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Retinex theory deals with compensation for illumination effects in images, which has a number of applications including Retinex illusion, medical image intensity inhomogeneity and color image shadow effect etc.. Such ill-posed problem has been studied by researchers for decades. However, most exiting methods paid little attention to the noises contained in the images and lost effectiveness when the noises increase. The main aim of this paper is to present a general Retinex model to effectively and robustly restore images degenerated by both illusion and noises. We propose a novel variational model by incorporating appropriate regularization technique for the reflectance component and illumination component accordingly. Although the proposed model is non-convex, we prove the existence of the minimizers theoretically. Furthermore, we design a fast and efficient alternating minimization algorithm for the proposed model, where all subproblems have the closed-form solutions. Applications of the algorithm to various gray images and color images with noises of different distributions yield promising results.

The performance of image segmentation highly relies on the original inputting image. When the image is contaminated by some noises or blurs, we can not obtain the efficient segmentation result by using direct segmentation methods. In order to efficiently segment the contaminated image, this paper proposes a two step method based on the hybrid total variation model with a box constraint and the K-means clustering method. In the first step, the hybrid model is based on the weighted convex combination between the total variation functional and the high-order total variation as the regularization term to obtain the original clustering data. In order to deal with non-smooth regularization term, we solve this model by employing the alternating split Bregman method. Then, in the second step, the segmentation can be obtained by thresholding this clustering data into different phases, where the thresholds can be given by using the K-means clustering method. Numerical comparisons show that our proposed model can provide more efficient segmentation results dealing with the noise image and blurring image.The difficulty in studying the MS model is that it involves two unknowns: the intensity function and the set of edges. So the MS model is hard to implement in practice since the discretization of the unknown set of edges is a very complex task. In order to effectively solve the MS model, Chan and Vese (CV) [14] proposed an easily handle model by assuming that the segmentation result was a piecewise constant image with two different constant values. The CV model has achieved good performance in image segmentation task because of its ability to obtain a larger convergence range and handle topological changes naturally [41]. However, more work needs to be done on effective representation of regions and their boundaries for multi-phase segmentation. Vese and Chan extended the work in [40] to utilize multi-phase level set functionals to represent multiple regions. Similar to the two-phase case, the model is non-convex and thus the global minimization can not be guaranteed. In order to overcome these drawbacks, convex relaxation methods [12,4,5] and graph cut [6,2,16,19,21,48] method were proposed in this field.During some phases of obtaining a real image, we can only get a contaminated image due to the interference of some random noises and blurs. This interference leads not to obtain an expected segmentation results by using classical methods such as the MS model and clustering methods. So it is very important to suppress these contaminated information before segmenting the image [21,18,35,33]. Recently a two step method based on the MS model was proposed in [8]. The first step is through restoring the contaminated image to obtain a smooth solution by using the modified MS model under the space W 1,2 (Ω) for the image domain Ω ⊂ R 2 . Once the restored image is obtained, the second stage is to threshold it into different phases by using the K-means clustering method. However, the image space is not continuous in the region of ed...

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