Reduced-order modeling and classification of hydrodynamic pattern formation in gravure printing

TL;DR

The paper tackles understanding and classifying hydrodynamic pattern formation in gravure printing by integrating reduced-order modeling with supervised learning on the HYPA-p dataset. It demonstrates that applying a fast Fourier transform before randomized SVD concentrates energy into a small number of modes, enabling an effective low-rank representation $X\approx U_r\,\Sigma\,V^*$ with $r$ as low as about 7, and that a kNN classifier trained on this reduced data achieves a test error as low as $E_{test}\approx 3\%$, outperforming human experts. The study also shows that FFT preprocessing, data balancing, and normalization influence performance in nuanced ways, and it uses the trained models to generate regime maps linking pattern classes to printing parameters, providing actionable insight for process tuning. Overall, the approach offers an interpretable, scalable framework for pattern classification in complex printing systems and lays groundwork for real-time digital-twin-type analyses of gravure processes.

Abstract

Hydrodynamic pattern formation phenomena in printing and coating processes are still not fully understood. However, fundamental understanding is essential to achieve high-quality printed products and to tune printed patterns according to the needs of a specific application like printed electronics, graphical printing, or biomedical printing. The aim of the paper is to develop an automated pattern classification algorithm based on methods from supervised machine learning and reduced-order modeling. We use the HYPA-p dataset, a large image dataset of gravure-printed images, which shows various types of hydrodynamic pattern formation phenomena. It enables the correlation of printing process parameters and resulting printed patterns for the first time. 26880 images of the HYPA-p dataset have been labeled by a human observer as dot patterns, mixed patterns, or finger patterns; 864000 images (97%) are unlabeled. A singular value decomposition (SVD) is used to find the modes of the labeled images and to reduce the dimensionality of the full dataset by truncation and projection. Selected machine learning classification techniques are trained on the reduced-order data. We investigate the effect of several factors, including classifier choice, whether or not fast Fourier transform (FFT) is used to preprocess the labeled images, data balancing, and data normalization. The best performing model is a k-nearest neighbor (kNN) classifier trained on unbalanced, FFT-transformed data with a test error of 3%, which outperforms a human observer by 7%. Data balancing slightly increases the test error of the kNN-model to 5%, but also increases the recall of the mixed class from 90% to 94%. Finally, we demonstrate how the trained models can be used to predict the pattern class of unlabeled images and how the predictions can be correlated to the printing process parameters, in the form of regime maps.

Figures (18)

• Figure 1: We investigate hydrodynamic pattern formation phenomena during gravure printing (a). The patterns are mainly formed during the sub-process of fluid transfer (dashed box) and can be assigned to one of three classes (b): point splitting, transition regime, and lamella splitting. The three classes are represented in the printed images as dot patterns (A), mixed patterns (B), and finger patterns (C). We use a large image dataset of gravure-printed patterns, including 26880.0 labeled images. The labeled images are randomly divided into train and test dataset and the train dataset is rearranged to a data matrix $\mathbf{X}$ and a singular value decomposition (SVD) is performed (c). For dimensionality reduction, the matrix $\mathbf{X}$ is projected onto the truncated SVD modes $\mathbf{U_{\text{r}}}$. This yields reduced-order data for training of several classifiers (d). The trained classifiers can be used to classify the unlabeled images of the dataset and to enable application-oriented scientific analysis on hydrodynamic pattern formation in gravure printing (e).
• Figure 2: Industrial gravure printing machines used for creation of the HYPA-p dataset. Bobst Rotomec MW 60-600/250 (Bobst, Mex, Switzerland) (a) and Gallus RCS 330-HD (Gallus Ferd. Rüesch AG, St. Gallen, Switzerland) (b). The gravure printing unit of the latter is shown in (c). Examples for dot pattern images (d), mixed pattern images (e), and finger pattern images (f) from the labeled dataset. Both machines can create all types of patterns. Each printed example has a size of 260 x 260 (2.75 x 2.75). More examples of printed patterns can be found in \ref{['fig:patterns_many']} in the Appendix.
• Figure 3: Schematic workflow. Variant (a) of the workflow performs randomized SVD (rSVD) directly on the data, which has been rearranged to a data matrix $\mathbf{X}$. The rSVD modes $\mathbf{U}$ are truncated to $\mathbf{U_{\text{r}}}$ and projected onto $\mathbf{X}$. The projected rSVD modes $\mathbf{U_{\text{r}}^*X}$ are used for training of several classifiers, leading to a minimum test error of 13 %. A normalization of $\mathbf{U_{\text{r}}^*X}$ to mean zero and unity standard deviation is optional and not performed by default. Variant (b) first performs a fast Fourier transform (FFT) on $\mathbf{X}$. Second, rSVD is performed on the magnitude of the FFT-transformed data $\mathbf{|\hat{X}|}$. The rest of the workflow is the same. This variant leads to a decreased minimum test error of 3 %. $\mathbf{X}$ can be chosen as the complete dataset $\mathbf{X_{\text{c}}}$ or as a balanced dataset $\mathbf{X_{\text{b}}}$. The train-test-split of $\mathbf{X}$ is always 80 %-20 % and leads to $\mathbf{X_{\text{train}}}$ and $\mathbf{X_{\text{test}}}$, which are fed separately through the workflow. For testing of the trained classifier, the rSVD modes from the training are reused to ensure equal transformation of $\mathbf{X_{\text{train}}}$ and $\mathbf{X_{\text{test}}}$.
• Figure 4: Normalized singular values $\sigma_{k,n}$ and cumulative energy in % over first $r$ modes. In the complete view (a), only every $500^{\text{th}}$ data point is plotted for better clarity. The zoomed in view (b) shows all data points for the first 20 modes.
• Figure 5: Selected modes $\mathbf{U}$ (a) and selected FFT modes $\mathbf{\hat{U}}$ (b). The modes were obtained by using SVD on the complete dataset $\mathbf{X_{\text{c}}}$ and on the complete, FFT-transformed dataset $|\mathbf{\hat{X}_{\text{c}}}|$, respectively. No train-test-split was performed. The FFT modes $\mathbf{\hat{U}}$ are displayed in the spatial domain to aid in interpretation; this is achieved by taking the magnitude of the inverse FFT with phase zero for each FFT mode in $\mathbf{\hat{U}}$.