With the increasing complexity of industrial systems and process data, deep learning has achieved superior performance in soft sensing but remains constrained by limited interpretability. Most existing interpretability techniques are correlation based, capturing statistical dependencies but providing little insight into underlying mechanisms. Causal modeling, by contrast, offers stronger interpretability by revealing directional and temporal influences, thereby improving both reliability and understanding. Although some recent methods consider time delays, their treatment of lags remains coarse and limited, and cannot adequately capture heterogeneous cross variable delay patterns in industrial time series. To address these limitations, we propose spatio-temporal causal learning with delay annotation (STCLD), which introduces a spatio-temporal delay attention (STDA) module to explicitly learn delay annotated spatio temporal causal graphs for soft sensing. STDA minimizes a maximum mean discrepancy objective to discover causal relations with edge specific delays, while attention path strength and multidimensional dynamic complexity are used to infer causal directions in a model-based way. The learned causal graph and delay information then guide a delay aware prediction module to build a self-interpretable soft sensor. Experiments on two real-world industrial datasets show that STCLD consistently outperforms strong baselines in both predictive accuracy and causal interpretability, providing a robust and general framework for interpretable soft sensor modeling in complex process industries.

Deep learning-based soft sensors (DLSSs) have been demonstrated to exhibit significantly improved sensing accuracy; however, their vulnerability to adversarial attacks affects their reliability, thus hindering their widespread application. To improve the reliability of DLSSs, in this article, we conducted a systematic investigation of the adversarial attack and defense of DLSSs. By considering the task requirements of DLSSs and the actual scenarios that attackers may encounter, a framework based on black-box attack and proactive defense was proposed to realize the adversarial attack and defense of soft sensors. The adversarial attack was implemented through the proposed knowledge-guided adversarial attack (KGAA) method. By reconstructing the optimization model and introducing the mechanism knowledge into the objective function, the KGAA method could overcome the ill-posed problem of adversarial attack optimization when attacking a regression model. Moreover, based on the KGAA, a corresponding KGAA adversarial training defense method was proposed to achieve proactive defense. The attack and defense methods were verified in terms of the thermal deformation sensing of an air preheater rotor. Compared to other attacks, the KGAA exhibited higher imperceptibility, rationality, and stability; it can thus be considered a practical attack. The implementation of KGAA adversarial training enhances the adversarial robustness of DLSSs, thus aiding the defense of DLSSs to various attacks and improving their reliability.

For deep learning-based soft sensors, the lack of interpretability and the consequent unreliability has become one of the most important problems. In this article, a neural network scheme called the deep multiple attention soft sensor (DMASS), which consists solely of attention mechanisms, is proposed to develop a self-interpretable soft sensor. DMASS was established to ensure the self-interpretability of data selection and sensor modeling and try to integrate these originally independent phases into the single scheme. First, the existing attention mechanisms’ core implementation steps are summarized as a unified form, and then the variable attention mechanism and time lag attention mechanism are proposed. When DMASS's training is completed, the obtained attention weights provide the self-interpretable data selection results. Then, a self-attention activation structure (SAAS) is proposed to extract the nonlinear spatio-temporal features of data. The mathematical expression for the extracted feature, the SAAS's attention matrix, the information path diagram for DMASS's training, and the uncertainty-aware interval prediction show the self-interpretability of sensor modeling. Finally, DMASS was applied to predict the thermal deformation of the air preheater rotor, and the validity of DMASS's self-interpretability is verified by the known mechanism analysis and information bottleneck theory. Meanwhile, DMASS's great sensing performance was confirmed through comparison with other novel soft sensors.

Abstract Image prior plays a decisive role in the performance of widely studied model‐based restoration methods. To further improve restoration performance, this paper proposes an exponential scale mixture‐based simultaneous sparse prior (ESM‐SSP) to accurately characterize image prior information. Specifically, first, two structured dictionaries are adaptively learned to explore the local and non‐local sparsity of similar patch groups simultaneously. Then, the exponential scale mixture (ESM) is employed to model simultaneous sparse coefficients. The adoption of ESM enables us to accurately estimate simultaneous sparse coefficients by adaptively adjusting the regularization parameters. With the aid of ESM‐SSP, an effective image restoration algorithm is developed to preserve more image details. Extensive experimental results on image denoising and deblocking demonstrate that compared with many state‐of‐the‐art model‐based methods, the proposed ESM‐SSP‐based restoration algorithm not only has competitive peak signal‐to‐noise ratio, but also produces higher structural similarity index and better visuals. More importantly, the proposed method can also compete favourably with the superior deep learning‐based restoration methods.

In order to overcome the potential distortion problem, an improved haze-corrected version of multiplicative-based component substitution pansharpening method is proposed. Our improvement is to acquire the optimal spatial detail maps in detail extraction procedure with sparse structural manifold embedding (SSME) and guided filter. First, a major spatial detail image is estimated by using SSME to well preserve the structural details, such as textures, edges, and contours. Second, a detail map is obtained by using the guided filter compensates for the local information loss of the major detail image due to inaccurate recovery for the patches with weaken structural information. Finally, the pansharpened image is acquired by proportionally injecting two detail images into the low-resolution multispectral image. The performance evaluations on degraded and real QuickBird data sets by using quality assessment metrics and visual analysis demonstrate that the proposed method achieves better results as compared with other existing methods.

Dual-tree transforms have recently received much attention for the properties of shift-invariance and directional-selectivity. However, their designs generally encounter fractional-delay constraints, and become more complicated for providing linear-phase (LP) individual filters and flexible directional-selectivity, two important properties in image processing. In this paper, we propose an alternative shift-invariant and directional-selective transform-the dual-tree cosine-modulated filter bank (DTCMFB). In the proposed DTCMFB, its primal and dual filter banks are derived by cosine-modulating one LP prototype filter, and thus its design involves no fractional-delay constraints. Meanwhile, the derived modulation technique guarantees each individual filter to be LP and the LP condition is satisfied without any constraint on the prototype filter. By separable operations, the DTCMFB is extended to two-dimensions. The resulting 2D DTCMFB can provide much more flexible directional-selectivity. Finally, several simulations are given to verify the proposed DTCMFB, and the experiments on nonlinear approximation and image denoising are presented to demonstrate its potential in image processing.

Ellipse fitting is widely applied in the fields of computer vision and automatic industry control, in which the procedure of ellipse fitting often follows the preprocessing step of edge detection in the original image. Therefore, the ellipse fitting method also depends on the accuracy of edge detection besides their own performance, especially due to the introduced outliers and edge point errors from edge detection which will cause severe performance degradation. In this paper, we develop a robust ellipse fitting method to alleviate the influence of outliers. The proposed algorithm solves ellipse parameters by linearly combining a subset of ("more accurate") data points (formed from edge points) rather than all data points (which contain possible outliers). In addition, considering that squaring the fitting residuals can magnify the contributions of these extreme data points, our algorithm replaces it with the absolute residuals to reduce this influence. Moreover, the norm of data point errors is bounded, and the worst case performance optimization is formed to be robust against data point errors. The resulting mixed l1-l2 optimization problem is further derived as a second-order cone programming one and solved by the computationally efficient interior-point methods. Note that the fitting approach developed in this paper specifically deals with the overdetermined system, whereas the current sparse representation theory is only applied to underdetermined systems. Therefore, the proposed algorithm can be looked upon as an extended application and development of the sparse representation theory. Some simulated and experimental examples are presented to illustrate the effectiveness of the proposed ellipse fitting approach.