Herein, we propose a mathematical model for road crack segmentation in images, focusing on the difficulties of the real world road conditions, such as the lighting and color changes, complex crack shape etc. The proposed model belongs to the family of nonlinear partial differential equations (PDEs), involving edge-aware anisotropic diffusion, curvature-driven contour evolution, high order biharmonic regularization, and feature-driven attraction force for capturing the crack regions. A theoretical analysis is conducted to show the well-posedness of the model. In addition, a physics-informed neural network (PINN) version of the model is presented which allows us to discretize the PDEs in a mesh-free fashion and to approximate high order derivatives through the deep neural networks. Various numerical experiments on EdmCrack600 data are implemented for validating the proposed method. All the experimental results show that the proposed model is superior to the other segmentation models, and that our model achieves excellent performance in terms of the metrics, i.e., dice similarity, intersection over union, sensitivity, and specificity.

Image denoising remains a fundamental challenge in image processing, particularly when dealing with additive white gaussian noise (AWGN) that degrades visual quality and information content. This paper introduces a novel multi-stage denoising framework that uniquely combines Contourlet transform, radial basis function neural networks (RBFNN), and kalman filtering to effectively preserve important image features while removing noise. The contourlet transform first decomposes images into multi-resolution, directional subbands, providing a sparse representation that better captures geometric structures compared to traditional wavelet approaches. We then employ an RBFNN trained through back-propagation to adaptively threshold the contourlet coefficients based on local image characteristics and noise levels. Finally, kalman filtering is applied as a post-processing step to further suppress residual noise artifacts. Comprehensive experiments conducted on standard benchmark datasets demonstrate that our approach outperforms several state-of-the-art methods, including BM3D and recent deep learning-based techniques, particularly at moderate to high noise levels (σ ≥ 15). Quantitative evaluations show our method achieves superior PSNR improvements of up to 2.4dB and SSIM improvements of 0.12 compared to recent competing approaches, while qualitative results confirm better preservation of edges and textural details. The proposed framework offers an effective balance between computational efficiency and denoising performance, making it suitable for various practical applications.