一类四阶偏微分方程图像处理问题研究

发布时间：2018-10-04 21:07

【摘要】：偏微分方程(Partial Diffusion Equation,PDE)去噪模型有很好的去噪效果。PDE去噪模型主要分为变分PDE方法和扩散PDE方法两大类。但是二阶PDE是通过分段平面来逼近原图的,所以二阶去噪模型普遍存在“阶梯”效应,这就在原本平坦的区域产生了虚假的边界。而四阶方程是通过分段斜面来逼近原图的,其能够从本质上避免“阶梯”效应。但是现有的四阶去噪模型并不能保留图像的边缘。对于PDE图像分割,传统的水平集方法中,自然延拓并不能保证嵌入函数在演化过程中始终保持为带符号的距离函数,所以需要在演化过程中进行重新初始化,这对分割的效率和准确性都产生了影响。针对上述问题本文做了如下工作:本文通过对比经典Y-K模型与LLT模型的优缺点,转换思路,从扩散PDE的角度出发,针对四阶扩散方程的性质,直接构造一个新的四阶PDE。在上述思路下,本文提出一类基于图像特征的四阶PDE去噪模型。在LLT模型框架下,引入扩散系数,使得方程能够根据图像的特征自适应判断扩散的程度。本文将边缘检测能力更强、受噪声影响更小的一阶的梯度算子作为扩散系数。将所提出的四阶项应用到变分水平集分割模型中得到一种四阶项正则化水平集方法。直接从嵌入函数的演化方程出发,通过耦合四阶项达到避免重新初始化的目的。对于模型的数值实现,利用差分法对模型进行离散,提出了中心差分显格式,但是由于显格式对步长有很大的限制,严重影响了计算效率。所以设计了一种半隐格式,并引入了加法算子分裂算法(Addition Operator Splitting Algorithm,AOS),可以增加步长来在效率和精度上取得折中。最后,在数值实验部分,分别对光滑图像加入不同强度的高斯白噪声,应用提出的去噪模型进行去噪。并将本文的实验结果与其他模型进行对比。本文还将四阶项正则化水平集方法与经典的距离正则化水平集方法进行了对比,发现本文模型允许更大范围的初始值、并且避免了测地线活动轮廓模型分割结果过度光滑的缺点。
[Abstract]:PDE (partial differential equation (Partial Diffusion Equation,PDE) denoising model has good denoising effect. PDE denoising model can be divided into two categories: variational PDE method and diffusion PDE method. However, the second-order PDE approximates the original graph by piecewise plane, so the second-order denoising model has a "ladder" effect, which produces false boundaries in the original flat region. The fourth-order equation approximates the original graph by a piecewise oblique plane, which can essentially avoid the "ladder" effect. But the existing four-order denoising model can not preserve the edge of the image. For PDE image segmentation, in the traditional level set method, natural continuation can not guarantee that the embedded function is always a signed distance function in the evolution process, so it needs to be reinitialized in the evolution process. This has an impact on the efficiency and accuracy of segmentation. The work of this paper is as follows: by comparing the advantages and disadvantages between the classical Y-K model and the LLT model, and transforming ideas, a new fourth-order PDE. is constructed directly from the point of view of diffusive PDE and according to the properties of the fourth-order diffusion equation. Based on the above ideas, this paper presents a fourth order PDE denoising model based on image features. In the framework of LLT model, the diffusion coefficient is introduced, so that the equation can adaptively judge the degree of diffusion according to the characteristics of the image. In this paper, the first-order gradient operator with stronger edge detection ability and less noise effect is used as diffusion coefficient. The fourth order term is applied to the variational level set segmentation model to obtain a fourth order regularization level set method. Starting directly from the evolution equation of the embedded function, the fourth order term is coupled to avoid reinitialization. For the numerical realization of the model, the difference method is used to discretize the model, and a central difference explicit scheme is proposed. However, the explicit scheme has a great limitation on the step size, which seriously affects the calculation efficiency. So we design a semi-implicit scheme and introduce the addition operator splitting algorithm (Addition Operator Splitting Algorithm,AOS), which can increase the step size to achieve a compromise in efficiency and precision. Finally, in the part of numerical experiment, the white noise of Gao Si with different intensity is added to the smooth image, and the proposed denoising model is applied to de-noising. The experimental results are compared with other models. The fourth order regularization level set method is also compared with the classical distance regularization level set method. It is found that the model in this paper allows a larger range of initial values. It also avoids the disadvantage that the segmentation result of geodesic active contour model is too smooth.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP391.41;O175.2

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