as S can now be comprised of much bigger 2D patches. Figure 2.1: 1D current T , and flat norm decomposition T − ∂S at appropriate scale λ . The resulting current is shown slightly separated from the input current for clearer visualization. Figure 2.1 illustrates the utility of flat norm for deblurring or smoothing applications, e.g., in 3D terrain maps or 3D image denoising. But efficient methods for computing flat norm are known only for certain types of currents in two dimensions. For d = 1, Under the setting where T is a boundary, i.e., a loop, embedded in R 2 and the minimizing surface S ∈ R 2 as well, the flat norm could be calculated efficiently, for instance, using graph cut methods [28] – see the work of Goldfarb and Yin [22] and Vixie et al. [45], and references therein. Motivated by applications in image analysis, these approaches usually worked with a grid representation of the underlying space ( R $^{2}$). Pixels in the image readily provide such a representation. While it is computationally convenient that L $^{1}$TV minimizers give us the scaled flat norm for the input images, this approach restricts us to currents that are boundaries of codimension 1. Correspondingly, the calculation of flat norm for 1-boundaries embed- ded in higher dimensional spaces, e.g., R $^{3}$, or for input curves that are not necessarily boundaries has not received much attention so far. Similarly, flat norm calculations for higher dimensional input sets have also not been well-studied. Such situations often ap- pear in practice – for instance, consider the case of an input set T that is a curve sitting on a manifold embedded in R $^{3}$, with choices for S restricted to this manifold as well. 13 S T T − ∂S