基于物理总能量目标函数的稀疏重建模型

欠采样条件下的稀疏重建模型往往直接取稀疏约束项或保真项作为求解的目标函数, 却从未阐述其中的物理演化规律. 针对此问题, 从物理运动的角度出发, 提出了一种基于物理总能量目标函数的稀疏重建模型. 首先, 建立了微粒在黏性介质中的运动模型, 模型中粒子的重力势能函数为松弛变换后的l₂-l₁ 范数; 其次, 基于该微粒的物理总能量建立了新的稀疏重建模型, 该重建模型在保留l₂-l₁ 模型稀疏约束和保真项的基础上, 增加了对相邻两次迭代结果偏差的约束, 避免因该偏差过大引起的震荡; 第三, 提出了针对该模型的梯度投影算法, 并证明了算法的收敛性, 算法在新模型目标函数下的梯度方向总是包含上一步迭代的物理惯性, 从而达到加速收敛和避免局部最优解的目的; 最后, 将该模型应用于标准灰度图像的稀疏重建以及精密电子组装中微焦点X射线缺陷检测. 实验结果表明, 该算法不仅保证了图像的重建质量, 收敛速度还得到了大幅提升. 在精密电子组装内部缺陷检测应用中, 该算法在微焦点X射线图像的边缘细节保留方面有明显的优势, 可更准确地识别缺陷, 满足工业应用快速性和准确性要求.

Physical total energy based objective function model for sparse reconstruction

Image reconstruction from sparse data is one of the key technologies in physical imaging, and it can often be mathematically described as an underdetermined linear inverse problem. Mathematical models for sparse reconstruction often choose the sparseness constraints or data fidelity term directly as objective functions. However, physics concepts and laws for these modeling or solving processes have never been explored. In this paper, sparse reconstruction is investigated for the first time from the perspective of physical motion. Firstly, a physical model is created to describe a particle motion in viscous medium, in which the particle gravity potential energy function is the norm of l₂-l₁ after the relaxation transformation. In discrete calculations, the particle displacement is determined by the corresponding iterative result, and its velocity can be described as the change between two adjacent iterations. Then, a new mathematical model based on the physical motion model is studied for sparse reconstruction, in which the total energy of particle is chosen as a new objective function and nonnegative displacements as constraints. This new model preserves sparse constrains and fidelity term of original l₂-l₁ model, and adds the constrains of deviations between two adjacent iterations so as to avoid oscillations caused by large deviations. Furthermore, a targeted gradient projection technique is adopted to solve such a reconstruction model, and its convergence is discussed as well. Especially in this algorithm, the gradient of this new objective function contains the iterative step of previous iteration, and such iterative steps play the role of physical inertia property in iterative process, which can effectively enlarge the iterative steps to accelerate the convergence and avoid local optima. Finally, two sets of experimental results are presented, including natural grayscale image reconstruction and micro focus X-ray defect detection in precision electronic package. The results demonstrate that the proposed method outperforms its competitors distinctly in time efficiency on the basis of guaranteeing the reconstruction quality. Additionally, on detecting internal defects in solder joint of integrated circuit, the proposed method is well performed in retaining edge details of the reconstructed micro focus X-ray images. Therefore, the proposed method can identify the solder joint internal defects more accurately and is more suitable to rapid and precise micro focus X-ray defect detection in industry.