基于非负矩阵分解的印刷品原稿原色油墨光谱预测方法

引用本文

李玉梅, 刘传杰, 陈浩杰, 陈桥, 何颂华. 基于非负矩阵分解的印刷品原稿原色油墨光谱预测方法. 光谱学与光谱分析, 2018,38(12): 3864-3870
LI Yu-mei, LIU Chuan-jie, CHEN Hao-jie, CHEN Qiao, HE Song-hua. The Spectral Prediction Method of Primary Ink for Prints Manuscript Based on Non-Negative Matrix Factorization. Spectroscopy and Spectral Analysis, 2018,38(12): 3864-3870. 复制到剪切板

Doi: 10.3964/j.issn.1000-0593(2018)12-3864-07
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光谱学与光谱分析所有

基于非负矩阵分解的印刷品原稿原色油墨光谱预测方法

李玉梅^1,², 刘传杰¹, 陈浩杰¹, 陈桥^2,^*, 何颂华²

1. 曲阜师范大学工学院, 山东日照 276826

2. 深圳职业技术学院传播工程学院, 广东深圳 518000

*通讯联系人 e-mail: qiaochen@szpt.edu.cn

作者简介: 李玉梅, 女, 1990年生, 曲阜师范大学硕士研究生 e-mail: 1353228591@qq.com

收稿日期: 2017-11-01 接受日期: 2018-04-16

基金项目: 国家自然科学基金项目(61108087)资助

摘要

实现半色调印刷品原稿的光谱复制技术其前提要确定原稿所用油墨数目及油墨成份, 但目前应用于印刷品原稿原色油墨光谱预测的算法还有待研究, 且已有的基色色料光谱预测方法存在诸多弊端。针对这一问题在非负矩阵分解算法基础上结合印刷品原稿光学特性, 创新的提出了一种基于约束条件非负矩阵分解的油墨光谱预测算法ISPNMF, 和对黑色油墨光谱预测结果优化的算法。 ISPNMF算法克服了基本非负矩阵分解有多重最优解和局部极小值的缺陷, 实现了预测算法唯一的全局最优解。黑色油墨预测光谱优化的算法克服了彩色油墨对光线混合吸收给黑色油墨预测带来的干扰, 能优化得到逼近于实际黑色油墨光谱的预测值。使用Konica Minolta C1085和HP indigo5600两台四色数码印刷机及其自身配备的墨粉和墨膏来摸拟不同品牌的油墨, 在230 g白卡纸上打印 IT8.7/3色标, 并使用X-rite i1 Pro2获取两样张的光谱反射率作为实验数据样本, 来探究并验证算法的准确性和实用性。实验结果表明, 在印刷品原稿线性经验空间中能准确预测原稿所用原色油墨数目和油墨光谱, 且彩色油墨预测光谱与实际使用的油墨光谱相比其拟合度均高达99.9%, 光谱角距离均小于0.045, 黑色油墨的预测光谱经优化后与实际油墨光谱拟合度也高达99.9%。这说明该算法不仅能实现对印刷品原稿原色油墨的准确预测, 而且可以精确匹配实际使用的原色油墨, 对实现印刷品原稿的光谱复制技术有重要意义。

关键词: 印刷品原稿; 原色油墨光谱预测; 非负矩阵分解; 主成分分析; 光谱颜色复制

中图分类号:O432.3 文献标识码:A

The Spectral Prediction Method of Primary Ink for Prints Manuscript Based on Non-Negative Matrix Factorization

LI Yu-mei^1,², LIU Chuan-jie¹, CHEN Hao-jie¹, CHEN Qiao^2,^*, HE Song-hua²

1. School of Engineering, Qufu Normal University, Rizhao 276826, China

2. School of Communication, Shenzhen Polytechnic, Shenzhen 518000, China

Abstract

To achieve the spectral reproduction technology of halftone prints manuscript, the number of primary ink and the ink composition used in the manuscript should be specified before hand. However, there are still many problems to be solved in the primary ink spectral prediction for prints manuscript, and existing methods of spectral prediction have many disadvantages. To solve this problem, the algorithm of primary ink spectral prediction based on constrained non negative matrix factorization ISPNMF, and the optimizing algorithm for black ink spectral prediction have been put forward innovatively. The short comings of multiple optimal solutions and local minima of the basic non-negative matrix factorization were overcome, and the unique global optimal solution was realized by the algorithm of ISPNMF. The interference of prediction black inkcaused by the colored inks mixed absorption was eliminated by the optimizing algorithm for black ink spectral prediction, and the optimized result was close to the actual black ink spectrum. The accuracy of the algorithm was verified by using the samples of simulating different brands ink. In the experiments, Konica Minolta C1085 and HP indigo 5600, two kinds of four-color digital printing machine, with its toner and ink paste mimicking different brands of ink, were used. And the IT8.7/3 color card was printed in 230 g white cardboard, then X-rite I1 Pro2 was used to obtain the spectral reflectance data of two proofs as the experimental samples, to explore and verify the accuracy and practicability of the algorithms. The experimental results showed that, the number and spectrum of primary inks used in the printed manuscript can be accurately predicted in the linear empirical space. The GFC of prediction results of color inks were all up to 99.9%, and the SAD were all less than 0.045. The GFC of prediction results of black inks, which were optimized, were also up to 99.9%. The algorithms can not only predict theprimary ink of prints manuscript accurately, but also can match the actualprimary ink precisely. It is of great significance to the realization of the spectral replication of prints manuscript.

Key words: Prints manuscript; Primary ink spectral prediction; Non-negative matrix factorization; Principal component analysis; Spectral color reproduction

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引言

基于光谱的印刷复制技术有别于传统的基于色度匹配的印刷技术, 图像色彩信息以光谱信息形式传输复制, 这样能很大程度上减小甚至消除同色异谱现象^[1]。要实现半色调印刷品原稿的光谱复制技术, 前提必须确定原稿所使用的油墨光谱成份, 如不知已有的半色调印刷品的油墨类型却想得到和原图像色彩再现的效果, 原色油墨的光谱预测尤为关键, 理想的做法便是找到和原稿相同的原色油墨来完成印刷, 实现光谱曲线的最佳匹配^[2]。

但是目前应用于原稿基色光谱预测的方法并不多, 大量研究仍在进行, 主要有主成分分析法PCA、 Tzeng提出的基于旋转主成分分析的光谱预测方法, 以及何博士提出的基于有约束条件的非负矩阵分解SCNMF的光谱预测方法。其中PCA法进行基色光谱预测时, 得到的基向量和对应的系数矩阵会出现负值, 没有物理意义^[3]; 旋转主成分分析的光谱预测算法受旋转初值影响其精度具有不稳定性。何博士提出的有约束的非负矩阵分解方法, 虽解决了负值问题, 但其预测结果受约束条件影响, 具体约束条件受原稿光学特性影响^[4], 且中性灰的光谱预测还不准确。应用于半色调印刷品原稿原色油墨的光谱预测算法还需进一步探索。本研究在非负矩阵分解算法基础上结合油墨半透明的光学特性, 创新性的提出了一种基于约束条件非负矩阵分解的油墨光谱预测算法ISPNMF(ink spectral prediction non-negative matrix factorization), 和一种对黑色油墨光谱预测结果的优化算法, 实现了对印刷品原色油墨光谱的准确预测。

1 原色油墨线性经验空间模型的建立

印刷品原稿油墨预测算法是建立在能准确反映数据集物理维度的线性空间基础上, 而在光谱反射率空间用主成分分析法进行光谱分析得到的主要物理维度大于原稿实际使用的油墨数, 因此完成原稿油墨的光谱预测要将其转换到一个能够完全反映油墨混合状态的线性空间模型中。

半色调印刷品使用半透明的油墨其阶调变化由离散分布的不同网点面积率的网点表示, 其光学特性不同于透明或不透明色料的连续调原稿^[2]。油墨混合状态可借助Neugebauer方程描述光谱反射率与网点百分比之间的关系^[2], 但该方程基于纽介堡色元光谱反射率的线性加权, 空间维度将远超出实际使用的原色油墨数目。基于半色调原稿的这些特性, 建立适用于原稿油墨光谱预测的线性空间模型, 其思想相似于非负矩阵分解:

(1) 该空间模型以印刷品原稿色彩光谱反射率为基础, 其主要维度等于原色油墨数目;

(2) 该空间表示印刷品原稿的整体光谱特性, 由各原色油墨光谱特性与其浓度百分比的线性组合;

(3) 印刷品原稿每种颜色的光谱特性是混合该颜色的各原色油墨光谱特性的线性叠加效果;

(4) 每个颜色混合所需的各原色油墨浓度百分比之和为1。

$\begin{array}{l} Θ = \overset{k}{\sum_{i = 1}} a_{i} c_{i} = AC \\ 0 \leq a_{i}; 0 \leq c_{i} \leq 1; \overset{k}{\sum_{i = 1}} C_{ij} = 1, j = 1, 2, 3, \dots, m (1) \end{array}$

空间模型中Θ 为n× m矩阵表示印刷品原稿线性经验空间的光学特性, n为矩阵维数, 即光谱采样的空间维度, m为原稿颜色采样样本数。 k表示原稿所用原色油墨数目, 向量a_i表示第i个原色油墨的光谱特性, 用向量组A表示A_n_×_k=(a₁, a₂, a₃, …, a_k), 向量c_i表示第i个原色油墨对应的浓度百分比, 用矩阵C表示C_k_×_m=(c₁, c₂, c₃, …, c_m), $\overset{k}{\sum_{i = 1}}$ C_ij=1表示矩阵C中每列和为1, 即每个采样颜色的各油墨浓度和为1。

上述空间经大量实验推导, 得出一个由光谱反射率空间转换得到的转换优化模型^[2]

$\begin{array}{l} Θ = R_{p}^{\frac{1}{n}} - R_{m}^{\frac{1}{n}}; n \geq 2 (2) \\ R_{m} = {(R}_{p}^{\frac{1}{n}} {- Θ)}^{n} (3) \end{array}$

式中, R_m为测得原稿的光谱反射率, R_p为纸张光谱反射率, 考虑到油墨覆盖到纸基以后油墨及纸张对光线的实际反射作用, 用转换因数n进行了修正, 转换后的经验空间除去了纸张对油墨光谱反射的影响。转换因数n值决定着线性经验空间关于原色油墨浓度及光谱特性的线性相关程度, 通过实验及优化算法确定最佳n值, 以此建立的经验空间用主成分分析法确定主要物理维度, 其主要物理维度等于实际使用的油墨数目, 并将主要维度数目作为以下油墨光谱预测算法的基底数目。

2 印刷品原稿原色油墨光谱预测方法

2.1 基于约束条件NMF的光谱预测算法

基于有约束条件的非负矩阵分解的原色油墨光谱预测算法ISPNMF是在基本非负矩阵分解算法基础上建立的, 能使其准确预测印刷品原色油墨光谱曲线。该预测算法不同于常用的一些基色组分预测算法, 其分解结果完全没有负分量的存在具有明确的实际意义^[5]。

基本非负矩阵分解算法即在已知目标矩阵W的前提下, 同时分解得到主要因素V矩阵和在V基底下的系数矩阵H, 采用合适的目标函数和迭代规则不断迭代分解找到使得逼近误差E最小的最优近似解。这种算法思想与印刷品原色油墨混合状态经验空间模型相一致^[4]。非负矩阵分解中引入了两种目标函数, 其中欧氏距离目标函数的研究应用较为广泛, 也更接近油墨预测光谱逼近实际光谱的求解过程。

$\begin{array}{l} W = \overset{p}{\sum_{i = 1}} v_{i} h_{i} \approx VH (4) \\ E = \min \sum_{ij} (W_{ij} - {(VH)}_{ij})^{2} (5) \\ v_{i}, h_{i} \in R^{+}; V \in R_{+}^{n \times p}; H \in R_{+}^{p \times m}; W \in R_{+}^{n \times m} \end{array}$

2.1.1 半透明油墨预测的约束条件

ISPNMF算法在对预测结果有具体期望值的情况下在基本非负矩阵算法上添加除非负限制以外的限制条件, 包括全加性约束、平滑性约束和稀疏性约束, 以此使得分解结果满足印刷品原稿原色油墨的光谱特性和油墨浓度分布特性。

(1)非负约束, 是此算法最基本的约束条件, 保证了分解结果的可解释性和明确的物理意义^[5], 具体的保证了分解得到的油墨光谱矩阵和浓度系数矩阵非负性, 非负性约束在一定程度上导致了相应的稀疏性描述^[5]。

(2)全加性约束, 即要求浓度系数矩阵C的列向量加和为1, 该约束条件满足半色调印刷品油墨的混合状态, 每个叠印网点单元的各色油墨混合浓度的加和为100%, 转化为如下函数求解优化问题

$\begin{array}{l} \min F (Θ, AC) \\ s.t. A \in R_{+}^{n \times k}; C \in R_{+}^{k \times m}; Θ \in R_{+}^{n \times m} (6) \\ \overset{k}{\sum_{i = 1}} C_{ij} = 1, j = 1, 2, \dots, m \end{array}$

其中, Θ 表示印刷品原稿线性经验空间的光学特性, A指经验空间中预测的原色油墨光谱矩阵, C为油墨浓度系数矩阵, F函数是近似最优解预测值极度逼近原始值时的最小目标函数, k表示原稿中原色油墨数, 这恰好符合非负矩阵分解算法的思想, 唯一不同的是多了全加性约束条件, 这可在迭代过程中引入, 每次迭代都强制系数矩阵列向量符合加和为1的特性直至算法收敛, 迭代公式如式(7)

$c_{j} \leftarrow \frac{c_{ij}}{\overset{k}{\sum_{i = 1}} c_{ij}}, C^{n + 1} \leftarrow \frac{c^{n}}{\sum c}, j = 1, 2, \dots, m (7)$

(3)平滑性约束, 要求算法分解得到的A矩阵元素分布符合油墨光谱值的实际分布情况, 具有平滑且无窄带凸起的特点, 数据分布特点与稀疏分布恰好相反, 非负矩阵分解中一个矩阵不稀疏必然导致另一个矩阵稀疏分布^{[4, 6]}, 该约束条件与欧氏距离目标函数相结合作为平滑性罚项引入, 得到新的目标函数具体如式(8)

$\begin{array}{l} F (A, C) = \min [\frac{1}{2} \sum_{ij} (Θ_{ij} - {(AC)}_{ij})^{2} + \frac{α}{2} G_{1} (A)] (8) \\ s . t . A \in R_{+}^{n \times k}; C \in R_{+}^{k \times m}; Θ \in R_{+}^{n \times m} \end{array}$

(4)稀疏性约束, 非负约束虽然一定程度导致了矩阵分布的相对稀疏, 但算法在不同用途中对稀疏程度的要求不同。原色油墨光谱预测中, 矩阵C的数据分布表示了各原色油墨在呈现色彩时的浓度分布情况, 印刷品图像中的颜色是由几种原色油墨不同浓度混合而成, 不可能均匀分布于整个空间, 故对矩阵C的数据分布有一定稀疏程度的要求。采用Hoyer的增强稀疏性编码^[7], 将稀疏性罚项引入目标函数实现对系数矩阵不同程度稀疏性的控制, 转为如式(9)优化问题

$\begin{array}{l} F (A, C) = \min [\frac{1}{2} \sum_{ij} (Θ_{ij} - {(AC)}_{ij})^{2} + β G_{1} (C)] (9) \\ s . t . A \in R_{+}^{n \times k}; C \in R_{+}^{k \times m}; Θ \in R_{+}^{n \times m} \end{array}$

2.1.2 目标函数及迭代规则

综合以上约束条件, 同时将条件(3)和(4)作为目标函数的罚项导入, 得到新的目标函数其解符合数据分布特征

$\begin{array}{l} F (A, C) = \frac{1}{2} \sum_{ij} (Θ_{ij} - {(AC)}_{ij})^{2} + \frac{α}{2} G_{1} (A) + β G_{2} (C) (10) \\ G_{1} (A) = \sum_{ij} (A_{ij})^{2}, G_{2} (C) = \sum_{ij} C_{ij} (11) \end{array}$

α G₁(A)和β G₂(C)是导入目标函数的罚项用来约束油墨光谱矩阵A和浓度系数矩阵C, α 和β 是根据实验经验得到的常数参数受原稿类型光学特性影响, 透明、不透明和半透明色料原稿其值均不相同。对目标函数求解时, 采用乘性迭代规则对未知A和C交替迭代直至找到误差最小时的最优解, 分别求其偏导, 得迭代梯度如式(12)和式(13)

$\begin{array}{l} \frac{\partial F (A, C)}{\partial A} = - (Θ C^{T} - AC C^{T} - αA) (12) \\ \frac{\partial F (A, C)}{\partial C} = - (A^{T} θ - A^{T} AC - β) (13) \end{array}$

为动态调整迭代规则中的下降梯度其迭代步长如式(14)和式(15)

$\begin{array}{l} ρ_{1} = \frac{A^{n} .}{A^{n} C C^{T} + αA} (14) \\ ρ_{2} = \frac{C^{n} .}{A^{T} A C^{n} + 0.5 β} (15) \end{array}$

对原色油墨光谱矩阵A和系数矩阵C从梯度负方向开始迭代, 迭代规则如式(16)和式(17)

$\begin{array}{l} A^{n + 1} \leftarrow A^{n} + ρ_{1} (Θ C^{T} - AC C^{T} - αA) (16) \\ C^{n + 1} \leftarrow C^{n} + ρ_{2} (A^{T} Θ - A^{T} AC - β) (17) \end{array}$

将调整梯度下降的迭代步长分别代入式(16)和式(17), 为得到预期的分解结果, 并将以上约束条件中对浓度系数C加和为100%的约束置于迭代过程中, 得到最终具体迭代规则如式(18)— 式(20)

$\begin{array}{l} A \leftarrow A . \frac{Θ C^{T}}{AC C^{T} - αA} (18) \\ C \leftarrow C . \frac{A^{T} Θ}{A^{T} AC - β} (19) \\ C \leftarrow C . / (ones (k, 1) sum (C)) (20) \end{array}$

2.2 黑色油墨的优化算法

实验中通过上述算法对印刷品原稿原色油墨光谱曲线进行预测, 彩色油墨的预测光谱与实际光谱拟合程度较高, 但黑色油墨预测光谱形状与实际光谱曲线有一定差距, 其形状类似于其他彩色油墨光谱的线性叠加^[4], 黑色似乎是由其他三彩色叠印得到, 这与印刷品的减色呈色原理相关, 对真正使用的K色油墨预测有一定程度的干扰, 需要对预测得到的黑色油墨光谱曲线进行优化使其逼近实际光谱曲线。

黑色油墨预测光谱优化方法采用非线性有约束的优化函数, 具体流程如图1。理想状态下, 建立的印刷品原稿光谱线性经验空间是由实际使用的原色油墨相关光谱特性向量组和对应浓度系数矩阵的线性组合, 而预测的黑色油墨光谱特性曲线与实际黑色油墨光谱存在偏差, 由此假设存在一个更准确的黑色油墨光谱向量, 使得新组成的原色油墨光谱向量组与对应系数的线性组合θ ₁与Θ 的误差ε 更小, 即{ε ₂=‖ Θ -θ ₁‖ ²}≤ {ε ₁=‖ Θ -θ ‖ ²}, 或者θ ₁与Θ 的误差收敛。

	Figure Option View Download New Window
	图1 黑色油墨预测光谱优化流程Fig.1 The optimization process of black ink prediction spectrum

上述黑色油墨预测光谱的优化方法中, 首先在线性经验空间运用油墨光谱预测算法(ISPNMF)得到原色油墨光谱预测值A及其对应系数C, 并将其逼近误差记为ε ₁, 如若原稿使用黑色油墨, 将黑色油墨预测光谱向量作为优化向量black, 并与其他(k-1)个彩色油墨预测向量组合成新的油墨光谱向量组A_new, 对黑色油墨预测光谱进行优化处理, 每处理一次对应得到与原始线性经验空间相似的θ ₁, 并将θ ₁与原始线性空间Θ 的逼近误差记为ε ₂, 当ε ₂收敛时最小值赋予f。直至ε ₂≤ ε ₁或f有最小值时输出黑色油墨光谱的优化向量black。

2.3 算法评价标准

对光谱拟合情况的评价主要从拟合度GFC、光谱角距离SAD和均方根误差RMSE这三个应用广泛的指标来衡量预测光谱与实际光谱匹配的精确度^[8]。其中均方根误差表示预测值与真实值之间的偏差来评价两光谱曲线形状的相似程度, 偏差越小相似程度越高, 光谱角指将预测光谱与真实光谱看作两个空间向量, 计算两光谱曲线的广义夹角, 夹角越小即SAD越小, 两曲线相似程度越高^[8], 另一个光谱匹配指标拟合度基于施瓦茨不等式建立, 用于评价两光谱曲线的精度, 当其值达到99%时, 认为光谱匹配已经达到了理想匹配。

均方根误差RMSE

$RMSE = \sqrt[]{\frac{\sum (R_{m} - R_{recon})^{2}}{n}} (21)$

拟合度GFC

$GFC = \frac{|\sum (R . \dot{R})|}{\sqrt[]{\sum (R^{2})} \sqrt[]{\sum ({\dot{R}}^{2})}} (22)$

光谱角SAD

$SAD = co s^{- 1} \frac{|\sum (R . \dot{R})|}{\sqrt[]{\sum (R^{2})} \sqrt[]{\sum ({\dot{R}}^{2})}} (23)$

3 结果与讨论

分别使用Konica Minolta C1085和HP indigo5600两台数码印刷机及其自身配备的墨粉和墨膏来摸拟不同品牌的油墨, 均为四色油墨印刷机, 在230 g白卡纸上打印IT8.7/3色标(1 120色块), 并使用X-rite i1 Pro2获取两样张的光谱反射率作为实验数据样本, 对比原色油墨光谱预测算法和黑色油墨光谱预测优化算法的准确性和实用性。

其中两台数码印刷机均使用KCMY四原色, 在线性经验空间中使用惠普数码印刷机得到样张数据, 原色实际光谱特性曲线记为HPaic (actual ink curve), 使用上述ISPNMF算法得到的原色预测光谱曲线记为HPpic (predicted ink curve), 使用黑色油墨预测光谱优化方法对预测黑优化后得到的光谱曲线记为HPopt(optimization)。同理用柯尼卡美能达数码印刷机获得的分别记为KMaic, KMpic和KMopt。经验空间Θ 中两样张使用的原色其实际光谱特性曲线如图2和图3。

	Figure Option View Download New Window
	图2 柯美数码印刷机4原色光谱特性曲线Fig.2 4 primary color spectral property curves of KM

	Figure Option View Download New Window
	图3 惠普数码印刷机4原色光谱特性曲线Fig.3 4 primary color spectral property curves of HP

将两样本光谱反射率数据分别转到线性经验空间Θ 中, 并用主成分分析法PCA分别在光谱反射率R空间和线性经验Θ 空间, 获取数据的前6个特征值对应的贡献率和累计贡献率, 列于表1中, 分析经验空间数据的主要物理维度及印张原稿所使用的原色油墨数目。

表1 经验空间中两数据样本的贡献率和累计贡献率 Table 1 Contribution rate and cumulative contribution rate of two samples in empirical space

特征值的贡献率对应于特征向量对数据集信息量的贡献程度, 其累计贡献率指前几个特征向量代表整个数据集的信息量^[6], 故将特征向量对应的贡献率和累计贡献率作为判断空间主要维度的标准, 取累计贡献率大于99.9%且增长速度即贡献率小于0.03%, 即Aa_j≥ 99.9%且Aa_j₊₁-Aa_j≤ 0.03%时, j作为数据样本的主要维度数目。表1中, 两样张在线性经验空间的前4个基向量的累计贡献率均大于99.9%, 且增长速度分别为0.008%和0.02%均小于0.03%, 但在光谱反射率空间前4个基向量的累计贡献率虽然大于99.9%, 但其增长速度均大于0.03%, 且需要5个或6个基向量才能达到经验空间中4个基向量同等的信息贡献程度, 故经验空间两数据样本分别用4个基向量可充分表征整个数据样本的信息量, 即两空间的主要维度数为4, 这与样张实际使用的原色数目也一致。

确定基于非负矩阵分解算法的光谱预测算法中使用的基底数后, 在Θ 空间中对两样张原稿数据样本分别进行原色的光谱预测, 并将预测结果和实际原色光谱特性曲线相对比于图4评价参数列于表2, 图中可以看出两台印刷机自身原色模拟的两类油墨存在明显差异, 预测结果中除中性灰的直接预测存在较大误差外, 其他彩色原色的预测结果分别与实际使用的原色光谱曲线匹配较为理想。两原稿样本数据直接预测的黑原色光谱特性曲线形状类似于三彩色原色曲线的线性叠加, 这与半色调原稿的减色呈色原理相关, 中性灰的呈色可以看作是其他三彩色对光线吸收反射的混合结果, 预测得到的K像是运用了底色去除GCR印刷工艺, 这对真正使用的K原色油墨光谱的预测有一定程度的干扰。进一步对黑原色的直接预测结果运用上述优化算法, 图4(a)中可以看出K原色油墨预测光谱的优化结果(KMopt和HPopt)分别与实际光谱特性曲线十分逼近。

	Figure Option View Download New Window
	图4 经验空间中IT8.7/3色标原色和预测曲线匹配图 (a): 黑原色油墨; (b): 青原色油墨; (c): 品原色油墨; (d): 黄原色油墨Fig.4 The curves matching graph of IT8.7/3 standard color andprediction color in experiential space (a): Black ink; (b): Cyan ink; (c): Magenta ink; (d): Yellow ink

表2 经验空间中IT8.7/3色标预测光谱与实际光谱的比较 Table 2 Comparison of the prediction and actual spectra of the IT8.7/3 color card in experiential space

表2中包含两原稿原色预测结果的评价以及HPpic与KMaic的匹配评价, 定量描述了算法预测的准确性和油墨匹配的精确性。可以看出两样张中预测的黑原色曲线优化结果远优于直接预测的结果, 且各彩色预测曲线与实际使用的原色光谱特性曲线相比拟合度均大于99.9%, HPpic与KMaic的匹配结果劣于实际匹配结果, 说明本原色油墨光谱预测算法不仅能对印刷品原稿原色油墨实现准确预测, 而且可以精确匹配得到实际使用的原色油墨, 这对实现印刷品原稿的光谱复制有重要意义。

4 结论

利用提出的ISPNMF算法和黑色油墨预测光谱的优化算法对半色调印刷品原色油墨光谱的预测估计值与实际油墨光谱相比其拟合度均高达99.9%, 提出的基于有约束条件的非负矩阵分解的原色油墨光谱预测算法ISPNMF及黑色油墨预测光谱优化算法根据半色调印刷品的光学特性和油墨混合状态, 针对非负矩阵分解算法局部最优的弊端, 为能在线性经验空间中实现原色油墨的光谱预测, 通过添加符合油墨光谱和浓度分布情况的约束条件包括浓度系数全加性约束、浓度分布稀疏约束和预测光谱平滑约束的条件, 合理构造目标函数, 以此克服了基本非负矩阵分解算法中多重最优解和局部极小值的缺陷, 实现了预测算法唯一的全局最优解, 并增强了原色油墨光谱预测的准确性。黑色油墨预测光谱的优化算法克服了彩色油墨对光线混合吸收给预测黑色油墨光谱带来的干扰, 能很好的优化得到逼近于实际黑色油墨光谱的预测值, 这对实现印刷品原稿的光谱复制技术有重要实用价值。

The authors have declared that no competing interests exist.

参考文献

文献选项

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2010

0.0

... 引言基于光谱的印刷复制技术有别于传统的基于色度匹配的印刷技术, 图像色彩信息以光谱信息形式传输复制, 这样能很大程度上减小甚至消除同色异谱现象^[1] ...

2016

0.0

... 要实现半色调印刷品原稿的光谱复制技术, 前提必须确定原稿所使用的油墨光谱成份, 如不知已有的半色调印刷品的油墨类型却想得到和原图像色彩再现的效果, 原色油墨的光谱预测尤为关键, 理想的做法便是找到和原稿相同的原色油墨来完成印刷, 实现光谱曲线的最佳匹配^[2] ...

... 半色调印刷品使用半透明的油墨其阶调变化由离散分布的不同网点面积率的网点表示, 其光学特性不同于透明或不透明色料的连续调原稿^[2] ...

... 油墨混合状态可借助Neugebauer方程描述光谱反射率与网点百分比之间的关系^[2], 但该方程基于纽介堡色元光谱反射率的线性加权, 空间维度将远超出实际使用的原色油墨数目 ...

... 上述空间经大量实验推导, 得出一个由光谱反射率空间转换得到的转换优化模型^[2] ...

2015

0.0

... 其中PCA法进行基色光谱预测时, 得到的基向量和对应的系数矩阵会出现负值, 没有物理意义^[3] ...

2016

0.0

... 何博士提出的有约束的非负矩阵分解方法, 虽解决了负值问题, 但其预测结果受约束条件影响, 具体约束条件受原稿光学特性影响^[4], 且中性灰的光谱预测还不准确 ...

... 这种算法思想与印刷品原色油墨混合状态经验空间模型相一致^[4] ...

... (3)平滑性约束, 要求算法分解得到的A矩阵元素分布符合油墨光谱值的实际分布情况, 具有平滑且无窄带凸起的特点, 数据分布特点与稀疏分布恰好相反, 非负矩阵分解中一个矩阵不稀疏必然导致另一个矩阵稀疏分布^{[4, 6]}, 该约束条件与欧氏距离目标函数相结合作为平滑性罚项引入, 得到新的目标函数具体如式(8) ...

... 2 黑色油墨的优化算法实验中通过上述算法对印刷品原稿原色油墨光谱曲线进行预测, 彩色油墨的预测光谱与实际光谱拟合程度较高, 但黑色油墨预测光谱形状与实际光谱曲线有一定差距, 其形状类似于其他彩色油墨光谱的线性叠加^[4], 黑色似乎是由其他三彩色叠印得到, 这与印刷品的减色呈色原理相关, 对真正使用的K色油墨预测有一定程度的干扰, 需要对预测得到的黑色油墨光谱曲线进行优化使其逼近实际光谱曲线 ...

2008

0.0

, ZHANG

Yu-jin

(李乐, 章毓晋). Journal of Electronics(电子学报), 2008, 4: 737.

The fundamentals and properties of non-negative matrix factorization (NMF) are introduced,and available NMF algorithms are classified into two categories:basic NMF model-based algorithms and improved NMF model-based algorithms.Based on these,the design principles,application characteristics,and existing problems of the algorithms are systematically discussed.Besides,some open problems in the development of NMF algorithms are presented and analyzed.

1. Tsinghua National Laboratory for Information Science and Technology,Tsinghua University,Beijing 100084,China;2. Department of Electronic Engineering,Tsinghua University,Beijing 100084,China

本文介绍了非负矩阵分解(Non-negative Matrix Factorization,NMF)的基本原理和性质,将现有NMF算法分为了基于基本NMF模型的算法和基于改进NMF模型的算法两大类,在此基础上较为系统地分析、总结和比较了它们的构造原则、应用特点以及存在的问题,最后预测和分析了未来NMF算法研究的可能方向.

... 该预测算法不同于常用的一些基色组分预测算法, 其分解结果完全没有负分量的存在具有明确的实际意义^[5] ...

... (1)非负约束, 是此算法最基本的约束条件, 保证了分解结果的可解释性和明确的物理意义^[5], 具体的保证了分解得到的油墨光谱矩阵和浓度系数矩阵非负性, 非负性约束在一定程度上导致了相应的稀疏性描述^[5] ...

1999

0.0

... 特征值的贡献率对应于特征向量对数据集信息量的贡献程度, 其累计贡献率指前几个特征向量代表整个数据集的信息量^[6], 故将特征向量对应的贡献率和累计贡献率作为判断空间主要维度的标准, 取累计贡献率大于99 ...

2011

0.0

LIU

Xue-song

, WANG

Bin

, ZHANG

Li-ming

(刘雪松, 王斌, 张立明). Journal of Infrared and Millimeter Waves(红外与毫米波学报), 2011, 30(1): 27.

Because of the local minima in the objective function, the traditional Nonnegative Matrix Factorization (NMF) algorithm is sensitive to the initial value when being applied to hyperspectral unmixing. In order to solve the problem, a new approach based on constrained NMF was proposed for decomposition of mixed pixels by introducing constraints of abundance separation and smoothness into the objective function of NMF. The algorithm can also satisfy the abundance nonnegative and sum-to-one constraints, which are necessary for hyperspectral unmixing. Experimental results on simulated and real hyperspectral data demonstrate that the proposed approach can overcome the shortcoming of local minima, and obtain better results. Meanwhile, the algorithm performs well for noisy data, and can also be used for the unmixing of hyperspectral data in which pure pixels do not exist.

[Liu Xue-Song; Wang Bin; Zhang Li-Ming] Fudan Univ, Dept Elect Engn, Shanghai 200433, Peoples R China.;[Wang Bin] Fudan Univ, Minist Educ, Key Lab Wave Scattering & Remote Sensing Informat, Shanghai 200433, Peoples R China.

... 采用Hoyer的增强稀疏性编码^[7], 将稀疏性罚项引入目标函数实现对系数矩阵不同程度稀疏性的控制, 转为如式(9)优化问题 ...

2015

0.0

SUN

Yan-li

, ZHANG

Xia

, SHUAI

Tong

(孙艳丽, 张霞, 帅通). Journal of Remote Sensing(遥感学报), 2015, 4: 618.

This paper proposes an automated radiometric normalization method based on Spectral Angle Distance (SAD) and Euclidean Distance (ED). The method is implemented on two hyperspectral images taken by the Compact High Resolution Imaging Spectrometer (CHRIS) sensor. Experimental results confirm that the proposed method is not only superior to the Mean Absolute Difference (MAD)-based normalization of radiation characteristics, but also has the advantage of commendably preserving the spectral characteristics. Radiometric normalization minimizes the radiometric differences between two images caused by unstable factors in the acquisition conditions rather than by changes in surface reflectance, which is crucial to land-cover change detection. Radiometric normalization is known as relative radiometric correction. In contrast to absolute correction, the relative method does not need atmospheric data at the moment of image acquisition. This method uses one of the images as reference and then adjusts the radiometric characteristics of the other image, known as subject image, to match the reference image.This paper proposes an automated radiometric normalization method based on Spectral Angle Distance (SAD) and Euclidean Distance (ED). The method is implemented on two hyperspectral images taken by the Compact High Resolution Imaging Spectrometer (CHRIS) sensor. Experimental results confirm that the proposed method is not only superior to the Mean Absolute Difference (MAD)-based normalization of radiation characteristics, but also has the advantage of commendably preserving the spectral characteristics.Radiometric normalization minimizes the radiometric differences between two images caused by unstable factors in the acquisition conditions rather than by changes in surface reflectance, which is crucial to land-cover change detection. Radiometric normalization is known as relative radiometric correction. In contrast to absolute correction, the relative method does not need atmospheric data at the moment of image acquisition. This method uses one of the images as reference and then adjusts the radiometric characteristics of the other image, known as subject image, to match the reference image.This paper proposes an automated radiometric normalization method based on SAD and ED. The proposed method selects unchanged pixels by SAD and ED by considering that the same feature has a similar spectrum shape on hyperspectral images. Therefore, we make an attempt to validate the SAD-ED radiometric normalization of multitemporal hyperspectral satellite images. The Mean Absolute Difference (MAD)-based normalization method is also applied for comparison. In the essay, common evaluation index Root Mean Square Error (RMSE) and Relative Deviation Index (RDI) are used to verify the normalized results. Considering the features of hyperspectral remote sensing images, we also apply the spectral fidelity indexes, i.e., Pearson Correlation Coefficient (PCC) and spectral distortion degree.The method is implemented on two hyperspectral images taken by the CHRIS sensor. Common evaluation index RMSE and (RDI) are used to verify the normalized results. PCC and spectral distortion degree are also applied to evaluate spectral fidelity for radiometric normalization of multitemporal hyperspectral satellite imagery. The evaluation results of RMSE and deviation D show that the SAD-ED normalization of CHRIS images is feasible and more effective than the MAD-based normalization. In addition, the evaluation results of PCC and DD show that the SAD-ED normalization performs better in keeping the spectral-dimensional information of hyperspectral images compared with the MAD-based normalization.Experimental results confirm that the proposed method in this essay is not only superior to the MAD-based normalization of radiation characteristics, but also has the advantage of commendably preserving the spectral characteristics of hyperspectral images, thereby having good application prospects.This paper proposes an automated radiometric normalization method based on Spectral Angle Distance (SAD) and Euclidean Distance (ED). The method is implemented on two hyperspectral images taken by the Compact High Resolution Imaging Spectrometer (CHRIS) sensor. Experimental results confirm that the proposed method is not only superior to the Mean Absolute Difference (MAD)-based normalization of radiation characteristics, but also has the advantage of commendably preserving the spectral characteristics.Radiometric normalization minimizes the radiometric differences between two images caused by unstable factors in the acquisition conditions rather than by changes in surface reflectance, which is crucial to land-cover change detection. Radiometric normalization is known as relative radiometric correction. In contrast to absolute correction, the relative method does not need atmospheric data at the moment of image acquisition. This method uses one of the images as reference and then adjusts the radiometric characteristics of the other image, known as subject image, to match the reference image.This paper proposes an automated radiometric normalization method based on SAD and ED. The proposed method selects unchanged pixels by SAD and ED by considering that the same feature has a similar spectrum shape on hyperspectral images. Therefore, we make an attempt to validate the SAD-ED radiometric normalization of multitemporal hyperspectral satellite images. The Mean Absolute Difference (MAD)-based normalization method is also applied for comparison. In the essay, common evaluation index Root Mean Square Error (RMSE) and Relative Deviation Index (RDI) are used to verify the normalized results. Considering the features of hyperspectral remote sensing images, we also apply the spectral fidelity indexes, i.e., Pearson Correlation Coefficient (PCC) and spectral distortion degree.The method is implemented on two hyperspectral images taken by the CHRIS sensor. Common evaluation index RMSE and (RDI) are used to verify the normalized results. PCC and spectral distortion degree are also applied to evaluate spectral fidelity for radiometric normalization of multitemporal hyperspectral satellite imagery. The evaluation results of RMSE and deviation D show that the SAD-ED normalization of CHRIS images is feasible and more effective than the MAD-based normalization. In addition, the evaluation results of PCC and DD show that the SAD-ED normalization performs better in keeping the spectral-dimensional information of hyperspectral images compared with the MAD-based normalization.Experimental results confirm that the proposed method in this essay is not only superior to the MAD-based normalization of radiation characteristics, but also has the advantage of commendably preserving the spectral characteristics of hyperspectral images, thereby having good application prospects. This paper proposes an automated radiometric normalization method based on SAD and ED. The proposed method selects unchanged pixels by SAD and ED by considering that the same feature has a similar spectrum shape on hyperspectral images. Therefore, we make an attempt to validate the SAD-ED radiometric normalization of multitemporal hyperspectral satellite images. The Mean Absolute Difference (MAD)-based normalization method is also applied for comparison. In the essay, common evaluation index Root Mean Square Error (RMSE) and Relative Deviation Index (RDI) are used to verify the normalized results. Considering the features of hyperspectral remote sensing images, we also apply the spectral fidelity indexes, i.e., Pearson Correlation Coefficient (PCC) and spectral distortion degree. The method is implemented on two hyperspectral images taken by the CHRIS sensor. Common evaluation index RMSE and (RDI) are used to verify the normalized results. PCC and spectral distortion degree are also applied to evaluate spectral fidelity for radiometric normalization of multitemporal hyperspectral satellite imagery. The evaluation results of RMSE and deviation D show that the SAD-ED normalization of CHRIS images is feasible and more effective than the MAD-based normalization. In addition, the evaluation results of PCC and DD show that the SAD-ED normalization performs better in keeping the spectral-dimensional information of hyperspectral images compared with the MAD-based normalization. Experimental results confirm that the proposed method in this essay is not only superior to the MAD-based normalization of radiation characteristics, but also has the advantage of commendably preserving the spectral characteristics of hyperspectral images, thereby having good application prospects.

辐射归一化旨在减小不同时相遥感影像间因获取条件不一致而导致的非地表辐射变化的差异,是土地覆盖变化监测的重要前提条件.本文根据高光谱图像上同类地物的谱形及数值的相似性,利用光谱角距离(SAD)和欧氏距离(ED)双重判定选取不变特征点,提出了一种基于光谱角—欧氏距离的辐射归一化方法.在评价指标中除了常用的均方根误差和相对偏差,更增加了高光谱特色的衡量光谱保真性指标:皮尔森系数、光谱扭曲程度.利用高光谱遥感CHRIS图像对本文提出方法进行验证,并与基于多元变化检测(MAD)的辐射归一化方法比较.结果表明,本文方法不仅在辐射特性上优于基于多元变化检测(MAD)的方法,而且具有保持光谱特性的优势,具有较好的应用前景.

... 3 算法评价标准对光谱拟合情况的评价主要从拟合度GFC、光谱角距离SAD和均方根误差RMSE这三个应用广泛的指标来衡量预测光谱与实际光谱匹配的精确度^[8] ...

... 其中均方根误差表示预测值与真实值之间的偏差来评价两光谱曲线形状的相似程度, 偏差越小相似程度越高, 光谱角指将预测光谱与真实光谱看作两个空间向量, 计算两光谱曲线的广义夹角, 夹角越小即SAD越小, 两曲线相似程度越高^[8], 另一个光谱匹配指标拟合度基于施瓦茨不等式建立, 用于评价两光谱曲线的精度, 当其值达到99%时, 认为光谱匹配已经达到了理想匹配 ...