Data processing has to deal with many practical difficulties. Data is often corrupted by artifacts or noise and acquiring data can be expensive and difficult. Thus, the given data is often incomplete and inaccurate. To overcome these problems, it is often assumed that the data is sparse or low-dimensional in some domain. When multiple measurements are taken, this sparsity often appears in a structured manner. We propose a new model that assumes the data only contains a few relevant objects, i.e., it is sparse in some object domain. We model an object as a structure that can only change slightly in form and continuously in position over different measurements. This can be modeled by a matrix with highly correlated columns and a column shift operator that we introduce in this work. We present an efficient algorithm to solve the object reconstruction problem based on a K-approximation graph. We prove optimal approximation bounds and perform a numerical evaluation of the method. Examples from applications including Geophysics, video processing, and others will be given.

翻译：数据处理必须处理许多实际困难。 数据往往被文物或噪音腐蚀,而且获取数据可能费用昂贵和困难。 因此, 给的数据往往是不完整和不准确的。 为了克服这些问题, 人们常常假设数据在某些领域是稀少或低维的。 当进行多重测量时, 这种宽度通常会以结构化的方式出现。 我们提议一个新的模型, 假设数据只包含一些相关对象, 即它在某些对象域中是稀疏的。 我们把一个对象模型作为结构, 其形式只能稍有变化, 并且在不同测量中处于位置。 可以通过一个与我们在此工作中引入的高度关联的列和列转移操作器的矩阵进行模型。 我们提出一种有效的算法, 以解决基于 K- 组合图的物体重建问题。 我们证明最理想的近似边框, 并对方法进行数字评价。 将给出一些应用的例子, 包括 Geophiccs、 视频处理和其他应用的例子 。