  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.
