(S) \\ {\Phi ({S^c})}{\boldsymbol{f}}({S^c}) \end{array} \right] = \left[ \begin{array}{l} {{\boldsymbol{U}}_L}(S){\rm{ }}\;\,\quad{{\boldsymbol{U}}_H}(S) \\ {{\boldsymbol{U}}_L}({S^c}){\rm{ }}\quad{{\boldsymbol{U}}_H}({S^c}) \end{array} \right]\left[ \begin{array}{l} {{{\tilde{\boldsymbol f}}}_L} \\ {{{\tilde{\boldsymbol f}}}_H} \end{array} \right] $$ (5) 在ILSR算法中[19], Narang等根据图带限特性——
${{\tilde{\boldsymbol f}}_H} = {\mathbf{0}}$ , 将式(5)表示为式(6). 该算法的重构准则是通过中间变量——带限图信号的低频分量${{\tilde{\boldsymbol f}}_L}$ 恒定, 建立了采样信号${\boldsymbol{f}}(S)$ 和未采样信号${\boldsymbol{f}}({S^c})$ 之间的联系, 得到重构信号的闭式解, 如式(6)所示.$$\begin{split} {\boldsymbol{f}}({S^c}) =\;&\Phi^{\rm{T}} ({S^c}){{\boldsymbol{U}}_L}({S^c}){[{{\boldsymbol{U}}^{\rm{T}}_L}{(S)}{{\boldsymbol{U}}_L}(S)]^{ - 1}}\times \\ &{{\boldsymbol{U}}^{\rm{T}}_L}{(S)}\Phi (S){\boldsymbol{f}}(S) \end{split} $$ (6) 与ILSR算法的重构准则不同, 本文提出的重构准则是基于采样信号的高频分量和未采样信号的图高频分量之和为零; 即图带限分量的恒等不变性, 如式(7)所示. 根据此特性, 可得重构信号
${\boldsymbol{f}}({{{S}}^c})$ 的闭式解, 如式(8)所示.$$ {{\tilde{\boldsymbol f}}_H} = [{{\boldsymbol{U}}^{\rm{T}}_H}{(S)}{\rm{ }}\quad{{\boldsymbol{U}}^{\rm{T}}_H}{({S^c})}]\left[ \begin{array}{l} {\Phi (S)}{\boldsymbol{f}}(S) \\ {\Phi ({S^c})}{\boldsymbol{f}}({S^c}) \end{array} \right] $$ (7) $$ \begin{split} {\boldsymbol{f}}({S^c}) =\;& - \Phi^{\rm{T}} {({S^c})}{[{{\boldsymbol{U}}_H}({S^c}){{\boldsymbol{U}}^{\rm{T}}_H}{({S^c})}]^{ - 1}} \times\\ &{{\boldsymbol{U}}_H}({S^c}){{\boldsymbol{U}}^{\rm{T}}_H}{(S)}\Phi (S){\boldsymbol{f}}(S) \end{split} $$ (8) 然而, 由于闭式解中涉及到矩阵逆运算, 导致求解的计算开销大. 尤其是当处理大规模图网络数据时, 计算和存储的成本都较高. 为了避免此问题, 本文基于图带限分量的恒等不变性提出如式(9)所示的重构模型, 采用迭代求解实现重构带限图信号的目的. 该模型的目标函数利用了图带限分量的恒等不变性, 将其建模为最小二乘模型. 进而估计出未采样信号
${\boldsymbol{f}}({S^c})$ .$$ \begin{split} & \mathop {{\rm{min}}}\limits_{{\boldsymbol{f}}({S^c})} {\rm{ }}\left\| {{\boldsymbol{Y}} - {{\boldsymbol{U}}^{\rm{T}}_H}{{({S^c})}}\Phi ({S^c}){\boldsymbol{f}}({S^c})} \right\|_2^2 \\ & {\rm{ }}{\rm{s.t}}\;\;\;{\rm{ }}{\boldsymbol{Y}} = - {{\boldsymbol{U}}^{\rm{T}}_H}{(S)}\Phi (S){\boldsymbol{f}}(S) \end{split} $$ (9) 通过设计该重构模型的求解算法, 本文提出了基于谱移位的带限图信号重构算法(Reconstruction algorithm of band-limited graph signals based graph frequency shifting, BGSR-GFS). BGSR-GFS算法流程如算法1所示.
算法 1. BGSR-GFS算法
输入.
${{\boldsymbol{U}}_1} = {{\boldsymbol{U}}_H}({S^c}){{\boldsymbol{U}}^{\rm{T}}_H}{({S^c})},$ $\sigma, $ $\Phi (S)$ $\Phi ({S^c}),$ ${\boldsymbol{f}}(S),$ $K,$ ${{\boldsymbol{U}}_2} = {{\boldsymbol{U}}_{{H}}}({S^c}){{\boldsymbol{U}}^{\rm{T}}_{{H}}}{(S)}$ 输出.
${{\boldsymbol{f}}_{{R}}}({S^c})$ 初始化.
${\boldsymbol{f}}({S^c}) = {\mathbf{0}},$ ${{\boldsymbol{H}}^{(1)}} = {\bf {I}},$ $k = 1$ ${{\boldsymbol{G}}^{(1)}} = {{\boldsymbol{U}}_2}{\boldsymbol{\Phi }}(S){\boldsymbol{f}}(S)$ 当
$k \leq K,$ 则:步骤 1.
${{\boldsymbol{d}}^{(k)}} = - {{\boldsymbol{H}}^{(k)}}{{\boldsymbol{G}}^{(k)}}$ ;步骤 2.
${\alpha ^{(k)}} = \frac{{{({\boldsymbol{G}}^{(k)})}^{{\rm{ T}}}{{\boldsymbol{d}}^{(k)}}}}{{{({\boldsymbol{d}}^{(k)})}^{{\rm{ T}}}{{\boldsymbol{U}}_1}{{\boldsymbol{d}}^{(k)}}}}$ ;步骤 3.
${\boldsymbol{f}}^{(k + 1)}{({S^c})} = {\boldsymbol{f}}^{(k)}{({S^c})} + {\alpha ^{(k)}}{{\boldsymbol{d}}^{(k)}} ;$ 步骤 4.
${{\boldsymbol{G}}^{(k + 1)}} = {{\boldsymbol{U}}_1}{\boldsymbol{f}}^{(k + 1)}{({S^c})} + {{\boldsymbol{G}}^{(1)}}$ ;步骤 5.
${{\boldsymbol{p}}^{(k)}} = {\boldsymbol{f}}^{(k + 1)}{({S^c})} - {\boldsymbol{f}}{({S^c})^{(k)}}$ ;步骤 6.
${{\boldsymbol{q}}^{(k)}} = {{\boldsymbol{G}}^{(k + 1)}} - {{\boldsymbol{G}}^{(k)}}$ ;步骤 7.
$$\Delta {{\boldsymbol{H}}^{(k)}} = \frac{{{{\boldsymbol{p}}^{(k)}}{({\boldsymbol{p}}^{(k)})}^{\rm{T}}}}{{{({\boldsymbol{p}}^{(k)})}^{\rm{T}}{{\boldsymbol{q}}^{(k)}}}} - \frac{{{{\boldsymbol{H}}^{(k)}}{{\boldsymbol{q}}^{(k)}}{({\boldsymbol{q}}^{(k)})}^{\rm{T}}{{\boldsymbol{H}}^{(k)}}}}{{{({\boldsymbol{p}}^{(k)})}^{\rm{T}}{{\boldsymbol{H}}^{(k)}}{{\boldsymbol{q}}^{(k)}}}}$$ 步骤 8.
${{\boldsymbol{H}}^{(k + 1)}} = {{\boldsymbol{H}}^{(k + 1)}} + \Delta {{\boldsymbol{H}}^{(k)}} ;$ 步骤 9. 若
$\big\| {{\boldsymbol{f}}^{(k + 1)}{{({S^c})}} - {\boldsymbol{f}}^{(k)}{{({S^c})}}}\big\|_2^2$ , 小于门限阈值$\sigma $ 或达到最大迭代次数$K\,(k > K)$ , 则终止迭代, 输出参数${\boldsymbol{f}}{({S^c})^{(k + 1)}},$ 否则, 继续迭代, 跳转至步骤1;步骤 10.
${{\boldsymbol{f}}_{{R}}}({S^c}) = {\boldsymbol{f}}^{(k + 1)}{({S^c})}$ .该重构算法基于拟牛顿法进行迭代求解. 在高效利用其重构模型二阶梯度信息的同时, 避免了海森矩阵的求解.
当且仅当
${{\boldsymbol{U}}^{\rm{T}}_H}{({S^c})}$ 满足其行数大于等于列数时, 该最小二乘问题有唯一解. 可知BGSR-GFS重构算法的适用条件为图信号具有带限特性且带宽$B$ 小于等于采样节点个数$M$ .2. 基于残差谱移位的重构算法
由于ILSR重构算法并没有对迭代过程中的残差信号进行分析和处理, 所以无论在还是迭代效率上都较为有限. 因此, 针对如何根据迭代残差信号的相关特性提升和迭代效率, 研究者们先后提出了IPR/IWR[20]和IGDR重构算法[21]. 此类基于残差估计的重构算法的关键在于如何根据采样节点的残差信号
${{\boldsymbol{f}}^{(k)}_{{{\rm{Re}}{\rm{s}}}}}(S)$ 估计未采样节点的残差信号${{\boldsymbol{f}}^{(k)}_{{{\rm{Re}} {\rm{s}}}}}({S^c})$ .基于残差估计的重构算法的迭代步骤归纳为公式(10), 不同算法之间的差异在于如何更好地估计采样残差信号
${{\boldsymbol{f}}^{(k)}_{{\rm{Re}} {\rm{s}}}}{(S)}$ 和未采样残差信号${{\boldsymbol{f}}^{(k)}_{{\rm{Re}} {\rm{s}}}}{({{\rm{S}}^c})}$ .$$ \begin{split} {{\boldsymbol{f}}^{(k + 1)}} = \;&\Phi^{\rm{T}} {({S^c})}{\boldsymbol{U}}^{{S^c}}_L\Phi ({S^c}){\boldsymbol{f}}^{(k)}{({S^c})} + \\ &\Phi^{\rm{T}}{(S)}{\boldsymbol{U}}^S_L\Phi (S){\boldsymbol{f}}^{(k)}{(S)} +\\ & {{\boldsymbol{f}}^{(k)}_{{{\rm{Re}}{\rm{s}}}}}{({S^c})} + {{\boldsymbol{f}}^{(k)}_{{{\rm{Re}} {\rm{s}}}}}{(S)} \end{split} $$ (10) $$ \begin{split} &{\boldsymbol{U}}_L^{{S^c}} = {{\boldsymbol{U}}_L}({S^c}){{\boldsymbol{U}}^{\rm{T}}_L}{({S^c})} \qquad\qquad\quad\quad \\ &{\boldsymbol{U}}_L^S = {{\boldsymbol{U}}_L}({S}){{\boldsymbol{U}}^{\rm{T}}_L}{(S)} \\ & {\boldsymbol{f}}^{(k)}{({{S}})} = ({\boldsymbol{I}} - {{\boldsymbol{P}}_S}){{\boldsymbol{f}}^{(k)}} \\ &{\boldsymbol{f}}^{(k)}{({{{S}}^c})} = {{\boldsymbol{P}}_S}{{\boldsymbol{f}}^{(k)}} \end{split} $$ (11) Wang等[20]基于局部聚合的处理方法, 提出了基于局部集采样的IPR和IWR重构算法. 在IWR重构算法中, 采样残差首先进行相应权重的扩大(权重矩阵
${{\boldsymbol{W}}_{{\rm{IPR}}}}$ ), 然后再进行图带限滤波, 如式(12)所示.$$ \begin{split} & {{\boldsymbol{f}}^{(k)}_{{\rm{Res}}}}{(S)} = \Phi^{\rm{T}} {(S)}{\boldsymbol{U}}_L^S\Phi (S) \times\\ &\qquad\qquad\;\;{{\boldsymbol{W}}_{{\rm{IPR}}}}[{\boldsymbol{f}}(S) - {\boldsymbol{f}}^{(k)}{(S)}] \\ & {{\boldsymbol{f}}^{(k)}_{{\rm{Res}}}}{({S^c})} = {\boldsymbol{0}} \end{split} $$ (12) 不同于IWR算法, IPR重构算法通过采样残差
${{\boldsymbol{f}}_{{\rm{Re}}{\rm{ s}}}}({{S}})$ 和网络拓扑特性, 估计未采样残差${{\boldsymbol{f}}_{{\rm{Re}} {\rm{s}}}}({{{S}}^c})$ . 具体而言, 首先是基于局部集内平滑特性, 将未采样残差设置为局部集内的采样残差, 然后再进行图带限滤波. 如式(13)所示, 其中${\boldsymbol{V}}({v_d})$ 为采样节点${v_d}$ 的未采样邻居节点集.$$\begin{split} &{{\boldsymbol{f}}^{(k)}_{{\rm{Res}}}}{(S)} = \Phi^{\rm{T}} {(S)}{\boldsymbol{U}}_L^S\Phi (S)[{\boldsymbol{f}}(S) - {\boldsymbol{f}}^{(k)}{(S)}] \\ &{{\boldsymbol{f}}^{(k)}_{{\rm{Res}}}}{({S^c})} = \Phi^{\rm{T}} {({S^c})}{\boldsymbol{U}}_L^{{S^c}}\Phi ({S^c})[{{{\boldsymbol{f}}^{{\rm{Prop}}}}{(S)}]^{(k)}} \\ & [{{{\boldsymbol{f}}^{{\rm{Prop}}}}{(S)}]^{(k)}}\{ {v_i}\} = {\boldsymbol{f}}(S)\{ {v_d}\} - {\boldsymbol{f}}^{(k)}{(S)}\{ {v_d}\} , \\ &\qquad\qquad\qquad\qquad\qquad\qquad\quad\forall {v_i} \in {\boldsymbol{V}}({v_d}) \end{split} $$ (13) 由于IPR/IWR算法在迭代过程中, 都对采样残差进行相应的预处理工作; 所以相比于ILSR算法, 两种算法的和迭代效率均有提升. 然而, 由于IWR和IPR重构算法对于未采样图信号的迭代残差估计是基于平滑准则, 导致会出现过平滑现象[27]. 为了缓解“过平滑” 问题, Yang等[21]提出基于局部扩散算子的IGDR重构算法, 如式(14)所示. 其中,
$J$ 为采样节点和未采样节点之间的最大跳数,${\delta _j}$ 表示与采样节点集$S$ 和未采样节点${v_j}$ 之间的最短路径相关的指示函数. IGDR算法通过将采样残差经过图带限滤波后得到的全局未采样残差${\boldsymbol{f}}_{{\rm{Re}}{\rm{ s}}}^G({S^c})$ 和采样残差基于随机游走策略得到的局部未采样残差${\boldsymbol{f}}_{{\rm{Re}} {\rm{s}}}^L({S^c})$ 相加, 得到最终的未采样残差${{\boldsymbol{f}}_{{\rm{Re}}{\rm{ s}}}}({S^c})$ .$$ \begin{split} & {{\boldsymbol{f}}^{(k)}_{{\rm{Res}}}}{(S)} = \Phi^{\rm{T}} {(S)}{\boldsymbol{U}}_L^S\Phi (S)[{\boldsymbol{f}}(S) - {\boldsymbol{f}}^{(k)}{(S)}] \\ & {{\boldsymbol{f}}^{(k)}_{{\rm{Res}}}}{({S^c})} = [{\boldsymbol{f}}_{{\rm{Re}} {\rm{s}}}^G{({S^c})]^{(k)}} + [{\boldsymbol{f}}_{{\rm{Re}} {\rm{s}}}^L{({S^c})]^{(k)}} =\\ & \qquad\qquad\qquad{{\boldsymbol{U}}_{\rm{H}}}({S^c}){{\boldsymbol{U}}_{\rm{H}}^{\rm{T}}{(S)}}[{\boldsymbol{f}}(S) - {\boldsymbol{f}}^{(k)}{(S)}]+ \\ & \qquad\qquad\qquad{{\boldsymbol{P}}_S}\sum\limits_{j = 1}^J {{\delta _j}{{\boldsymbol{D}}^{ - 1}}{{\boldsymbol{A}}^j}[{\boldsymbol{f}}(S) - {\boldsymbol{f}}^{(k)}{{(S)}}]} \end{split} $$ (14) 综上所述, IPR/IWR重构算法是基于图平滑滤波估计残差信号, 而IGDR算法是基于图带限特性的原则而设计的. 两种残差重构法都是基于重构信号的低频分量相似性而设计的, 对于高频分量缺乏相应的分析和处理, 导致迭代效率和相比于ILSR算法的提升有限.
根据ILSR算法以及凸集映射原理[18]可知, 在第
$k$ 次迭代中的信号${{\boldsymbol{f}}^{(k)}}$ 满足图带限特性[28]. 因为${{\boldsymbol{f}}^{(k)}}$ 满足图带限特性以及图傅里叶变换具有线性特征, 所以可知残差信号${\boldsymbol{f}}_{{\rm{Re}} s}^{(k)} = {\boldsymbol{f}} - {{\boldsymbol{f}}^{(k)}}$ 同样满足图带限分量的恒等不变性. 由此, 本文设计了一种基于残差谱移位的图信号重构模型, 如式(15)所示.$$ \begin{split} &{{\boldsymbol{f}}^{(k)}_{{\rm{Res}}}}{(S)} = \Phi^{\rm{T}} {(S)}{\boldsymbol{U}}_L^S{\Phi _S}[{\boldsymbol{f}}(S) - {\boldsymbol{f}}^{(k)}{(S)}] \\ & {{\boldsymbol{f}}^{(k)}_{{\rm{Res}}}}{({S^c})} = \Phi^{\rm{T}} {({S^c})}{{\boldsymbol{f}}^*} \\ &\mathop {\min }\limits_{{{\boldsymbol{f}}^ * }}\;\;{\rm{ }}\left\| {{\boldsymbol{Y}} - {{\boldsymbol{U}}^{\rm{T}}_H}{{({S^c})}}{{\boldsymbol{f}}^ * }} \right\|_2^2 \\ &{\rm{s.t}}\qquad{\rm{ }}{\boldsymbol{Y}} = - {{\boldsymbol{U}}^{\rm{T}}_H}{(S)}\Phi (S)[{\boldsymbol{f}}(S) - {\boldsymbol{f}}^{(k)}{(S)}]{\rm{ }} \end{split} $$ (15) 基于此重构模型, 本文提出基于残差谱移位的重构算法(Band-limited graph signals reconstruction based graph frequency shifting of residual signals, BGSR-GFS-R), 算法流程如算法2所示.
算法 2. BGSR-GFS-R算法
输入.
${{\boldsymbol{U}}_1} = {{\boldsymbol{U}}_H}({S^c}){{\boldsymbol{U}}^{\rm{T}}_H}{({S^c})},$ $K,$ $M$ 输出.
${{\boldsymbol{f}}_{{R}}}({S^c})$ .${{\boldsymbol{H}}_{{\rm{BL}}}} = {[{{\boldsymbol{U}}^{\rm{T}}_{{L}}}{(S)}{\rm{ }}\;\;\;{{\boldsymbol{U}}^{\rm{T}}_{{L}}}{(S^c)}]^{\rm{T}}}[{{\boldsymbol{U}}^{\rm{T}}_{{L}}}{(S)}{\rm{ }}\;\;\;{{\boldsymbol{U}}^{\rm{T}}_{{L}}}{(S^c)}]$ ${{\boldsymbol{U}}_2} = {{\boldsymbol{U}}_H}({S^c}){{\boldsymbol{U}}^{\rm{T}}_H}{(S)},$ $\Phi (S),$ $\Phi ({S^c}),$ ${\boldsymbol{f}}(S),$ ${{\boldsymbol{P}}_{{S}}},$ $\sigma $ 步骤 1. 初始化
$k = 1,$ ${{\boldsymbol{f}}^{(1)}} = {{\boldsymbol{H}}_{{\rm{BL}}}}{\boldsymbol{f}}(S)$ ;当
$k \leq K,$ 则:步骤 2.
${{\boldsymbol{f}}_{{\rm{Res}}}}(S) = {\boldsymbol{f}}(S) - {{\boldsymbol{P}}_S}{{\boldsymbol{f}}^{(k)}}$ ;步骤 3. 设置零向量
${{\boldsymbol{f}}^{(1)}_{{\rm{Res}}}}{({S^c})}$ , 单位矩阵${{\boldsymbol{H}}^{(1)}}$ ,${{\boldsymbol{G}}^{(1)}} = $ $ {{\boldsymbol{U}}_{\rm{2}}}{\boldsymbol{\Phi }}(S){{\boldsymbol{f}}_{{\rm{Res}}}}(S),$ 初始化$m=1 $ ;当
$m \leq M$ , 则:步骤 4.
${{\boldsymbol{d}}^{(m)}} = - {{\boldsymbol{H}}^{(m)}}{{\boldsymbol{G}}^{(m)}}$ ;步骤 5.
${\alpha ^{(m)}} = \frac{{{({\boldsymbol{G}}^{(m)})}^{{\rm{ T}}}{{\boldsymbol{d}}^{(m)}}}}{{{({\boldsymbol{d}}^{(m)})}^{{\rm{ T}}}{{\boldsymbol{U}}_1}{{\boldsymbol{d}}^{(m)}}}}$ ;步骤 6.
${{\boldsymbol{f}}^{(m + 1)}_{{\rm{Res}}}}{({S^c})} = {\boldsymbol{f}}^{(m)}{({S^c})} + {\alpha ^{(m)}}{{\boldsymbol{d}}^{(m)}}$ ;步骤 7.
${{\boldsymbol{G}}^{(m + 1)}} = {{\boldsymbol{U}}_1}{\boldsymbol{f}}^{(m + 1)}{({S^c})} + {{\boldsymbol{G}}^{(1)}}$ ;步骤 8.
${{\boldsymbol{p}}^{(m)}} = {{\boldsymbol{f}}^{(m + 1)}_{{\rm{Res}}}}{({S^c})} - {{\boldsymbol{f}}^{(m)}_{{\rm{Res}}}}{({S^c})}$ ;步骤 9.
${{\boldsymbol{q}}^{(m)}} = {{\boldsymbol{G}}^{(m + 1)}} - {{\boldsymbol{G}}^{(m)}}$ ;步骤 10.
$$\Delta {{\boldsymbol{H}}^{(m)}} = \frac{{{{\boldsymbol{p}}^{(m)}}{({\boldsymbol{p}}^{(m)})}^{\rm{T}}}}{{{({\boldsymbol{p}}^{(m)})}^{\rm{T}}{{\boldsymbol{q}}^{(m)}}}} - \frac{{{{\boldsymbol{H}}^{(m)}}{{\boldsymbol{q}}^{(m)}}{({\boldsymbol{q}}^{(m)})}^{\rm{T}}{{\boldsymbol{H}}^{(m)}}}}{{{({\boldsymbol{p}}^{(m)})}^{\rm{T}}{{\boldsymbol{H}}^{(k)}}{{\boldsymbol{q}}^{(m)}}}}$$ 步骤 11.
${{\boldsymbol{H}}^{(m + 1)}} = {{\boldsymbol{H}}^{(m + 1)}} + \Delta {{\boldsymbol{H}}^{(m)}}$ ;步骤 12. 若
$\big\| {{{\boldsymbol{f}}^{(m + 1)}_{{\rm{Res}}}}{{({S^c})}} - {{\boldsymbol{f}}^{(m)}_{{\rm{Res}}}}{{({S^c})}}} \big\|_2^2$ 小于门限阈值$\sigma $ 或达到最大迭代次数$M\,(m > M),$ 则终止迭代, 输出参数${{\boldsymbol{f}}^{(k + 1)}_{{\rm{Res}}}}{({S^c})},$ 否则, 继续迭代, 跳转至步骤4;步骤 13.
$[{{\boldsymbol{f}}^ * _{{\rm{Res}}}}{({S^c})}]^{(k)} = \Phi^{\rm{T}} {({S^c})}{{\boldsymbol{f}}^{(m + 1)}_{{\rm{Res}}}}{({S^c})}$ ;步骤 14.
${{\boldsymbol{f}}^{(k + 1)}} = {{\boldsymbol{H}}_{{\rm{BL}}}}{{\boldsymbol{f}}^{(k)}} + {{\boldsymbol{H}}_{{\rm{BL}}}}{{\boldsymbol{f}}_{{\rm{Res}}}}(S) + [{{\boldsymbol{f}}^ *_{{\rm{Res}}}}{({S^c}) }]^{(k)}$ 步骤 15. 若
$\left\| {{{\boldsymbol{f}}^{(k + 1)}} - {{\boldsymbol{f}}^{(k)}}} \right\|_2^2$ 小于门限阈值$\sigma $ 或达到最大迭代次数$K\,(k >K),$ 则终止迭代, 输出参数${{\boldsymbol{f}}^{(k + 1)}},$ 否则, 继续迭代, 跳转至步骤2;步骤 16.
${{\boldsymbol{f}}_{{R}}}({S^c}) = ({\boldsymbol{I}} - {{\boldsymbol{P}}_{{S}}}){{\boldsymbol{f}}^{(k + 1)}}$ .本文提出的BGSR-GFS-R重构算法基于迭代中的采样残差信号和谱移位策略, 估计得到未采样残差信号. 然后将未采样残差信号与经过带限图滤波后的未采样信号相加, 最终得到重构后的未采样信号. 相比于其他基于残差处理的重构算法(IWR/IPR和IGDR), 本算法对于残差信号的处理不依赖与图网络的子图集合. 并且, 由于本算法利用的是其残差信号的图带限分量的恒等不变性, 将其建模为最小二乘问题后进行迭代求解, 避免了“过平滑”现象.
由于其残差信号的图傅里叶变换的变换矩阵同样为
${{\boldsymbol{U}}^{\rm{T}}_{{H}}}{({S^c})} ,$ 所以要求矩阵${\boldsymbol{U}}_1={\boldsymbol{U}}_0({S^c}){\boldsymbol{U}}^{\rm{T}}_0({S^c})$ 为满秩矩阵, 即采样节点个数不小于带限信号的带宽.3. 分段带限图信号重构算法
在现有的图信号重构算法中, 所针对的图信号往往具备平滑或者是低频段受限的信号特征; 即各节点的信号值与其邻居节点的信号值差异较小, 在图谱域上呈现出能量较为集中在低频区域内. 除此以外, 由于在实际情况中由于物理设备及传输手段的限制, 采集得到的图信号中往往存在着少量的异常节点数据[7]. 将上述的数据集基于地理距离建模为图信号后, 本文发现由于其中存在着少量节点信号值与其邻居节点的信号值差异较大, 其在图谱域上所呈现的是类似于分段带限的信号特性, 如图2所示.
$$ \left[ \begin{array}{l} {{{\tilde{\boldsymbol f}}}_L} \\ {{{\tilde{\boldsymbol f}}}_0} \\ {{{\tilde{\boldsymbol f}}}_H} \end{array} \right] = \left[ \begin{array}{l} {{\boldsymbol{U}}^{\rm{T}}_L}{(S)}{\rm{ }}\quad{{\boldsymbol{U}}^{\rm{T}}_L}{({S^c})} \\ {{\boldsymbol{U}}^{\rm{T}}_0}{(S)}{\rm{ }}\quad{{\boldsymbol{U}}^{\rm{T}}_0}{({S^c})} \\ {{\boldsymbol{U}}^{\rm{T}}_H}{(S)}{\rm{ }}\quad{{\boldsymbol{U}}^{\rm{T}}_H}{({S^c})} \end{array} \right]\left[ \begin{array}{l} {\Phi (S)}{\boldsymbol{f}}(S) \\ {\Phi ({S^c})}{\boldsymbol{f}}({S^c}) \end{array} \right] $$ (16) 针对分段带限图信号的重构问题, 本文上述两种重构算法同样适用. 由
${{\tilde{\boldsymbol f}}_0} = {\mathbf{0}}$ , 可知分段带限图信号在图频率${\lambda _i} \in ({\omega _1},{\omega _2})$ 内, 同样满足图带限分量的恒等不变性, 如式(16)所示. 基于上述分析, 本文提出分段带限图信号重构的优化模型, 如式(17)所示.$$\begin{split} & \mathop {\min }\limits_{{\boldsymbol{f}}({S^c})} {\rm{ }}\left\| {{\boldsymbol{Y}} - {{\boldsymbol{U}}^{\rm{T}}_0}{{({S^c})}}\Phi ({S^c}){\boldsymbol{f}}({S^c})} \right\|_2^2 \\ &{\rm{ s}}{\rm{.t }}\;\;\;\;\;\;{\boldsymbol{Y}} = - {{\boldsymbol{U}}^{\rm{T}}_0}{(S)}\Phi (S){\boldsymbol{f}}(S) \end{split} $$ (17) 基于上述模型, 只需更改重构算法BGSR-GFS/BGSR-GFS-R中的部分输入变量, 便可实现分段带限图信号的重构. 具体而言, 在BGSR-GFS中更改的输入变量
${{\boldsymbol{U}}_1} = {{\boldsymbol{U}}_0}({S^c}){{\boldsymbol{U}}^{\rm{T}}_0}{({S^c})}$ /${{\boldsymbol{U}}_2} = {{\boldsymbol{U}}_0}({S^c}){{\boldsymbol{U}}^{\rm{T}}_0}{(S)}$ , 在GBSR-GFS-R算法中除了同样更新矩阵${{\boldsymbol{U}}_1}$ 和${{\boldsymbol{U}}_2}$ , 还需要将${{\boldsymbol{H}}_{{\rm{BL}}}}$ 更新为${\boldsymbol{H}}_{{\rm{BL}}}^ *$ .$${\boldsymbol{H}}_{{\rm{BL}}}^ * = \left[ \begin{array}{l} {{\boldsymbol{U}}_{{L}}}(S){\rm{ }}\;\;\;\;\;\,{{\boldsymbol{U}}_{{H}}}(S) \\ {{\boldsymbol{U}}_{{L}}}({S^c}){\rm{ }}\;\;\;\;{{\boldsymbol{U}}_{{H}}}({S^c}) \end{array} \right]\left[ \begin{array}{l} {{\boldsymbol{U}}^{\rm{T}}_{{L}}}{(S)}{\rm{ }}\;\;\;\,\;\;{{\boldsymbol{U}}^{\rm{T}}_{{H}}}{(S)} \\ {{\boldsymbol{U}}^{\rm{T}}_{{L}}}{({S^c})}{\rm{ }}\;\;\;\;{{\boldsymbol{U}}^{\rm{T}}_{{H}}}{({S^c})} \end{array} \right]$$ (18) 4. 实验仿真及分析
本文将BGSR-GFS和BGSR-GFS-R重构算法与4种重构算法(ILSR、O-PGIR、IPR和IGDR)进行对比. 由于IPR算法性能优于IWR算法,故实验中只对比了IPR算法; 其次, 由于GBSR-IHT和GBSR-GFT是将ILSR算法变换至图谱域上进行重构, 其迭代效率和ILSR算法一致, 故本文未将其加入对比算法. 本文的实验仿真是在3.40 GHz的Intel i7-6700处理器和16 GB RAM的个人计算机上运行, 使用的软件为MATLAB R2019b.
实验中采用的数据集分别为美国明尼苏达州交通网络(
${G_1}$ )和美国部分主要城市温度网络(${G_2}$ ), 如图3所示.${G_1}$ 是由2642个节点和6608条边构成的, 节点和边分别表示交通网中的十字路口和实际的州际公路[29];${G_2}$ 中的节点个数为218, 节点表示美国主要城市[30], 本文采用$K$ 近邻法构建节点之间的边连接$(K = 5).$ 数据集中的带限图信号是由服从高斯分布的随机信号经过带限图滤波后构成的. 数据集${G_1}$ 的截止频率为0.4077, 数据集${G_2}$ 的截止频率为0.3698. 迭代阈值$\sigma $ 设置为$1 \times {10^{ - 8}}$ .本文采用的采样策略为贪婪采样[20]和随机采样. 基于贪婪采样策略,
${G_1}$ 和${G_2}$ 分别得到的采样节点数为873和33, 如图3所示. 为了公平地比较各算法在不同采样情况下的重构效果, 仿真中随机采样的节点个数与贪婪采样一致. 本文采用重构信号和原始信号之间的相对误差(Relative error, RE) 评估算法的重构精度, 如式(19)所示. 其中${\boldsymbol{f}}_S^R$ 和${{\boldsymbol{f}}_S}$ 分别表示重构信号和原始信号.$${{RE}} = \frac{{\left\| {{{\boldsymbol{f}}_S} - {\boldsymbol{f}}_S^R} \right\|_2^2}}{{\left\| {{{\boldsymbol{f}}_S}} \right\|_2^2}}$$ (19) 在无噪情况中, 不同算法的重构性能如图4所示, 其中图4(a)和图4(c)表示基于随机采样的
${G_1}$ 和${G_2}$ 数据集的重构性能, 图4(b)和图4(d)表示基于贪婪采样的${G_1}$ 和${G_2}$ 数据集的重构性能. 相比于ILSR和O-PGIR算法, 由于BGSR-GFS算法利用了信号的图高频分量特征; 无论采用随机采样或贪婪采样, 新算法都具有更优的迭代效率和重构误差. 此外, 由于BGSR-GFS-R算法基于残差信号的图高频分量特征进行重构, 该算法能够高效地估计未采样节点的残差信号, 相比于IPR和IGDR重构算法, 其迭代效率和重构精度均有提升.
图 4 无噪环境下带限图信号重构性能对比Fig. 4 Comparison of graph band-limited signals reconstruction performances in noiseless environment如图4(a)所示, 本文将重构算法应用于基于随机采样的
${G_1}$ 数据集中, BGSR-GFS和BGSR-GFS-R的重构精度分别为$3.75 \times {10^{ - 15}}$ 和$2.07 \times {10^{ - 15}}$ , 算法ILSR, OPGIR, IPR和IGDR的重构精度分别为$1.05 \times {10^{ - 7}}$ ,$7.82 \times {10^{ - 13}}$ ,$8.13 \times {10^{ - 15}}$ 和$6.51 \times $ $ {10^{ - 15}}$ . 而基于贪婪采样, BGSR和BGSR-GFS-R的重构精度分别为$3.79 \times {10^{ - 15}}$ 和$2.70 \times {10^{ - 15}}$ , ILSR, OPGIR, IPR和IGDR的重构精度分别为$1.47 \times $ $ {10^{ - 14}}$ ,$9.01 \times {10^{ - 15}}$ ,$7.06 \times {10^{ - 15}}$ 和$6.51 \times {10^{ - 15}}$ . 新算法的重构精度提升40 % ~ 70 %.如图4(b)所示, 本文将重构算法应用于数据集
${G_2}$ 中, 新算法BGSR-GFS的重构精度分别为$1.97 \times {10^{ - 15}}$ (随机)和$2.47 \times {10^{ - 15}}$ (贪婪), 新算法BGSR-GFS-R的重构精度分别为$5.53 \times {10^{ - 16}}$ (随机)和$7.51 \times {10^{ - 16}}$ (贪婪). 在随机采样中, 算法ILSR, OPGIR, IPR和IGDR的重构精度分别为$7.59 \times $ $ {10^{ - 6}}$ ,$6.73 \times {10^{ - 11}}$ ,$2.76 \times {10^{ - 15}}$ 和$3.42 \times {10^{ - 15}}$ . 基于贪婪采样,算法ILSR, OPGIR, IPR和IGDR的重构精度分别为$1.65 \times {10^{ - 9}}$ ,$4.65 \times {10^{ - 15}}$ ,$1.43 \times {10^{ - 15}}$ 和$3.40 \times {10^{ - 15}}$ . 相比于其他算法, 新算法的重构精度提升约60 %.如表1和表2所示, 相比于ILSR和O-PGIR算法, BGSR-GFS算法的重构效率提升70 %. 相比于ILSR和O-PGIR算法, BGSR-GFS-R算法的重构效率提升75 %.
表 1 无噪情况下基于随机采样的${G_1}$ 重构效率Table 1${G_1}$ reconstruction efficiency of random sampling in noiseless算法 迭代次数 运行时间 (s) ILSR 220 139.99 OPGIR 114 108.78 IPR 96 61.87 IGDR 33 20.47 BGSR-GFS 27 5.73 BGSR-GFS-R 8 8.97 表 2 无噪情况下基于随机采样的${G_2}$ 重构效率Table 2${G_2}$ reconstruction efficiency of random sampling in noiseless算法 迭代次数 运行时间 (s) ILSR 269 0.1509 OPGIR 139 0.1291 IPR 64 0.0405 IGDR 34 0.0271 BGSR-GFS 7 0.0065 BGSR-GFS-R 5 0.0146 为了对比噪声环境中的算法的鲁棒性, 本文在采样信号中分别加入信噪比为
$20\;{\rm{dB}}$ 和$40\;{\rm{dB}}$ 的随机高斯噪声. 信号重构性能对比如图5所示, 本文提出的重构算法和对比算法的抗噪鲁棒性相同, 然而BGSR-GFS和BGSR-GFS-R的迭代效率更高. 无论是本文算法还是对比算法均没有进行噪声抑制或消除的步骤, 导致无法消除噪声对于重构性能的影响.
图 5 含噪环境下带限图信号重构性能对比Fig. 5 Comparison of graph band-limited signals reconstruction performances in noisy environment在第3组仿真中, 本文将针对分段带限图信号进行重构性能对比. 本文将第1组仿真实验中的图信号加入高频分量. 即随机选取
${{Q}}$ 个连续的高频分量后, 再通过图傅里叶逆变换得到分段带限图信号(${G_1}$ 和${G_2}$ 的${{Q}}$ 值分别为10和3). 为了确保对比试验的公平性, 本文将对比算法中的低通图滤波器调整为带通图滤波器.如图6所示, 无论是基于随机采样或贪婪采样, 本文算法都具有良好的重构精度和迭代效率. 由于ILSR和O-PGIR算法都是利用图信号的低频分量相似性原则设计重构算法, 而没有考虑到图信号的高频段分量的差异性, 所以迭代效率十分有限. 算法IPR在ILSR的基础上, 基于相邻节点残差信号等值传递的原则进行迭代过程中增量的估计, 而算法IGDR在IPR的基础上增加了扩散策略, 进一步提高了迭代效率; 两种基于残差法的重构策略实质上都是利用了残差信号低频分量之间的相似性, 同样无法实现高效的信号重构. 与上述4种算法不同的是, 由于本文提出的两种算法同时考虑了图低频相似性和图高频差异性, 通过图谱域移位策略重构分段带限图信号, 具有较高的重构精度和迭代效率.
图 6 分段带限图信号重构性能对比Fig. 6 Comparison of graph separate band-limited signals reconstruction performances5. 结束语
本文针对带限图信号的重构问题, 提出了基于图带限分量恒等特性的重构模型. 通过将该重构模型转化为最小二乘问题, 本文提出了两种基于图谱域移位的重构算法. 此外, 本文所提出的新算法同样适用于分段带限图信号的重构问题. 最后, 数值仿真表明, 相比于其他重构算法, 本文算法的重构性能更优.
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图 2 基于空域/频域的NR-IQA方法分类
Fig. 2 Classification of spatial/spectral domain-based NR-IQA methods
图 4 不同类型NR-IQA方法在不同人工模糊数据集中平均性能评价指标值比较
Fig. 4 Average performance evaluation result comparison through different types of NR-IQA methods for different artificial blur databases
图 5 不同类型NR-IQA方法在不同自然模糊数据集中平均性能评价指标值比较
Fig. 5 Average performance evaluation result comparison through different types of NR-IQA methods for different natural blur databases
表 1 含有模糊图像的主要图像质量评价数据集
Table 1 Main image quality assessment databases including blurred images
数据集 时间 参考图像 模糊图像 模糊类型 主观评价 分值范围 IVC[28] 2005 4 20 高斯模糊 MOS 模糊−清晰 [1 5] LIVE[22] 2006 29 145 高斯模糊 DMOS 清晰−模糊 [0 100] A57[30] 2007 3 9 高斯模糊 DMOS 清晰−模糊 [0 1] TID2008[26] 2009 25 100 高斯模糊 MOS 模糊−清晰 [0 9] CSIQ[25] 2009 30 150 高斯模糊 DMOS 清晰−模糊 [0 1] VCL@FER[29] 2012 23 138 高斯模糊 MOS 模糊−清晰 [0 100] TID2013[27] 2013 25 125 高斯模糊 MOS 模糊−清晰 [0 9] KADID-10k 1[31] 2019 81 405 高斯模糊 MOS 模糊−清晰 [1 5] KADID-10k 2[31] 2019 81 405 镜头模糊 MOS 模糊−清晰 [1 5] KADID-10k 3[31] 2019 81 405 运动模糊 MOS 模糊−清晰 [1 5] MLIVE1[33] 2012 15 225 高斯模糊和高斯白噪声 DMOS 清晰−模糊 [0 100] MLIVE2[33] 2012 15 225 高斯模糊和JEPG压缩 DMOS 清晰−模糊 [0 100] MDID2013[32] 2013 12 324 高斯模糊、JEPG压缩和白噪声 DMOS 清晰−模糊 [0 1] MDID[34] 2017 20 1600 高斯模糊、对比度变化、高斯噪声、
JPEG或JPEG2000MOS 模糊−清晰 [0 8] BID[21] 2011 — 586 自然模糊 MOS 模糊−清晰 [0 5] CID2013[35] 2013 — 480 自然模糊 MOS 模糊−清晰 [0 100] CLIVE[36-37] 2016 — 1162 自然模糊 MOS 模糊−清晰 [0 100] KonIQ-10k [38] 2018 — 10073 自然模糊 MOS 模糊−清晰 [1 5] 
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表 2 基于空域/频域的不同方法优缺点对比
Table 2 Advantage and disadvantage comparison for different methods based on spatial/spectral domain
方法分类 优点 缺点 边缘信息 概念直观、计算复杂度低 容易因图像中缺少锐利边缘而影响评价结果 再模糊理论 对图像内容依赖小, 计算复杂度低 准确性依赖 FR-IQA 方法 奇异值分解 能较好地提取图像结构、边缘、纹理信息 计算复杂度较高 自由能理论 外部输入信号与其生成模型可解释部分之间的
差距与视觉感受的图像质量密切相关计算复杂度高 DFT/DCT/小波变换 综合了图像的频域特性和多尺度特征, 准确性和鲁棒性更高 计算复杂度高
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表 3 基于学习的不同方法优缺点对比
Table 3 Advantage and disadvantage comparison for different methods based on learning
方法分类 优点 缺点 SVM 在小样本训练集上能够取得比其他算法更好的效果 评价结果的好坏由提取的特征决定 NN 具有很好的非线性映射能力 样本较少时, 容易出现过拟合现象, 且
计算复杂度随着数据量的增加而增大深度学习 可以从大量数据中自动学习图像特征的多层表示 对数据集中数据量要求大 字典/码本 可以获得图像中的高级特征 字典/码本的大小减小时, 性能显著下降 MVG 无需图像的 MOS/DMOS 值 模型建立困难, 对数据集中数据量要求较大
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表 4 用于对比的不同NR-IQA方法
Table 4 Different NR-IQA methods for comparison
方法类别 方法 特征 模糊/通用 空域/频域 空域 边缘信息 JNB[43] 计算边缘分块所对应的边缘宽度 模糊 边缘信息 CPBD[44] 计算模糊检测的累积概率 模糊 边缘信息 MLV[47] 计算图像的最大局部变化得到反映图像对比度信息的映射图 模糊 自由能理论 ARISM[63] 每个像素 AR 模型系数的能量差和对比度差 模糊 边缘信息 BIBLE[49] 图像的梯度和 Tchebichef 矩量 模糊 边缘信息 Zhan 等[14] 图像中最大梯度及梯度变化量 模糊 频域 DFT变换 S3[65] 在频域测量幅度谱的斜率, 在空域测量空间变化情况 模糊 小波变换 LPC-SI[81] LPC 强度变化作为指标 模糊 小波变换 BISHARP[77] 计算图像的均方根来获取图像局部对比度信息,
同时利用小波变换中对角线小波系数模糊 HVS滤波器 HVS-MaxPol[85] 利用 MaxPol 卷积滤波器分解与图像清晰度相关的有意义特征 模糊 学习 机器学习 SVM+SVR BIQI[86] 对图像进行小波变换后, 利用 GGD 对得到的子带系数进行参数化 通用 SVM+SVR DIIVINE[87] 从小波子带系数中提取一系列的统计特征 通用 SVM+SVR SSEQ[88] 空间−频域熵特征 通用 SVM+SVR BLIINDS-II[91] 多尺度下的广义高斯模型形状参数特征、频率变化系数特征、
能量子带特征、基于定位模型的特征通用 SVR BRISQUE[96] GGD 拟合 MSCN 系数作为特征, AGGD 拟合 4 个相邻元素乘积系数作为特征 通用 SVR RISE[107] 多尺度图像空间中的梯度值和奇异值特征, 以及多分辨率图像的熵特征 模糊 SVR Liu 等[109] 局部模式算子提取图像结构信息, Toggle 算子提取边缘信息 模糊 SVR Cai 等[110] 输入图像与其重新模糊版本之间的 Log-Gabor 滤波器响应差异和基于方向
选择性的模式差异, 以及输入图像与其 4 个下采样图像之间的自相似性模糊 深度学习 CNN Kang＇s CNN[116] 对图像分块进行局部对比度归一化 通用 浅层CNN+GRNN Yu＇s CNN[127] 对图像分块进行局部对比度归一化 模糊 聚类技术+RBM MSFF[139] Gabor 滤波器提取不同方向和尺度的原始图像特征,
然后由 RBMs 生成特征描述符通用 DNN MEON[132] 原始图像作为输入 通用 CNN DIQaM-NR[131] 使用 CNN 提取失真图像块和参考图像块的特征 通用 CNN DIQA[118] 图像归一化后, 通过下采样及上采样得到低频图像 通用 CNN SGDNet[133] 使用 DCNN 作为特征提取器获取图像特征 通用 秩学习 Rank Learning[141] 选取一定比例的图像块集合作为输入, 梯度信息被用来指导图像块选择过程 模糊 DCNN+SFA SFA[128] 多个图像块作为输入, 并使用预先训练好的 DCNN 模型提取特征 模糊 DNN+NSS NSSADNN[134] 每个图像块归一化后用 CNNs 提取特征, 得到 1024 维向量 通用 CNN DB-CNN[123] 用预训练的 S-CNN 及 VGG-16 分别提取合成失真与真实图像的相关特征 通用 CNN CGFA-CNN[124] 用 VGG-16 以提取失真图像的相关特征 通用 字典/码本 聚类算法+码本 CORNIA[145] 未标记图像块中提取局部特征进行 K-means 聚类以构建码本 通用 聚类算法+码本 QAC[147] 用比例池化策略估计每个分块的局部质量,
通过 QAC 学习不同质量级别上的质心作为码本通用 稀疏学习+字典 SPARISH[143] 以图像块的方式表示模糊图像, 并使用稀疏系数计算块能量 模糊 MVG MVG模型 NIQE[150] 提取 MSCN 系数, 再用 GGD 和 AGGD 拟合得到特征 通用
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表 5 基于深度学习的方法所采用的不同网络结构
Table 5 Different network structures of deep learning-based methods
方法 网络结构 Kang＇s CNN[116] 包括一个含有最大/最小池化的卷积层, 两个全连接层及一个输出结点 Yu＇s CNN[127] 采用单一特征层挖掘图像内在特征, 利用 GRNN 评价图像质量 MSFF[139] 图像的多个特征作为输入, 通过端到端训练学习特征权重 MEON[132] 由失真判别网络和质量预测网络两个子网络组成, 并采用 GDN 作为激活函数 DIQaM-NR[131] 包含 10 个卷积层和 5 个池化层用于特征提取, 以及 2 个全连接层进行回归分析 DIQA[118] 网络训练分为客观失真部分及与人类视觉系统相关部分两个阶段 SGDNet[133] 包括视觉显著性预测和图像质量预测的两个子任务 Rank Learning[141] 结合了 Siamese Mobilenet 及多尺度 patch 提取方法 SFA[128] 包括 4 个步骤: 图像的多 patch 表示, 预先训练好的 DCNN 模型提取特征,
通过 3 种不同统计结构进行特征聚合, 部分最小二乘回归进行质量预测NSSADNN[134] 采用多任务学习方式设计, 包括自然场景统计 (NSS) 特征预测任务和质量分数预测任务 DB-CNN[123] 两个卷积神经网络分别专注于两种失真图像特征提取, 并采用双线性池化实现质量预测 CGFA-CNN[124] 采用两阶段策略, 首先基于 VGG-16 网络的子网络 1 识别图像中的失真类型, 而后利用子网络 2 实现失真量化
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表 6 基于空域/频域的不同NR-IQA方法在不同数据集中比较结果
Table 6 Comparison of different spatial/spectral domain-based NR-IQA methods for different databases
方法 发表时间 LIVE CSIQ PLCC SROCC RMSE MAE PLCC SROCC RMSE MAE JNB[43] 2009 0.843 0.842 11.706 9.241 0.786 0.762 0.180 0.122 CPBD[44] 2011 0.913 0.943 8.882 6.820 0.874 0.885 0.140 0.111 S3[65] 2012 0.919 0.963 8.578 7.335 0.894 0.906 0.135 0.110 LPC-SI[81] 2013 0.907 0.923 9.177 7.275 0.923 0.922 0.111 0.093 MLV[47] 2014 0.959 0.957 6.171 4.896 0.949 0.925 0.091 0.071 ARISM[63] 2015 0.962 0.968 5.932 4.512 0.944 0.925 0.095 0.076 BIBLE[49] 2016 0.963 0.973 5.883 4.605 0.940 0.913 0.098 0.077 Zhan 等[14] 2018 0.960 0.963 6.078 4.697 0.967 0.950 0.073 0.057 BISHARP[77] 2018 0.952 0.960 6.694 5.280 0.942 0.927 0.097 0.078 HVS-MaxPol[85] 2019 0.957 0.960 6.318 5.076 0.943 0.921 0.095 0.077 方法 发表时间 TID2008 TID2013 PLCC SROCC RMSE MAE PLCC SROCC RMSE MAE JNB[43] 2009 0.661 0.667 0.881 0.673 0.695 0.690 0.898 0.687 CPBD[44] 2011 0.820 0.841 0.672 0.524 0.854 0.852 0.649 0.526 S3[65] 2012 0.851 0.842 0.617 0.478 0.879 0.861 0.595 0.480 LPC-SI[81] 2013 0.861 0.896 0.599 0.478 0.869 0.919 0.621 0.507 MLV[47] 2014 0.858 0.855 0.602 0.468 0.883 0.879 0.587 0.460 ARISM[63] 2015 0.843 0.851 0.632 0.492 0.895 0.898 0.558 0.442 BIBLE[49] 2016 0.893 0.892 0.528 0.413 0.905 0.899 0.531 0.426 Zhan 等[14] 2018 0.937 0.942 0.410 0.320 0.954 0.961 0.374 0.288 BISHARP[77] 2018 0.877 0.880 0.564 0.439 0.892 0.896 0.565 0.449 HVS-MaxPol[85] 2019 0.853 0.851 0.612 0.484 0.877 0.875 0.599 0.484
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表 7 基于学习的不同NR-IQA方法在不同人工模糊数据集中比较结果
Table 7 Comparison of different learning-based NR-IQA methods for different artificial blur databases
方法 发表
时间LIVE CSIQ TID2008 TID2013 PLCC SROCC PLCC SROCC PLCC SROCC PLCC SROCC BIQI[86] 2010 0.920 0.914 0.846 0.773 0.794 0.799 0.825 0.815 DIIVINE[87] 2011 0.943 0.936 0.886 0.879 0.835 0.829 0.847 0.842 BLIINDS-II[91] 2012 0.939 0.931 0.886 0.892 0.842 0.859 0.857 0.862 BRISQUE[96] 2012 0.951 0.943 0.921 0.907 0.866 0.865 0.862 0.861 CORNIA[145] 2012 0.968 0.969 0.781 0.714 0.932 0.932 0.904 0.912 NIQE[150] 2013 0.939 0.930 0.918 0.891 0.832 0.823 0.816 0.807 QAC[147] 2013 0.916 0.903 0.831 0.831 0.813 0.812 0.848 0.847 SSEQ[88] 2014 0.961 0.948 0.871 0.870 0.858 0.852 0.863 0.862 Kang＇s CNN[116] 2014 0.963 0.983 0.774 0.781 0.880 0.850 0.931 0.922 SPARISH[143] 2016 0.960 0.960 0.939 0.914 0.896 0.896 0.902 0.894 Yu＇s CNN[127] 2017 0.973 0.965 0.942 0.925 0.937 0.919 0.922 0.914 RISE[107] 2017 0.962 0.949 0.946 0.928 0.929 0.922 0.942 0.934 MEON[132] 2018 0.948 0.940 0.916 0.905 — — 0.891 0.880 DIQaM-NR[131] 2018 0.972 0.960 0.893 0.885 — — 0.915 0.908 DIQA[118] 2019 0.952 0.951 0.871 0.865 — — 0.921 0.918 SGDNet[133] 2019 0.946 0.939 0.866 0.860 — — 0.928 0.914 Rank Learning[141] 2019 0.969 0.954 0.979 0.953 0.959 0.949 0.965 0.955 SFA[128] 2019 0.972 0.963 — — 0.946 0.937 0.954 0.948 NSSADNN[134] 2019 0.971 0.981 0.923 0.930 — — 0.857 0.840 CGFA-CNN[124] 2020 0.974 0.968 0.955 0.941 — — — — MSFF[139] 2020 0.954 0.962 — — 0.925 0.928 0.921 0.928 DB-CNN[123] 2020 0.956 0.935 0.969 0.947 — — 0.857 0.844 Liu 等[109] 2020 0.980 0.973 0.955 0.936 — — 0.972 0.964 Cai 等[110] 2020 0.958 0.955 0.952 0.923 — — 0.957 0.941
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表 8 基于学习的不同NR-IQA方法在不同自然模糊数据集中比较结果
Table 8 Comparison of different learning-based NR-IQA methods for different natural blur databases
方法 发表
时间BID CID2013 CLIVE PLCC SROCC PLCC SROCC PLCC SROCC BIQI[86] 2010 0.604 0.572 0.777 0.744 0.540 0.519 DIIVINE[87] 2011 0.506 0.489 0.499 0.477 0.558 0.509 BLIINDS-II[91] 2012 0.558 0.530 0.731 0.701 0.507 0.463 BRISQUE[96] 2012 0.612 0.590 0.714 0.682 0.645 0.607 CORNIA[145] 2012 — — 0.680 0.624 0.665 0.618 NIQE[150] 2013 0.471 0.469 0.693 0.633 0.478 0.421 QAC[147] 2013 0.321 0.318 0.187 0.162 0.318 0.298 SSEQ[88] 2014 0.604 0.581 0.689 0.676 — — Kang＇s CNN[116] 2014 0.498 0.482 0.523 0.526 0.522 0.496 SPARISH[143] 2016 0.356 0.307 0.678 0.661 0.484 0.402 Yu＇s CNN[127] 2017 0.560 0.557 0.715 0.704 0.501 0.502 RISE[107] 2017 0.602 0.584 0.793 0.769 0.555 0.515 MEON[132] 2018 0.482 0.470 0.703 0.701 0.693 0.688 DIQaM-NR[131] 2018 0.476 0.461 0.686 0.674 0.601 0.606 DIQA[118] 2019 0.506 0.492 0.720 0.708 0.704 0.703 SGDNet[133] 2019 0.422 0.417 0.653 0.644 0.872 0.851 Rank Learning[141] 2019 0.751 0.719 0.863 0.836 — — SFA[128] 2019 0.840 0.826 — — 0.833 0.812 NSSADNN[134] 2019 0.574 0.568 0.825 0.748 0.813 0.745 CGFA-CNN[124] 2020 — — — — 0.846 0.837 DB-CNN[123] 2020 0.475 0.464 0.686 0.672 0.869 0.851 Cai 等[110] 2020 0.633 0.603 0.880 0.874 — —
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