have ∂T ( φ ) = T ( dφ ) for every differential form φ ∈ D $^{d}$. Note that when T represents a smooth oriented manifold with boundary, this corresponds to the usual definition of boundary. We restrict our attention to integral currents T that are rectifiable currents with a rectifiable boundary (i.e., T ∈ R m and ∂T ∈ R m − $_{1}$). The flat norm of a d - dimensional current T is given by where E$_{d}$ is the set of d -dimensional currents with compact support. One also uses flat norm to measure the “distance” between two d -currents. More precisely, the flat norm distance between two d -currents T and P is given by Morgan and Vixie [34] showed that the L 1 total variation functional ( L $^{1}$TV) introduced by Chan and Esedo¯ glu [9] computes the flat norm for boundaries T with integer mul- tiplicity. Given this correspondence, and the use of scale in L $^{1}$TV, Morgan and Vixie defined [34] the flat norm with scale λ ∈ [0 , ∞ ) of an oriented d -dimensional set T as where S varies over oriented ( d +1)-dimensional sets, and V$_{d}$ is the d -dimensional volume, used in place of mass. Figure 2.1 illustrates this definition. Flat norm of the 1D current T is given by the sum of the length of the resulting oriented curve T − ∂S (shown separated from the input curve for clarity) and the area of the 2D patch S shown in red. Large values of λ , above the curvature of both humps in the curve T , preserve both humps. Values of λ between the two curvatures eliminate the hump on the right. Even smaller values “smooth out” both humps as illustrated here, giving a more “flat” curve, 12 F$_{λ}$ ( T ) ≡ min S { V$_{d}$( T − ∂S ) + λ V$_{d}$$_{+1}$( S ) } , (2.3) F ( T, P ) = inf { M( Q ) + M( R ) ∣ $_{∣}$T − P = Q + ∂R, Q ∈ E$_{d}$ , R ∈ E$_{d}$$_{+1}$ } . (2.2) F ( T ) = min S { M( T − ∂S ) + M( S ) ∣ T − ∂S ∈ E$_{d}$ , S ∈ E$_{d}$$_{+1}$ } , (2.1)