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摘要: 基于深度学习的表面缺陷检测技术是工业上的一项重要应用, 而缺陷图像数据集质量对缺陷检测性能有重要影响. 为解决实际工业生产过程中缺陷样本获取成本高、缺陷数据量少的痛点, 提出了一种基于去噪扩散概率模型(Denoising diffusion probabilistic model, DDPM)的缺陷图像生成方法. 该方法在训练过程中加强了模型对缺陷部位和无缺陷背景的差异化学习. 在生成过程中通过缺陷控制模块对生成缺陷的类别、形态、显著性等特征进行精准控制, 通过背景融合模块, 能将缺陷在不同的无缺陷背景上进行迁移, 大大降低新背景上缺陷样本的获取难度. 实验验证了该模型的缺陷控制和缺陷迁移能力, 其生成结果能有效扩充训练数据集, 提升下游缺陷检测任务的准确率.Abstract: Surface defect detection technology based on deep learning is an important application in industry and the quality of defect image dataset has a significant impact on defect detection performance. A defect image generation method based on denoising diffusion probabilistic model (DDPM) is designed to address the pain points of high cost of obtaining defect samples and low amount of defect data in actual industrial production processes. This method enhances the model＇s differential learning of defect locations and defect free backgrounds during the training process. Through the defect control module during the generation process, this method accurately controls the category, morphology, saliency and other features of generated defects. Through the background fusion module, defects can be migrated on different defect free backgrounds, which greatly reducing the difficulty of obtaining defect samples on new backgrounds. The experiment has verified the defect control and defect migration capabilities of the model, and its generated results can effectively expand the training dataset and improve the accuracy of downstream defect detection tasks.
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Key words:
- Data augmentation /
- dataset expansion /
- defect image generation /
- deep learning
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近年来, 张量方法在控制理论及其应用的各个领域得到了广泛的应用, 如基于张量数据表示的深度学习模型[1]、基于非负张量分析的时序链路预测方法[2]、局部图像描述中基于结构张量的HDO (Histograms of dominant orientations)算法[3]、二维解析张量投票算法[4]等.
张量动力系统已经广泛应用于Volterra系统识别[5]、张量乘积TP (Tensor product)模型变换[6]、人体动作识别[7-8]和塑性模型[9]等各个领域.因此, 由于在张量常微分方程解中的关键作用, 张量指数函数的计算已经成为一个重要的研究领域.
考虑如下常微分方程[9]
$$ \begin{align}\label{equation1-1} \left\{ \begin{array}{ll} \dot{\mathcal{Y}}(t)=\mathcal{A}\mathcal{Y}(t) \\ {\mathcal{Y}}(t_0)=\mathcal{Y}_{0} \end{array} \right. \end{align} $$ (1) 这里$\mathcal{A}$和$\mathcal{Y}_{0}$是给定张量, 一般是非对称的, 那么关于系统(1)的张量指数函数${\rm exp}(\cdot)$有如下唯一解:
$$ \begin{align}\label{equation1-2} {\mathcal{Y}}(t)={\rm exp}[(t-t_0)\mathcal{A}]{\mathcal{Y}}_{0} \end{align} $$ (2) 对于一般的张量$\mathcal{A}$, 它的指数函数可以表示为它的级数形式:
$$ \begin{align*}{\rm exp}(\mathcal{A})=\sum\limits_{n=0}^\infty \frac{1}{n!}\mathcal{A}^n\end{align*} $$ 目前计算张量指数函数(2)通常使用的方法是截断法[10], 其实质就是将张量指数函数展开成无穷级数后, 取前$n_{\rm{max}}$项, 得到近似解, 即:
$$ \begin{align*}{\rm exp}(\mathcal{A}) \approx \sum\limits_{n=0}^{n_{\rm{max}}} \frac{1}{n!}\mathcal{A}^n\end{align*} $$ 张量指数函数选取的项数$n_{\rm{max}}$受限于所需的精度:
$$ \begin{align*}\frac{1}{n_{\rm{max}}!}\|\mathcal{A}^{n_{\rm{max}}}\|\le \epsilon_{\rm{tol}}\end{align*} $$ 显然有截断法的精度与张量指数函数选取的项数$n_{\rm{max}}$有关, 保留的项数越多, 精度越高, 但是需要进行的张量乘积次数也就越多; 保留的项数越少, 计算量越少, 但是精度也就越低.因此, 计算张量指数函数的截断法有待改进.
为研究上述问题, 本文首次定义了张量的一种广义逆, 并以此为基础构造了张量广义逆Padé逼近GITPA (Generalized inverse tensor Padé approximation)的一种$\varepsilon$-算法. GITPA方法的优势在于:在计算过程中, 不必用到张量的乘积, 也不要计算张量的逆, 另外, 该方法对奇异张量也是适用的.目前在国内外, 关于计算张量的逆还没有找到一种比较可行的计算方法.作为GITPA方法的一个重要应用, 本文在后面给出计算张量指数函数的数值实验, 来说明$\varepsilon$-算法的有效性.
本文组织如下:第1节简单介绍本文用到的张量基础知识, 并定义张量的一种广义逆; 第2节, 首先给出广义逆张量Padé逼近的定义, 并以此为基础给出张量$\varepsilon$-算法; 第3节将$\varepsilon$-算法用来计算张量指数函数值, 并与通常使用的的级数截断法相比较; 最后给出简单的小结.
1. 一种张量广义逆
1.1 张量基础知识
本节介绍本文将要用到的张量基础知识.张量是一种多维数组, 其中向量是一阶张量, 矩阵是二阶张量, 特别地, 一个$ p $阶张量拥有$ p $个下标, 是$ p $个拥有独立坐标系的向量空间的外积(张量积).一个$ p $阶$ n_1\times n_2\times \cdots \times n_p $维张量可以表示为:
$$ \mathcal{A} = (a_{i_{1}i_{2}\cdots i_{p}})\in {\bf C}^{n_{1}\times n_{2}\times\cdots \times n_{p}} $$ 文献[11]提出了张量的切片方法.对一个三阶张量, 可以固定其中任意一个下标, 从而得到该张量的一种表示形式.设$ \mathcal{A} = (a_{i_{1}i_{2}i_{3}})\in {\bf C}^{2\times 2\times3} $, 固定其第三个下标, 则张量可以表示为
$$ \begin{align*} \mathcal{A} = \, &\left[ \begin{array}{c|c|c} \mathcal{A}_{1} &\mathcal{A}_{2}& \mathcal{A}_{3} \end{array}\right] = \nonumber\\ &\left[ \begin{array}{cc|cc|cc} a_{111} & a_{121} & a_{112} & a_{122} & a_{113} & a_{123}\\ a_{211} & a_{221} & a_{212} & a_{222} & a_{213} & a_{223}\\ \end{array} \right] \end{align*} $$ 下面定义两个三阶张量的$ t $-积, 可以通过递归自然地推广到高阶张量的$ t $-积.
定义1[12]. (块循环矩阵)令$ \mathcal{A} \in {\bf R}^{l\times p\times n} $, 则$ \mathcal{A} $的块循环矩阵被定义为
$$ \begin{equation*} bcirc(\mathcal{A}) = {\left[ \begin{array}{ccccc} \mathcal{A}_{1} & \mathcal{A}_{n} & \mathcal{A}_{n-1} & \cdots & \mathcal{A}_{2} \\ \mathcal{A}_{2} & \mathcal{A}_{1} & \mathcal{A}_{n} & \cdots & \mathcal{A}_{3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \mathcal{A}_{n} & \mathcal{A}_{n-1} & \cdots & \mathcal{A}_{2} & \mathcal{A}_{1} \\ \end{array} \right]}_{ln\times pn} \end{equation*} $$ 定义一个展开算子: $ unfold(\cdot)$[12], 用以下方式将一个${p\times m\times n}$的张量展开成一个${pn\times m}$的矩阵:
$$ \begin{equation*} unfold(\mathcal{B}) = \left[ \begin{array}{c} \mathcal{B}_{1}^{\rm{T}} \mathcal{B}_{2}^{\rm{T}} \cdots \mathcal{B}_{n}^{\rm{T}}\\ \end{array} \right]^{\rm{T}} \end{equation*} $$ 这里$fold(\cdot)$[12]是它的逆算子, 它会将一个$ {pn\times m} $的矩阵转化成$ {p\times m\times n} $的张量.因此,
$$ \begin{equation*} fold(unfold(\mathcal{B})) = \mathcal{B} \end{equation*} $$ 定义2[12-13]. (张量$ t $-积)令$ \mathcal{A} $是一个$ l\times p\times n $的张量, $ \mathcal{B} $是一个$ p\times m\times n $的张量, 则张量$ t $-积$ \mathcal{A}*\mathcal{B} $将得到一个$ l\times m\times n $的张量, 定义如下:
$$ \begin{equation*} \mathcal{A}\ast \mathcal{B} = fold(bcirc(\mathcal{A}) \cdot unfold(B)) \end{equation*} $$ 例1. 设$ \mathcal{A}, \mathcal{B}\in {\bf R}^{2\times 2\times3} $, 现固定其第三个下标, 分别产生
$$ \begin{align*} \begin{array}{rl} \mathcal{A} = &\left[ \begin{array}{cc|cc|cc} 1 & 2 & 5 & 6 & 9 & 10 \\ 3 & 4 & 7 & 8 & 11 & 12 \\ \end{array} \right]\\ \mathcal{B} = &\left[ \begin{array}{cc|cc|cc} 1 & 2 & 4 & 3 & 1 & 0 \\ 3 & 4 & 2 & 1 & 0 & 1 \\ \end{array} \right]\end{array} \end{align*} $$ 则由定义2得到
$$ \begin{align*} \begin{array}{rl} \mathcal{A}\ast\! \mathcal{B} = \\&fold\left[ \begin{array}{c} \left[ \begin{array}{ccc} \mathcal{A}_{1} & \mathcal{A}_{3} & \mathcal{A}_{2} \\ \mathcal{A}_{2} & \mathcal{A}_{1} & \mathcal{A}_{3} \\ \mathcal{A}_{3} & \mathcal{A}_{2} & \mathcal{A}_{1} \\ \end{array} \right]\cdot\left[ \begin{array}{c} \mathcal{B}_{1} \\ \mathcal{B}_{2} \\ \mathcal{B}_{3} \\ \end{array} \right] \\ \end{array} \right] = \\ &fold\left[ \begin{array}{c} \mathcal{A}_{1}\mathcal{B}_{1}+\mathcal{A}_{3}\mathcal{B}_{2}+\mathcal{A}_{2}\mathcal{B}_{3} \\ \mathcal{A}_{2}\mathcal{B}_{1}+\mathcal{A}_{1}\mathcal{B}_{2}+\mathcal{A}_{3}\mathcal{B}_{3} \\ \mathcal{A}_{3}\mathcal{B}_{1}+\mathcal{A}_{2}\mathcal{B}_{2}+\mathcal{A}_{1}\mathcal{B}_{3} \\ \end{array} \right] = \\ &\left[ \begin{array}{cc|cc|cc} 68 & 53 & 40 & 49 & 72 & 81 \\ 90 & 75 & 62 & 71 & 94 & 103 \\ \end{array} \right] \end{array} \end{align*} $$ 下面定义张量的范数.令$ \mathcal{A}\in {\bf C}^{n_{1}\times n_{2}\times\cdots \times n_{p}} $, 张量的范数就等于它所有元素平方和的平方根[11], 即
$$ \begin{align} \|\mathcal{A}\| = \sqrt{\sum^{n_{1}}_{i_{1} = 1}\sum^{n_{2}}_{i_{2} = 1}\cdots\sum^{n_{p}}_{i_{p} = 1}a^2_{i_{1}i_{2}\cdots i_{p}}} \end{align} $$ (3) 这和矩阵的$ Frobenius $-模是类似的.两个大小相同的张量$ \mathcal{A} $, $ \mathcal{B}\in {\bf C}^{n_{1}\times n_{2}\times\cdots \times n_{p}} $的内积等于它们对应元素的乘积的和[11], 即
$$ \begin{align} (\mathcal{A}, \mathcal{B}) = \sum^{n_{1}}_{i_{1} = 1}\sum^{n_{2}}_{i_{2} = 1}\cdots\sum^{n_{p}}_{i_{p} = 1}a_{i_{1}i_{2}\cdots i_{p}}b_{i_{1}i_{2}\cdots i_{p}} \end{align} $$ (4) 于是显然成立$ (\mathcal{A}, \mathcal{A}^\ast) = \|\mathcal{A}\|^2 $, 其中记号"$ \ast $''表示取复共轭.
1.2 张量广义逆
参考复数、向量和矩阵的倒数, 容易得到以下结论:
1) 若$ b $是一个复数, $ b\in {\bf C} $, 则$ bb^\ast = |b|^2 $, $ {1}/{b} = b^{-1} = {b^\ast}/{|b|^2} $;
2) 若$ {\pmb v} $是一个向量, $ {\pmb v}\in {\bf C}^n $, 则$ {\pmb v}\cdot {\pmb v}^\ast = |{\pmb v}|^2 $, $ {1}/{{\pmb v}} = {\pmb v}^{-1} = {{\pmb v}^\ast}/{|{\pmb v}|^2} $ (见Graves-Morris [14]);
3) 若$ A = a_{ij} $, $ B = b_{ij}\in {\bf C}^{s\times t} $, 则$ A\cdot B = \sum^s_{i = 1}\sum^t_{j = 1}(a_{ij}b_{ij})\in {\bf C} $, $ {1}/{A} = A_r^{-1} = {A^\ast}{\|A\|^2} $ (见顾传青[15-17]).因此, 下面张量的广义逆可以看作是向量和矩阵的推广:
定义3. 令$ \mathcal{A}, \mathcal{B}\in {\bf C}^{n_1\times n_2\times \cdots\times n_p} $, 其内积为
$$ \begin{equation*} (\mathcal{A}, \mathcal{B}) = \sum\limits^{n_{1}}_{i_{1} = 1}\sum\limits^{n_{2}}_{i_{2} = 1}\cdots\sum\limits^{n_{p}}_{i_{p} = 1}a_{i_{1}i_{2}\cdots i_{p}b_{i_{1}i_{2}\cdots i_{p}}} \end{equation*} $$ $ \mathcal{A}\cdot\mathcal{A}^\ast = \parallel\mathcal{A}\parallel^2 $, 则张量$ \mathcal{A} $的广义逆被定义为
$$ \begin{align} \begin{aligned} \mathcal{A}_r^{-1}& = \frac{1}{\mathcal{A}} = \frac{\mathcal{A}^\ast}{\parallel\mathcal{A}\parallel^2}, \mathcal{A}\neq 0, \mathcal{A}\in {\bf C}^{n_1\times n_2\times \cdots\times n_p} \end{aligned} \end{align} $$ (5) 其中张量的范数由式(3)给出.
引理1. 令$ \mathcal{A}, \mathcal{B}\in {\bf C}^{n_1\times n_2\times \cdots\times n_p}, \mathcal{A}, \mathcal{B}\neq0 $且$ b\in {\bf R}, b\neq0 $, 则下列关系成立:
1) $ \frac{b}{\mathcal{A}} = \frac{1}{\mathcal{B}}\Longleftrightarrow\mathcal{A} = b\mathcal{B} $;
2) $ (\mathcal{A}^{-1}_r)^{-1}_r = \mathcal{A}, (b\mathcal{A})^{-1}_r = \frac{1}{b}\mathcal{A}^{-1}_r $.
证明. 根据定义3, 结论2)是显然的.现在证明结论1).因为$ \mathcal{A}, \mathcal{B}\neq0 $, 从$ \mathcal{A} = b\mathcal{B} $, 容易推得$ \frac{b}{\mathcal{A}} = \frac{1}{\mathcal{B}} $.利用定义3, 得到引理1的左端, $ \frac{b\mathcal{A}^\ast}{\|\mathcal{A}\|^2} = \frac{\mathcal{B}^\ast}{\|\mathcal{B}\|^2} $, 即
$$ \begin{align*} \mathcal{A}^\ast = \frac{\parallel\mathcal{A}\parallel^2}{b\parallel\mathcal{B}\parallel^2}\mathcal{B}^\ast, \mathcal{A} = \frac{\parallel\mathcal{A}\parallel^2}{b\parallel\mathcal{B}\parallel^2}\mathcal{B} \end{align*} $$ 因此
$$ \begin{align*} \parallel\mathcal{A}\parallel^2 = \mathcal{A}\cdot\mathcal{A}^\ast = \frac{\parallel\mathcal{A}\parallel^4\parallel\mathcal{B}\parallel^2}{b^2\parallel\mathcal{B}\parallel^4} \end{align*} $$ 即$ b^2 = \frac{\parallel\mathcal{A}\parallel^2}{\parallel\mathcal{B}\parallel^2} $, 所以$ \mathcal{A} = b\mathcal{B} $.
2. 张量广义逆$ \pmb{\varepsilon} $-算法
2.1 广义逆张量Padé逼近的定义
设$ f(t) $是给定的张量多项式, 其系数为张量, 即
$$ \begin{equation} f(t) = \mathcal{C}_0+\mathcal{C}_1t+\mathcal{C}_2t^2+\cdots+\mathcal{C}_nt^n+\cdots \end{equation} $$ (6) $$ \begin{equation*} \mathcal{C}_i = (c_i^{(i_1i_2\cdots i_p)})\in {\bf C}^{n_{1}\times n_{2}\times\cdots \times n_{p}}, t\in {\bf C} \end{equation*} $$ 定义4. 令$ C^{n_1\times n_2\times \cdots\times n_p}[t] $是一个$ p $阶张量的多项式集合, 维数分别为$ n_1\times n_2\times \cdots\times n_p $.一个张量多项式$ \mathcal{A}(t) = (a_{i_1i_2 \cdots i_p}(t))\in {\bf C}^{n_1\times n_2\times \cdots\times n_p}[t] $是$ m $阶的, 表示为$ \partial\mathcal{A}(t) = m $, 若$ \partial(a_{i_1i_2 \cdots i_p}(t))\leq m $对于所有$ i_i = 1, 2, \cdots, n_i, 1\leq i\leq p $都成立, 且$ \partial(a_{(i_1i_2 \cdots i_p)}(t)) = m $对于某些$ i_i = 1, 2, \cdots, n_i, 1\leq i\leq p $成立.
定义5. 定义$ [\frac{n}{2k}] $型广义逆张量Padé逼近(GITPA)是一个张量有理函数
$$ \begin{equation} \mathcal{R}(t) = \frac{\mathcal{P}(t)}{q(t)} \end{equation} $$ (7) 其中, $ \mathcal{P}(t) $是一个张量多项式, $ q(t) $是一个数量多项式, 满足以下条件:
1) $ \partial{ \mathcal{P}(t)}\leq n, \partial{q(t)} = 2k; $
2) $ q(t)\mid \parallel\mathcal{P}(t)\parallel^2; $
3) $ q(t)f(t)-\mathcal{P}(t) = O(t^{n+1}); $
其中, $ \mathcal{P}(t) = (p_{(i_1i_2 \cdots i_p)}(t))\in {\bf C}^{n_1\times n_2\times \cdots \times n_p} $, 其范数$ \|\mathcal{P}(t)\|^2 $由式(3)给出, 而整除性条件2)表明分母数量多项式$ q(t) $能够整除分子张量多项式$ \mathcal{P}(t) $范数的平方.
2.2 张量$ \pmb\varepsilon $-算法
给定张量多项式(6), 根据张量广义逆(5), 定义张量$ \varepsilon $-算法如下:
$$ \begin{align*} &\varepsilon_{-1}^{(j)} = 0 , j = 0, 1, 2, \cdots \end{align*} $$ (8) $$ \begin{align*} &\varepsilon_0^{(j)} = \sum\limits_{i = 0}^j C_i z^i , j = 0, 1, 2, \cdots \end{align*} $$ (9) $$ \begin{align*} &\varepsilon_{k+1}^{(j)} = \varepsilon_{k-1}^{(j+1)}+(\varepsilon_k^{(j+1)}-\varepsilon_k^{(j)})^{-1} , j, k\ge0 \end{align*} $$ (10) 类似于矩阵情况, 根据式(8)$ \, \sim\, $(10), 上述张量$ \varepsilon $-算法组成了一个称为$ \varepsilon $-表的二维数组:
其中, 根据$ \varepsilon $-算法产生的元素$ \varepsilon_{k+1}^{(j)} $, 它的上标$ j $表示斜行数, 下标$ k+1 $表示列数.每个元素的产生与一个菱形有关, 例如, 由$ \varepsilon_0^{(1)}, \varepsilon_1^{(0)}, \varepsilon_1^{(1)}, \varepsilon_2^{(0)} $所组成的菱形, 现在要计算$ \varepsilon_2^{(0)} $, 由式(10), 将$ \varepsilon_1^{(1)} $减去$ \varepsilon_1^{(0)} $, 再取广义逆(3), 然后将结果加上$ \varepsilon_0^{(1)} $, 从而得到$ \varepsilon_2^{(0)} $.
定理1. (恒等定理)利用张量广义逆(5), 根据$ \varepsilon $-算法(8)$ \, \sim\, $(10)构造$ \varepsilon_{2k}^{(j)} $, 如果在计算过程中没有出现分母为零的情形, 则广义逆Padé逼近$ [\frac{j+2k}{2k}]_f $存在, 且成立恒等式:
$$ \begin{equation} \varepsilon_{2k}^{(j)} = \left[\frac{j+2k}{2k}\right]_f, j, k\ge 0 \end{equation} $$ (11) 证明. 该证明与矩阵情况类似, 可以参考文献[15].
设张量指数函数为
$$ \begin{equation} {\rm exp}(\mathcal{A}t) = \mathcal{I}+\mathcal{A}t+\frac{1}{2!}\mathcal{A}^2t^2+\frac{1}{3!}\mathcal{A}^3t^3+\cdots \end{equation} $$ (12) 给出计算张量指数函数(9)的张量$ \varepsilon $-算法如下:
算法1. (计算张量指数函数的$ \varepsilon $-算法):
输入:张量$ \mathcal{A} $、自变量$ t $和需要逼近的阶数$ j_{\rm{max}} $和$ k_{\rm{max}} $ (偶数)的值.
1) 计算$ \varepsilon $-表的第一列,
$ \varepsilon_{-1}^{(j)} = 0, j = 0, 1, 2, \cdots, j_{\rm{max}}+1 $.
2) 计算$ \varepsilon $-表的第二列,
$ \varepsilon_0^{(j)} = \sum_{i = 0}^j \frac{1}{i!}\mathcal{A}^it^i, j = 0, 1, 2, \cdots, j_{\rm{max}} $.
3) 逐列计算$ \varepsilon $-表的第三列至第$ k_{\rm{max}}+2 $列,
$ \varepsilon_{k+1}^{(j)} = \varepsilon_{k-1}^{(j+1)}+(\varepsilon_k^{(j+1)}-\varepsilon_k^{(j)})^{-1}, j, k\ge0 $.
输出:计算结果$ \varepsilon_{2k}^{(j)} = [\frac{j+2k}{2k}]_e $, 即为所求张量指数的$ [\frac{j+2k}{2k}] $型广义逆Padé逼近, 其中在第2)步计算$ \mathcal{A}^i $时用到定义1给出的张量$ t $-积.
2.3 张量$ \pmb\varepsilon $-算法的常用格式
给定张量指数函数(12), 下面给出张量$ \varepsilon $-算法的几个常用格式:
格式Ⅰ: $ \varepsilon_2^{-1} = [\frac{1}{2}]_e = \varepsilon_{0}^{(0)}+(\varepsilon_1^{(0)}-\varepsilon_1^{(-1)})^{-1} $, 其中
$$ \begin{align*} &\varepsilon_{0}^{(0)} = \mathcal{I}, \varepsilon_{0}^{(1)} = \mathcal{I}+\mathcal{A}t\\ &\varepsilon_{1}^{(0)} = (\varepsilon_0^{(1)}-\varepsilon_0^{(0)})^{-1} = \frac{1}{\mathcal{A}t}\\ &\varepsilon_{0}^{(-1)} = 0, \varepsilon_{1}^{(-1)} = (\varepsilon_0^{(0)}-\varepsilon_0^{(-1)})^{-1} = \frac{1}{\mathcal{I}} \end{align*} $$ 格式Ⅱ: $ \varepsilon_2^{0} = [\frac{2}{2}]_e = \varepsilon_{0}^{(1)}+(\varepsilon_1^{(1)}-\varepsilon_1^{(0)})^{-1} $, 其中
$$ \begin{align*} &\varepsilon_{1}^{(1)} = (\varepsilon_0^{(2)}-\varepsilon_0^{(1)})^{-1} = \frac{2}{\mathcal{A}^2t^2}\\ &\varepsilon_{0}^{(2)} = \mathcal{I}+\mathcal{A}t+\frac{\mathcal{A}^2t^2}{2} \end{align*} $$ 格式Ⅲ: $ \varepsilon_2^{1} = [\frac{3}{2}]_e = \varepsilon_{0}^{(2)}+(\varepsilon_1^{(2)}-\varepsilon_1^{(1)})^{-1} $, 其中
$$ \begin{align*} &\varepsilon_{1}^{(2)} = (\varepsilon_0^{(3)}-\varepsilon_0^{(2)})^{-1} = \frac{6}{\mathcal{A}^3t^3}\\ &\varepsilon_{0}^{(3)} = \mathcal{I}+\mathcal{A}t+\frac{\mathcal{A}^2t^2}{2}+\frac{\mathcal{A}^3t^3}{6} \end{align*} $$ 格式Ⅳ: $ \varepsilon_4^{-1} = [\frac{3}{4}]_e = \varepsilon_{2}^{(0)}+(\varepsilon_3^{(0)}-\varepsilon_3^{(-1)})^{-1} $, 其中
$$ \begin{align*} &\varepsilon_{3}^{(0)} = \varepsilon_{1}^{(1)}+(\varepsilon_2^{(1)}-\varepsilon_2^{(0)})^{-1}\\ &\varepsilon_{3}^{(-1)} = \varepsilon_{1}^{(0)}+(\varepsilon_2^{(0)}-\varepsilon_2^{(-1)})^{-1} \end{align*} $$ 格式Ⅴ: $ \varepsilon_4^{0} = [\frac{4}{4}]_e = \varepsilon_{2}^{(1)}+(\varepsilon_3^{(1)}-\varepsilon_3^{(0)})^{-1} $, 其中
$$ \begin{align*} &\varepsilon_{3}^{(1)} = \varepsilon_1^{(2)}+(\varepsilon_2^{(2)}-\varepsilon_2^{(1)})^{-1} \end{align*} $$ 格式Ⅵ: $ \varepsilon_4^{1} = [\frac{5}{4}]_e = \varepsilon_{2}^{(2)}+(\varepsilon_3^{(2)}-\varepsilon_3^{(1)})^{-1} $, 其中
$$ \begin{align*} &\varepsilon_{3}^{(2)} = \varepsilon_1^{(3)}+(\varepsilon_2^{(3)}-\varepsilon_2^{(2)})^{-1} \end{align*} $$ 例2. 设张量$ \mathcal{A}\in {\bf R}^{2\times 2\times 2} $, 其张量指数函数为
$$ \begin{equation*} \label{equation3-8} \begin{aligned} {\rm exp}(\mathcal{A}t) = \, &\left[ \begin{array}{cc|cc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} \right] +\\ &\left[ \begin{array}{cc|cc} 0 & 1 & 0 & 2 \\ 0 & -2 & 0 & -1 \\ \end{array} \right]t+\\ &\left[ \begin{array}{cc|cc} 0 & -2 & 0 & -\dfrac{5}{2} \\ 0 & \dfrac{5}{2} & 0 & 2 \\ \end{array} \right]t^2 +\end{aligned} \end{equation*} $$ $$ \begin{align} &\left[ \begin{array}{cc|cc} 0 & \dfrac{13}{6} & 0 & \dfrac{7}{3} \\ 0 & -\dfrac{7}{3} & 0 & -\dfrac{13}{6} \\ \end{array} \right]t^3+\\ &\left[ \begin{array}{cc|cc} 0 & -\dfrac{5}{3} & 0 & -\dfrac{41}{24} \nonumber\\ 0 & \dfrac{41}{24} & 0 & \dfrac{5}{3} \nonumber\\ \end{array} \right]t^4+\cdots = \\ &\mathcal{I}+\mathcal{A}t+\frac{1}{2!}\mathcal{A}^2t^2+ \frac{1}{3!}\mathcal{A}^3t^3+\\ &\frac{1}{4!}\mathcal{A}^4t^4+\cdots \end{align} $$ (13) 计算$ [\frac{2}{2}] $型GITPA.
由常用格式Ⅱ得
$$ \begin{equation*} \begin{aligned} &\varepsilon_2^{(0)} = \left[\frac{2}{2}\right]_e = \varepsilon_{0}^{(0)}+(\varepsilon_1^{(0)}-\varepsilon_1^{(-1)})^{-1} = \\ &\mathcal{I}+\mathcal{A}t+\frac{1}{\frac{2}{\mathcal{A}^2t^2}-\frac{1}{\mathcal{A}t}} = \\ &\frac{\left[ \begin{array}{cc|cc} a_1 & a_2 & 0 & a_3\\ 0 & a_1-a_3 & 0 & -a_2\\ \end{array} \right]}{a_1} = \frac{\mathcal{P}_2(t)}{q_2(t)} \end{aligned} \end{equation*} $$ 其中
$$ \begin{array}{rl} a_1 = &41+140t+125t^2\\ a_2 = &40t^2+41t\\ a_3 = &155t^2+82t \end{array} $$ 由定义5, $ \varepsilon_2^{(0)} = \frac{\mathcal{P}_2(t)}{q_2(t)} $是式(13)的$ [\frac{2}{2}] $型GITPA, 满足以下条件:
1) $ \partial{\mathcal{P}_2(t)} = 2, \partial{q_2(t)} = 2; $
2) $ q_2(t)\mid \parallel\mathcal{P}_2(t)\parallel^2; $这里
$$ \begin{align*} \|\mathcal{P}_2(t)\|^2 = \, &2(a_1^2+a_2^2+a_3^2)-2a_1a_3 = \\ &2(a_1^2+205t^2a_1-a_1a_3) = \\ &2a_1(a_1+205t^2-a_3) \end{align*} $$ 3) $ q_2(t){\rm exp}(\mathcal{A}t)-\mathcal{P}_2(t) = O(t^{3}) $.
3. 数值实验
本节计算张量$\varepsilon$-算法的近似值和精确值的误差, 并将本文的方法与计算张量指数函数的截断法进行比较.首先给出目前通常使用的无穷序列截断法:
算法2. (无穷序列截断法[10])
输入:张量$\mathcal{A}$、自变量$t$和误差限$\epsilon_{\rm{tol}}$的值.
1) 初始化$n=0$和${\rm exp}(\mathcal{A}t):=\mathcal{I}$.
2) $n:=n+1$.
3) 计算$\frac{t^n}{n!}$和$\mathcal{A}^n$.
4) 将该项计算结果加到
$$ {\rm exp}(\mathcal{A}t):={\rm exp}(\mathcal{A}t)+\frac{t^n}{n!}\mathcal{A}^n $$ 5) 判断是否停止, 若
$$ \begin{equation*} \frac{\|\mathcal{A}^n\|t^n}{n!} <\epsilon_{\rm{tol}} \end{equation*} $$ 则停止, 否则回到第2)步.
输出: ${\rm exp}(\mathcal{A}t)$.
例3. 设张量$\mathcal{A}\in {\bf R}^{2\times 2\times 2}$, $\mathcal{A}$中的元素为
$$ a_{121}=\frac{1}{2}, a_{221}=-\frac{2}{3}, a_{122}=-\frac{1}{2}, a_{222}=\frac{2}{3} $$ 其余均为0, 它的指数函数展开式为
$$ \begin{equation}\label{equation4-1} \begin{aligned} {\rm exp}(\mathcal{A}t)=\, &\left[ \begin{array}{cc|cc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} \right] +\\ &\left[ \begin{array}{cc|cc} 0 & \dfrac{1}{2} & 0 & \dfrac{2}{3} \\[2mm] 0 & -\dfrac{2}{3} & 0 & -\dfrac{1}{2} \\ \end{array} \right]t+\\ &\left[ \begin{array}{cc|cc} 0 & -\dfrac{1}{3} & 0 & -\dfrac{15}{72} \\[2mm] 0 & \dfrac{15}{72} & 0 & \dfrac{1}{3} \\ \end{array} \right]t^2+\\ &\left[ \begin{array}{cc|cc} 0 & \dfrac{19}{144} & 0 & \dfrac{43}{324} \\[2mm] 0 & -\dfrac{43}{324} & 0 & -\dfrac{19}{144} \\ \end{array} \right]t^3+\\ &\left[ \begin{array}{cc|cc} 0&-\dfrac{25}{648}&0&-\dfrac{128}{3315}\\[2mm] 0&\dfrac{128}{3315}&0&\dfrac{25}{648}\\ \end{array} \right]t^4+\cdots=\\ &\mathcal{I}+\mathcal{A}t+\frac{1}{2!}\mathcal{A}^2t^2+ \frac{1}{3!}\mathcal{A}^3t^3+\\ &\frac{1}{4!}\mathcal{A}^4t^4+\cdots \end{aligned} \end{equation} $$ (14) 用$\varepsilon$-算法和截断法计算该指数函数.
3.1 $\pmb{[\frac{4}{4}]}$型GITPA的数值实验
由常用格式V计算$[\frac{4}{4}]$型GITPA, 数值实验结果见表 1.
表 1 $[\frac{4}{4}]$型GITPA-算法数值实验Table 1 The numerical experiment of $[\frac{4}{4}]$ type GITPA-algorithm$x$ $(1, 2, 1)$ $(2, 2, 1)$ $(1, 2, 2)$ $(2, 2, 2)$ $ RES $ 0.2 $E$值 0.08766299 0.87955329 0.12044671 -0.08766299 $5.69\times10^{-13}$ $A$值 0.08766327 0.87955283 0.12044717 -0.08766327 0.4 $E$值 0.15420167 0.78130960 0.21869040 -0.15420167 $3.74\times10^{-10}$ $A$值 0.15420895 0.78129804 0.21870196 -0.15420895 0.6 $E$值 0.20408121 0.70078192 0.29921808 -0.20408121 $1.40\times10^{-8}$ $A$值 0.20412606 0.70071136 0.29928864 -0.20412606 0.8 $E$值 0.24081224 0.63444735 0.36555265 -0.24081224 $1.63\times10^{-7}$ $A$值 0.24096630 0.63420702 0.36579298 -0.24096630 1 $E$值 0.26715410 0.57953894 0.42046106 -0.26715410 $1.01\times10^{-6}$ $A$值 0.26753925 0.57894247 0.42105753 -0.26753925 表 1中由常用格式V计算的近似值记作$A$值, 根据截断法即算法2取指数函数展开式(14)前15项计算得到的结果为精确值, 记作$E$值.表 1中分别列出张量下标为(1, 2, 1), (2, 2, 1), (1, 2, 2), (2, 2, 2)在点0.2, 0.4, 0.6, 0.8, 1相应的近似值与精确值.表 1中最后一列表示近似值与精确值残差的范数的平方, 记作$RES$, 其计算公式如下:
$$ \begin{equation*} RES(t)=\|{\rm exp}(\mathcal{A}t)-\left[\frac{4}{4}\right]_{e^{\mathcal{A}t}}(t)\|^2 \end{equation*} $$ 其中, 模范数$\|\cdot\|$由式(3)定义.
表 1显示, 张量$\varepsilon$-算法得到的近似值能够达到较高的计算精度.可以发现, 自变量的值越接近0, 张量$\varepsilon$-算法的逼近效果越好, 计算结果基本符合理论预期.
3.2 两个算法数值实验的比较
首先, 在张量指数函数(14)中令$t=2$, 取前15项计算得到的精确值为
$$ \begin{equation*} \left[ \begin{array}{cc|cc} 1 & ~~0.3098~~&~~0 & ~~0.5932 \\ 0 & ~~0.4068~~&~~0 & ~~-0.3098 \\ \end{array} \right] \end{equation*} $$ 根据算法1分别计算$[\frac{2}{2}], [\frac{4}{4}], [\frac{6}{6}]$型GITPA, 并与截断算法2进行比较, 得到的数值实验结果见表 2.
表 2 算法1和算法2的数值实验比较Table 2 The numerical experiment comparison of Algorithm 1 and Algorithm 2$[\frac{j+2k}{2k}]$ 张量$\varepsilon$-算法 $n_{\rm{max}}$ $\sum^{n_{\rm{max}}}_{n=0}\frac{1}{n!}\mathcal{A}^nt^n $ $a_{121}$ $a_{221}$ $a_{122}$ $a_{222}$ $a_{121}$ $a_{221}$ $a_{122}$ $a_{222}$ $[\frac{2}{2}]$ 0.4235 0.3513 0.6487 -0.4235 1 1.0000 -0.3333 1.3333 -1.0000 $[\frac{4}{4}]$ 0.3049 0.4141 0.5859 -0.3049 2 -0.3333 1.0556 -0.0556 0.3333 $[\frac{6}{6}]$ 0.3098 0.4068 0.5932 -0.3098 3 0.7222 -0.0062 1.0062 -0.7222 - - - - 4 0.1049 0.6116 0.3884 -0.1049 - - - - 5 0.3931 0.3234 0.6766 -0.3931 - - - - 6 0.2810 0.4355 0.5645 -0.2810 - - - - 7 0.3184 0.3981 0.6019 -0.3184 - - - - 8 0.3075 0.4090 0.5910 -0.3075 - - - - 9 0.3103 0.4062 0.5938 -0.3103 - - - - 10 0.3097 0.4069 0.5931 -0.3097 - - - - 11 0.3098 0.4067 0.5933 -0.3098 - - - - 12 0.3098 0.4068 0.5932 -0.3098 表 2显示, $[\frac{6}{6}]$型GITPA计算的数值结果与截断法取前13项相加的数值结果的精确度相近, 由此说明本文给出的张量$\varepsilon$-算法是有效的.
3.3 两个算法计算复杂度的比较
下面从计算复杂度的角度来对两种算法进行比较.由于需要自乘, 前两维的维数必须相等.对于一个$l\times l\times n$的三阶张量$\mathcal{A}$, 1次张量$t$-积的计算复杂度为$ {\rm O}(l^3n^2)$; 而1次张量的广义逆等价于2次张量的数乘计算, 其计算复杂度为$ {\rm O}(l^2n)$.因此, 随着张量维数的增大, 计算张量$t$-积显然要比计算张量的广义逆付出更大的成本.
根据张量$\varepsilon$-算法计算$[\frac{6}{6}]$型GITPA所使用的是张量指数函数(14)的前7项, 需要进行5次张量$t$-积和21次广义逆计算, 而达到相同精度, 截断法需要取前13项, 需要进行11次张量$t$-积和12次数乘运算.两种算法的计算复杂度比较见表 3.从表 3可以看出:在计算过程中, 张量$\varepsilon$-算法计算复杂度为$5l^3n^2+42l^2n$, 截断法计算复杂度为$11l^3n^2+12l^2n$.
表 3 算法1和算法2的计算复杂度分析Table 3 The analysis of computational complexity of Algorithm 1 and Algorithm 2计算一次 $\varepsilon$-算法: $[\frac{6}{6}]$型 截断法:取13项 复杂度 计算 计算 $t$-积运算 $l^3n^2$ 5 11 数乘运算 $l^2n$ 21 12 范数运算 $l^2n$ 21 0 总和 $5l^3n^2+42l^2n$ $11l^3n^2+12l^2n$ 例4. 为了比较两种算法所需的时间, 在Matlab中选取$'seed~'=1:100$, 随机生成张量$\mathcal{A}$如下: $3\times 3\times 3, 10\times 10\times 10, 20\times 20\times 20, 30\times 30\times 30, 40\times 40\times 40$各100个张量, 分别使用算法1和算法2计算其张量指数函数(12), 其中, 本文给出的算法1计算的是$[\frac{6}{6}]$型GITPA, 算法2计算的是指数函数展开的前13项之和.两个算法计算所需时间见表 4, 表中数据均是由内存8 GB, 主频2.50 GHz, 操作系统Windows 10, 处理器Inter(R) Core(TM) i5-7300HQ的笔记本电脑的Matlab (R2015b)得到.
表 4 不同维数下两种算法的运行时间表(s)Table 4 The consuming time of two algorithms in different dimensions (s)张量维数 张量$\varepsilon$-算法运行时间 截断法运行时间 $ 3~\times ~3\times ~3$ 1.755262 1.103832 $10\times 10\times 10$ 2.272663 2.164860 $20\times 20\times 20$ 3.831094 4.805505 $30\times 30\times 30$ 6.419004 10.545785 $40\times 40\times 40$ 15.814063 30.012744 图 1显示, 算法1计算$[\frac{6}{6}]$型GITPA与算法2即截断法计算前13项之和相比, 在维数较低的情况下(维数低于$10\times 10\times 10$), GITPA的运行时间略高于截断法; 当维数较高时, 本文的算法1逐渐显示出优越性, 可以很好地降低计算时间.
图 1 不同维数下两种算法运行时间直方图(s)Fig. 1 The time-consuming comparison histogram of two algorithms in different dimensions (s)4. 结论
目前, 国内外还没有看到计算张量的逆和广义逆的有效算法, 本文提出的一种张量广义逆即定义3是一个实用的计算方法, 它是同一类型的向量广义逆(Graves-Morris [14])和矩阵广义逆(顾传青[15-16, 18])在张量上的推广.在该张量广义逆的基础上, 本文得到了计算张量指数函数(12)的张量$\varepsilon$-算法.从计算张量指数函数(14)的数值实验来看, 与目前通用的无穷序列截断法相比较, 在计算精度和计算复杂度上具有一定的优势, 在张量维数比较大时, 这个优势会更加明显.下面的研究工作从两个方面进行, 一是用本文提出的广义逆张量Padé逼近(GITPA)方法来考虑控制论中的模型简化问题, 二是对张量$\varepsilon$-算法的稳定性进行探讨.
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图 7 模型生成结果灰度分布统计
Fig. 7 Statistics of gray scale distribution of model generation results
表 1 评价指标统计
Table 1 Statistics of evaluation metrics
评价指标 Pix2pix StyleGAN2 DFMGAN RePaint模型 所提方法 IS↑ 1.388 1.368 1.301 1.474 1.541 FID↓ 59.056 124.748 96.783 72.750 57.650 KID↓ 0.024 0.098 0.053 0.044 0.020 MS-SSIM↓ 0.189 0.161 0.174 0.187 0.159 PSNR↓ 28.308 28.284 28.357 28.273 28.223 注: 1) RePaint模型使用裂纹掩码生成缺陷效果差, 改为区块掩码; 2) 箭头标识评价指标得分更好的方向. 
下载: 导出CSV
表 2 分类结果统计(%)
Table 2 Statistics of classification results (%)
测试集 Pix2pix StyleGAN2 DFMGAN RePaint模型 所提方法 缺陷检出率 总正确率 缺陷检出率 总正确率 缺陷检出率 总正确率 缺陷检出率 总正确率 缺陷检出率 总正确率 D1 30.77 65.38 26.92 63.46 7.69 53.85 13.46 54.81 88.46 94.23 D2 54.46 77.07 49.94 74.97 32.26 66.13 45.76 70.99 88.32 93.76 注: RePaint模型使用裂纹掩码生成缺陷效果差, 改为区块掩码.
下载: 导出CSV
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[1] 张辉, 张邹铨, 陈煜嵘, 吴天月, 钟杭, 王耀南. 工业铸件缺陷无损检测技术的应用进展与展望. 自动化学报, 2022, 48(4): 935−956Zhang Hui, Zhang Zou-Quan, Chen Yu-Rong, Wu Tian-Yue, Zhong Hang, Wang Yao-Nan. Application advance and prospect of nondestructive testing technology for industrial casting defects. Acta Automatica Sinica, 2022, 48(4): 935−956 [2] 罗东亮, 蔡雨萱, 杨子豪, 章哲彦, 周瑜, 白翔. 工业缺陷检测深度学习方法综述. 中国科学: 信息科学, 2022, 52(6): 1002−1039 doi: 10.1360/SSI-2021-0336Luo Dong-Liang, Cai Yu-Xuan, Yang Zi-Hao, Zhang Zhe-Yan, Zhou Yu, Bai Xiang. Survey on industrial defect detection with deep learning. Scientia Sinica Informationis, 2022, 52(6): 1002−1039 doi: 10.1360/SSI-2021-0336 [3] 陶晓天, 何博侠, 张鹏辉, 田德旭. 基于深度学习的航天密封圈表面缺陷检测. 仪器仪表学报, 2021, 42(1): 199−206Tao Xiao-Tian, He Bo-Xia, Zhang Peng-Hui, Tian De-Xu. Surface defect detection of aerospace sealing rings based on deep learning. Chinese Journal of Scientific Instrument, 2021, 42(1): 199−206 [4] 田娟秀, 刘国才, 谷珊珊, 鞠忠建, 刘劲光, 顾冬冬. 医学图像分析深度学习方法研究与挑战. 自动化学报, 2018, 44(3): 401−424Tian Juan-Xiu, Liu Guo-Cai, Gu Shan-Shan, Ju Zhong-Jian, Liu Jin-Guang, Gu Dong-Dong. Deep learning in medical image analysis and its challenges. Acta Automatica Sinica, 2018, 44(3): 401−424 [5] 王国力, 孙宇, 魏本征. 医学图像图深度学习分割算法综述. 计算机工程与应用, 2022, 58(12): 37−50 doi: 10.3778/j.issn.1002-8331.2112-0225Wang Guo-Li, Sun Yu, Wei Ben-Zheng. Systematic review on graph deep learning in medical image segmentation. Computer Engineering and Applications, 2022, 58(12): 37−50 doi: 10.3778/j.issn.1002-8331.2112-0225 [6] 李书林, 冯朝路, 于鲲, 刘鑫, 江鑫, 赵大哲. 基于深度学习的心脏磁共振影像超分辨率前沿进展. 中国图象图形学报, 2022, 27(3): 704−721 doi: 10.11834/j.issn.1006-8961.2022.3.zgtxtxxb-a202203005Li Shu-Lin, Feng Chao-Lu, Yu Kun, Liu Xin, Jiang Xin, Zhao Da-Zhe. Critical review of human cardiac magnetic resonance image super resolution reconstruction based on deep learning method. Journal of Image and Graphics, 2022, 27(3): 704−721 doi: 10.11834/j.issn.1006-8961.2022.3.zgtxtxxb-a202203005 [7] Redmon J, Divvala S, Girshick R, Farhadi A. You only look once: Unified, real-time object detection. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Las Vegas, USA: IEEE, 2016. 779−788 [8] Liu W, Anguelov D, Erhan D, Szegedy C, Reed S, Fu C Y, et al. SSD: Single shot MultiBox detector. In: Proceedings of the 14th European Conference on Computer Vision. Amsterdam, the Netherlands: Springer, 2016. 21−37 [9] Ren S Q, He K M, Girshick R, Sun J. Faster R-CNN: Towards real-time object detection with region proposal networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2017, 39(6): 1137−1149 doi: 10.1109/TPAMI.2016.2577031 [10] Redmon J, Farhadi A. YOLOv3: An incremental improvement. arXiv preprint arXiv: 1804.02767, 2018. [11] Santos C F G D, Papa J P. Avoiding overfitting: A survey on regularization methods for convolutional neural networks. ACM Computing Surveys (CSUR), 2022, 54(S10): Article No. 20 [12] 陶显, 侯伟, 徐德. 基于深度学习的表面缺陷检测方法综述. 自动化学报, 2021, 47(5): 1017−1034Tao Xian, Hou Wei, Xu De. A survey of surface defect detection methods based on deep learning. Acta Automatica Sinica, 2021, 47(5): 1017−1034 [13] Niu S L, Li B, Wang X G, Lin H. Defect image sample generation with GAN for improving defect recognition. IEEE Transactions on Automation Science and Engineering, 2020, 17(3): 1611−1622 [14] 伍麟, 郝鸿宇, 宋友. 基于计算机视觉的工业金属表面缺陷检测综述. 自动化学报, 2024, 50(7): 1261−1283Wu Lin, Hao Hong-Yu, Song You. A review of metal surface defect detection based on computer vision. Acta Automatica Sinica, 2024, 50(7): 1261−1283 [15] Rippel O, Müller M, Merhof D. GAN-based defect synthesis for anomaly detection in fabrics. In: Proceedings of the 25th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA). Vienna, Austria: IEEE, 2020. 534−540 [16] Zhang G J, Cui K W, Hung T Y, Lu S J. Defect-GAN: High-fidelity defect synthesis for automated defect inspection. In: Proceedings of the IEEE Winter Conference on Applications of Computer Vision (WACV). Waikoloa, USA: IEEE, 2021. 2523−2533 [17] Zhang H B, Pan D, Liu J H, Jiang Z H. A novel MAS-GAN-based data synthesis method for object surface defect detection. Neurocomputing, 2022, 499: 106−114 doi: 10.1016/j.neucom.2022.05.021 [18] Wang R Y, Hoppe S, Monari E, Huber M F. Defect transfer GAN: Diverse defect synthesis for data augmentation. In: Proceedings of the 33rd British Machine Vision Conference. London, UK: BMVA Press, 2023. Article No. 445 [19] 丁鹏, 卢文壮, 刘杰, 袁志响. 基于生成对抗网络的叶片表面缺陷图像数据增强. 组合机床与自动化加工技术, 2022, (7): 18−21Ding Peng, Lu Wen-Zhuang, Liu Jie, Yuan Zhi-Xiang. Image data augmentation of blade surface defects based on generative adversarial network. Modular Machine Tool and Automatic Manufacturing Technique, 2022, (7): 18−21 [20] Dhariwal P, Nichol A. Diffusion models beat GANs on image synthesis. In: Proceedings of the 35th International Conference on Neural Information Processing Systems. Curran Associates Inc., 2021. Article No. 672Dhariwal P, Nichol A. Diffusion models beat GANs on image synthesis. In: Proceedings of the 35th International Conference on Neural Information Processing Systems. Curran Associates Inc., 2021. Article No. 672 [21] Carlini N, Hayes J, Nasr M, Jagielski M, Sehwag V, Tramér F, et al. Extracting training data from diffusion models. In: Proceedings of the 32nd USENIX Security Symposium. Anaheim, USA: USENIX Association, 2023. 5253−5270 [22] Jain S, Seth G, Paruthi A, Soni U, Kumar G. Synthetic data augmentation for surface defect detection and classification using deep learning. Journal of Intelligent Manufacturing, 2022, 33(4): 1007−1020 doi: 10.1007/s10845-020-01710-x [23] Zhang H Y, Cisse M, Dauphin Y N, Lopez-Paz D. mixup: Beyond empirical risk minimization. In: Proceedings of the 6th International Conference on Learning Representations. Vancouver, Canada: ICLR, 2018. [24] Zhang L J, Deng Z, Kawaguchi K, Ghorbani A, Zou J. How does mixup help with robustness and generalization? In: Proceedings of the 9th International Conference on Learning Representations. Austria: ICLR, 2021.Zhang L J, Deng Z, Kawaguchi K, Ghorbani A, Zou J. How does mixup help with robustness and generalization? In: Proceedings of the 9th International Conference on Learning Representations. Austria: ICLR, 2021. [25] Chou H P, Chang S C, Pan J Y, Wei W, Juan D C. Remix: Rebalanced mixup. In: Proceedings of the Computer Vision-ECCV 2020 Workshops. Glasgow, UK: Springer, 2020. 95−110 [26] Ren X Y, Lin W Y, Yang X Q, Yu X H, Gao H J. Data augmentation in defect detection of sanitary ceramics in small and non-i.i.d datasets. IEEE Transactions on Neural Networks and Learning Systems, 2023, 34(11): 8669−8678 doi: 10.1109/TNNLS.2022.3152245 [27] Isola P, Zhu J Y, Zhou T, Efros A A. Image-to-image translation with conditional adversarial networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Honolulu, USA: IEEE, 2017. 5967−5976 [28] Duan Y X, Hong Y, Niu L, Zhang L Q. Few-shot defect image generation via defect-aware feature manipulation. In: Proceedings of the 37th AAAI Conference on Artificial Intelligence. Washington, USA: AAAI, 2023. 571−578 [29] Sohl-Dickstein J, Weiss E, Maheswaranathan N, Ganguli S. Deep unsupervised learning using nonequilibrium thermodynamics. In: Proceedings of the 32nd International Conference on Machine Learning. Lille, France: PMLR, 2015. 2256−2265 [30] Ho J, Jain A, Abbeel P. Denoising diffusion probabilistic models. In: Proceedings of the 34th International Conference on Neural Information Processing Systems. Vancouver, Canada: Curran Associates Inc., 2020. Article No. 574 [31] Ho J, Salimans T. Classifier-free diffusion guidance. arXiv preprint arXiv: 2207.12598, 2022.Ho J, Salimans T. Classifier-free diffusion guidance. arXiv preprint arXiv: 2207.12598, 2022. [32] Saharia C, Ho J, Chan W, Salimans T, Fleet D J, Norouzi M. Image super-resolution via iterative refinement. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022, 45(4): 4713−4726 [33] Lugmayr A, Danelljan M, Romero A, Yu F, Timofte R, Van Gool L. RePaint: Inpainting using denoising diffusion probabilistic models. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). New Orleans, USA: IEEE, 2022. 11451−11461 [34] Wang W L, Bao J M, Zhou W G, Chen D D, Chen D, Yuan L, et al. Semantic image synthesis via diffusion models. arXiv preprint arXiv: 2207.00050, 2022.Wang W L, Bao J M, Zhou W G, Chen D D, Chen D, Yuan L, et al. Semantic image synthesis via diffusion models. arXiv preprint arXiv: 2207.00050, 2022. [35] Nichol A Q, Dhariwal P, Ramesh A, Shyam P, Mishkin P, Mcgrew B, et al. GLIDE: Towards photorealistic image generation and editing with text-guided diffusion models. In: Proceedings of the 39th International Conference on Machine Learning. Baltimore, USA: PMLR, 2022. 16784−16804 [36] Ramesh A, Dhariwal P, Nichol A, Chu C, Chen M. Hierarchical text-conditional image generation with CLIP latents. arXiv preprint arXiv: 2204.06125, 2022.Ramesh A, Dhariwal P, Nichol A, Chu C, Chen M. Hierarchical text-conditional image generation with CLIP latents. arXiv preprint arXiv: 2204.06125, 2022. [37] Rombach R, Blattmann A, Lorenz D, Esser P, Ommer B. High-resolution image synthesis with latent diffusion models. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). New Orleans, USA: IEEE, 2022. 10674−10685 [38] Song Y, Sohl-Dickstein J, Kingma D P, Kumar A, Ermon S, Poole B. Score-based generative modeling through stochastic differential equations. In: Proceedings of the 9th International Conference on Learning Representations. Austria: ICLR, 2020.Song Y, Sohl-Dickstein J, Kingma D P, Kumar A, Ermon S, Poole B. Score-based generative modeling through stochastic differential equations. In: Proceedings of the 9th International Conference on Learning Representations. Austria: ICLR, 2020. [39] Tabernik D, Šela S, Skvarč J, Skočaj D. Segmentation-based deep-learning approach for surface-defect detection. Journal of Intelligent Manufacturing, 2020, 31(3): 759−776 doi: 10.1007/s10845-019-01476-x [40] Wieler M, Hahn T, Hamprecht F A. Weakly supervised learning for industrial optical inspection [Online], available: https://hci.iwr.uni-heidelberg.de/content/weakly-supervised-learning-industrial-optical-inspection, February 20, 2024 [41] Song K C, Yan Y H. A noise robust method based on completed local binary patterns for hot-rolled steel strip surface defects. Applied Surface Science, 2013, 285: 858−864 doi: 10.1016/j.apsusc.2013.09.002 [42] Karras T, Laine S, Aittala M, Hellsten J, Lehtinen J, Aila T. Analyzing and improving the image quality of StyleGAN. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). Seattle, USA: IEEE, 2020. 8107−8116 [43] Barratt S, Sharma R. A note on the inception score. arXiv preprint arXiv: 1801.01973, 2018.Barratt S, Sharma R. A note on the inception score. arXiv preprint arXiv: 1801.01973, 2018. [44] Heusel M, Ramsauer H, Unterthiner T, Nessler B, Hochreiter S. GANs trained by a two time-scale update rule converge to a local Nash equilibrium. In: Proceedings of the 31st International Conference on Neural Information Processing Systems. Long Beach, USA: Curran Associates Inc., 2017. 6629−6640 [45] Bińkowski M, Sutherland D J, Arbel M, Gretton A. Demystifying MMD GANs. In: Proceedings of the 6th International Conference on Learning Representations. Vancouver, Canada: ICLR, 2018. [46] Wang Z, Simoncelli E P, Bovik A C. Multiscale structural similarity for image quality assessment. In: Proceedings of the 37th Asilomar Conference on Signals, Systems and Computers. Pacific Grove, USA: IEEE, 2003. 1398−1402 [47] Huynh-Thu Q, Ghanbari M. Scope of validity of PSNR in image/video quality assessment. Electronics Letters, 2008, 44(13): 800−801 doi: 10.1049/el:20080522 期刊类型引用(1)
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