###### Abstract
However, the optimization problem with respect to this model is an L0-norm minimization problem, i.e., a non-convex optimization problem, which is NP-hard and very difficult to solve. To address this issue, a first-order primal-dual algorithm-based sparse time-frequency analysis method (STFA-PD) was developed. Firstly, the non-convex optimization problem of L0-norm was relaxed to an L1-norm minimization problem using a convex relaxation technique. Secondly, the first-order primal-dual algorithm was employed to solve the problem [41; 42]. Several time-frequency images (TFIs) reflecting the bearing condition can be obtained by conducting the time-frequency analysis of vibration signals. Currently, there are two types of bearing condition recognition methods based on TFIs [43; 44]. The first one is an artificial recognition method performed by skilled experts. This method relies heavily on individual experience and skill level and has low accuracy and efficiency for bearing condition recognition. The other is an automatic classification using computers, which is attracting increasing attention. Although TFIs can effectively characterize the local time-frequency properties of rolling bearing vibration signals, they cannot be directly used as features for an automatic classification, because the data dimensions of TFIs are too numerous for such a classification. Therefore, image dimensionality reduction and feature extraction need to be conducted before classification. Some algebraic feature extraction methods, such as two-direction two-dimensional principal component analysis (TD-2 DPCA) [45], two-direction two-dimensional linear discriminant analysis (TD-2 DLDA) [46], and two-dimensional non-negative matrix factorization (2 DNMF) [43], were proposed for the dimensionality reduction of TFIs. The low-dimensional features obtained through dimensionality reduction are directly used for automatic classification. However, as the horizontal distribution of the frequency properties in TFIs changes with time, the features will be inevitably affected by the influence of the time-domain truncation. As a result, even the TFIs with the same fault severity may have completely different features. Bearing vibration signals of different states can be used to express different characteristics in the time-frequency domain, resulting in TFIs with different textures. Additionally, the texture features can be expressed as global statistics of local features, which has little to do with the changes in the horizontal distribution with time. Therefore, texture-feature-based image classification can be used to distinguish the fault severity degrees of bearings. The most widely used texture descriptor for image recognition is the gray level co-occurrence matrix (GLCM), proposed by [PERSON] et al. [47] in 1973, as it can accurately describe the roughness, repetition direction, and complexity of the texture of an image. GLCM is a square matrix. GLCMs with different parameters can be easily obtained from a single image, while many statistical features can be easily obtained from each GLCM. Compared to GLCM, the statistical features are more convenient for classification. Consequently, GLCM-based texture features, i.e., statistical features, can be extracted from TFIs to obtain fault severity information. A new approach for the fault severity monitoring of rolling bearings based on the texture feature extraction of sparse TFIs is proposed in this paper. The novelty is that a new time-frequency analysis method, named STFA-PD, was developed for analyzing the bearing vibration signal. A high-qualityTFI without the cross-term interference, which we call sparse TFI, can be obtained from the vibration signal using the STFA-PD method. Thereafter, the GLCM can be used to extract the texture features from sparse TFTs as the input features for classifiers. Finally, the fault severity recognition can be realized. Figure 1 shows a flowchart of the proposed bearing fault severity monitoring approach.