## 1 Introduction  
Image processing has become more and more important in many fields, such as machine vision, geological prospecting, medical imaging, defect detection, and so on. In the past decades, many methods have been presented based on different views, and nonlinear diffusion is one of the most important techniques for the solid mathematical physics foundation [1; 2; 3]. Different from the uniform diffusion (holds the same velocity along any direction), nonlinear diffusion can be applied to describe lots of complex natural phenomenons. Nonlinear diffusion models are often driven by partial differential equations (PDEs) which can be derived by minimizing an energy functional and computing the Euler-Lagrange equation [4; 5]. [PERSON] and [PERSON] (PM) proposed a nonlinear diffusion model with a fixed edge stop velocity function related to the local gradient mode [6]. It can be applied to achieve some denoising tasks with homogeneous features and noise; however, it is difficult to efficiently govern the diffusion process for some more complex cases. [PERSON], [PERSON] and [PERSON] (ROF) proposed the classical total variation model to recover noisy images by minimizing a geometric energy functional [1]. The diffusion can be performed only along the edge to preserve the image features and remove the noise. Though these traditional methods have been reported to work well in many cases, they are still found to show some unfavorable phenomena, such as the staircase effect, loss of texture features, and so on [7; 8]. Especially in many industrial scenes, the lighting condition is limited, inhomogeneous and easy to be disturbed so that the observed images are often low-contrast, have an inhomogeneous background and are serious polluted. It is often not a good idea to directly apply the traditional models to these degraded images for defect detection since the mentioned unfavorable effects often become worse in such situations. Defects detection is often implemented on the images captured by some vision devices, which are effected by some objective factors, such as noise level, brightness, background gray distribution, size and shape of the defects. Lots of efforts have been made to present novel models or algorithms based on different starting points and proper assumptions. [PERSON] [9] presented a machine vision scheme for the inspection of brightness unevenness in LCD panel surfaces. An edge detection algorithm and a genetic algorithm were used to identify discontinuous points and extract the boundary of anomalous brightness region, respectively. [PERSON] et al. [10] presented an adaptive multiple-level threshold method based on the statistical characteristics of the local area to segment spot-type defects from the background surface. [PERSON] revised the Otsu method for selecting optimal threshold values for both unimodal and bimodal distributions and tested the performance on common defect detection applications [11]. [PERSON] proposed two machine learning methods to automatically detect the wheel defects, which automatically learn different types of wheel defects and predict during normal operation if a wheel has a defect or not [12]. In recent years, with different views or assumptions, several improved nonlinear diffusion models were presented for different tasks about image processing and defect detection. [PERSON] and [PERSON] took a non-negative decreasing function with an annealing gradient threshold as the diffusion coefficient, and then proposed a novel anisotropic diffusion model (TCAD) to detect subtle defects embedded in inhomogeneous textures [13]. [PERSON] and [PERSON] introduced a weighted diffusion function and a generalized diffusion coefficient function respectively to the classical PM model for defect detection in low-contrast surface images [14; 15]. These two models both carry out a smoothing process for faultless areas and perform a sharpening process for defect areas. Based on the directional Laplacian, [PERSON] et al. proposed a modified version (MPM) of the original Perona-Malik model [16].