DEGRADATION MODELING OF INK FADING AND DIFFUSION OF PRINTED IMAGES by ZIYI WANG A dissertation submitted to the School of Graduate Studies Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Doctor of Philosophy Graduate Program in Industrial and Systems Engineering Written under the direction of Elsayed A. Elsayed And approved by _______________________________ _______________________________ _______________________________ _______________________________ New Brunswick, New Jersey JANUARY, 2020 ABSTRACT OF THE DISSERTATION Degradation Modeling of Ink Fading and Diffusion of Printed Images By ZIYI WANG Dissertation Director: Elsayed A. Elsayed Color printing plays an important role in the modern society. It is known that the color of printed images degrades gradually due to the fading and diffusion of the inks. Color degradation leads to a distortion or loss of the original information in printed images. Therefore, it is desirable to understand how the color of printed images changes over time. In this dissertation, we present degradation models to predict the characteristics of the ink fading and diffusion of printed images. We begin by modeling the ink degradation from a physics-based perspective. Color images are printed by projecting small ink dots on medium, usually paper. This technique is called halftone printing. Halftone printing of color images results in a variety of ink mixtures and subsequently their potential catalytic fading. For the most commonly used Cyan-Magenta- Yellow-Black (CMYK) ink set, sixteen possible ink mixtures are generated during printing. A state transition diagram is then proposed for the ink fading in this multi-ink printing scenario. The ink area coverage is used as the performance indicator. Assuming constant fading and diffusion rates, we develop an ink fading model based on the differential ii equations according to the state transition diagram and an autoregressive ink diffusion model by discretizing the two-dimensional diffusion equation. The two models are then integrated into a single degradation model. Further examination of the developed degradation models reveals that the fading or diffusion rate is equivalent to the hazard rate in reliability engineering. It is known that the hazard rate of the exponential failure time distribution is constant. Hence, the developed degradation model with constant fading and diffusion rates is equivalent to the multistate Markov process model with exponential transition time distribution. By using non- exponential transition time distributions, the fading and diffusion rates become time- varying and a more general semi-Markov process degradation model is developed accordingly. Moreover, stochastic process models are investigated to provide stochastic area coverage prediction for the ink degradation. We first model the ink fading using the Hull- White/Vasicek (HWV) stochastic process. The HWV ink fading model considers that the variance of the ink area coverage shrinks as it approaches zero. Besides, spatial convolution is used to model ink diffusion. The two models are integrated into a spatio-temporal stochastic degradation model for the ink fading and diffusion of printed images. The cases of recurrent and non-recurrent time-varying fading and diffusion rates are investigated. Inks on the paper degrade, so does the paper. The degradation of paper condition may in turn affect the degradation of the inks. Therefore, the investigation of the degradation iii modeling of ink fading and ink diffusion with ink-paper interactions is needed. Two aspects of the ink-paper interactions are considered, i.e., the effect of paper aging such as depolymerization and yellowing, and the fiber orientation of the paper. The degradation process of printed images usually takes a very long time. An accelerated degradation model and the optimal design of accelerated degradation test planning is developed for accurate degradation prediction of printed images. The effects of three constant environmental stresses: temperature, humidity, and illumination (intensity), are investigated, and experimental data are used to validate the proposed model. The results show strong agreements between the proposed ink fading and ink diffusion prediction model and the actual experimental data. iv ACKNOWLEDGMENTS I would like to express my deepest gratitude to my adviser Dr. Elsayed A. Elsayed for his patient guidance and caring support throughout the years of my PhD study. It would not have been possible to complete this dissertation without his great effort and devotion. The education and training I received at Rutgers set high standards for me to follow in the future days of my life. My gratitude is extended to the committee members of my dissertation, Dr. Hoang Pham, Dr. Weihong Guo, and Dr. Minge Xie, for their valuable comments and advice. Besides, my special thanks go to Design Specialist Joe Lippencott for his help on the experimental set-up of this research. I would also like to thank my friends who accompanied me in the bittersweet days at this school. Finally, I sincerely thank my parents for their unconditional standing-by my side. v TABLE OF CONTENTS ABSTRACT OF THE DISSERTATION ........................................................................... ii ACKNOWLEDGMENTS .................................................................................................. v TABLE OF CONTENTS ................................................................................................... vi LIST OF TABLES .............................................................................................................. x LIST OF ILLUSTRATIONS ............................................................................................. xi CHAPTER 1 INTRODUCTION ....................................................................................... 1 1.1 Motivation of the Work ........................................................................................ 1 1.2 Color Printing Technologies ................................................................................ 2 1.3 Color Measurement .............................................................................................. 5 1.3.1 Colorimetry ................................................................................................... 6 1.3.2 Densitometry ................................................................................................. 8 1.3.3 Spectrum ....................................................................................................... 9 1.4 Problem Definition ............................................................................................. 10 1.5 Organization of the Dissertation ........................................................................ 13 CHAPTER 2 LITERATURE REVIEW .......................................................................... 15 2.1 Literature Review of Degradation of Printed Images ........................................ 15 2.1.1 Degradation Models of Printed Images ...................................................... 15 2.1.2 Accelerated Testing of Printed Images ....................................................... 20 vi 2.2 General Degradation Models ............................................................................. 23 2.2.1 Regression Models ...................................................................................... 23 2.2.2 Markov Models ........................................................................................... 24 2.2.3 Stochastic Process Models .......................................................................... 26 2.3 Accelerated Degradation Tests........................................................................... 28 2.4 Summary and Conclusions ................................................................................. 30 CHAPTER 3 PHYSICS-BASED DEGRADATION MODEL OF PRINTED IMAGES WITH CONSTANT FADING AND DIFFUSION RATES ............................................ 31 3.1 Performance Indicator ........................................................................................ 31 3.2 Degradation Model of Ink Fading ...................................................................... 32 3.2.1 Parameters Estimation ................................................................................ 35 3.2.2 Numerical Examples ................................................................................... 37 3.3 Degradation Model of Ink Diffusion .................................................................. 42 3.3.1 Numerical Examples ................................................................................... 44 3.4 Degradation Model of Ink Fading and Ink Diffusion ........................................ 47 3.4.1 Parameters Estimation ................................................................................ 48 3.4.2 Numerical Examples ................................................................................... 49 3.5 Summary and Conclusions ................................................................................. 52 CHAPTER 4 PHYSICS-BASED DEGRADATION MODEL OF PRINTED IMAGES WITH TIME-VARYING FADING AND DIFFUSION RATES .................................... 53 vii 4.1 Hazard Rate and Image Degradation ................................................................. 53 4.2 Degradation Model of Ink Fading .....................................................................