插值Bendlet框架下图像修复变分模型及快速算法

The microscopic slice image of the interaction tissue between the locusts and the biological pesticide plays an important role in the mechanism analysis of the biological pesticide, and it is also the necessary components of the 3D image model. The slice images possess complicated textures, blurry edges and multi-objects which are often covered with each other. In most cases, segmenting the specified object in the image precisely is necessary in the image analysis. To the specified object with damaged regions, it is very difficult to be segmented and restored by the common methods as the interference between the damaged region and the object contour. Aim at those problems above, the interpolative Bendlet transform theory is proposed in this project. Based on the Bendlet frame, the multi-scale sparse representation method and the variational model on the image processing are to be constructed, which includes 4 sections as follows. (1) Design the parametric interpolative Bendlet function and construct the corresponding adaptive bendlet dictionary. It would meet to the sparse representation requirement of the slice images with various textures and different issues. (2) Construct the target-controllable image segmentation variational model based on the parametric bendlet sparse representation to the images without damaged region. (3) Construct the coupled model of the target-controllable image segmentation and the image inpainting combining with the image exemplar-based inpainting method. The keys of the model are the uniqueness of the solution and the stability of the corresponding numerical algorithm. (4) Combining with the variational iteration method and the multi-scale interpolation bendlet theory, construct the fast numerical algorithm to the variational models.

蝗虫与生物农药互作组织序列切片显微图像对分析生物农药作用机理和三维重建具有非常重要的作用。切片图像具有纹理复杂、多目标彼此嵌套覆盖且边缘模糊等特点。为满足此类图像修复和分层次多目标分割的需要，提出插值Bendlet变换的概念，进而构造插值Bendlet框架下的图像多尺度稀疏表示方法及图像处理变分模型。具体研究内容包括：（1）构建插值Bendlet变换理论及算法,结合稀疏表示理论，设计自适应插值Bendlet字典，以适应不同纹理、不同组织图像的稀疏表达需要；（2）对目标物无破损图像，构造基于参数化Bendlet稀疏表示的目标可控分割变分模型；（3）对于目标物有破损的图像，结合图像修复变分方法，构造多目标同伦分割及修复的耦合多尺度变分模型；（4）结合变分迭代法、多尺度插值Bendlet理论构造变分模型的快速求解方法。

蝗虫与生物农药互作组织序列切片显微图像对分析生物农药作用机理和三维重建具有非常重要的作用。切片图像具有纹理复杂、多目标彼此嵌套覆盖且边缘模糊等特点。为满足此类图像修复和分层次多目标分割的需要，在多尺度小波插值算子的基础上，通过引入多方向小波系数和特征点拓扑关系分析，构造了二阶插值Shearlet（Bendlet）的概念，从而实现了图像多尺度稀疏表示，进而应用于图像处理变分模型的求解中。具体研究内容和重要结果包括：（1）构建插值Bendlet变换理论及算法,结合稀疏表示理论，设计自适应插值Bendlet字典，以适应不同纹理、不同组织图像的稀疏表达需要；（2）对目标物无破损图像，构造了两种目标可控图像分割方法。一种基于参数化Bendlet稀疏表示变分模型；一种是基于基于深度学习的分组拼接的PraNet目标可控分割。（3）对于目标物有破损的图像，结合流形理论和图像修复变分方法，构造了多目标同伦分割及修复的耦合多尺度变分模型并给出了对应的快速求解方法。

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