Shape Tube Metric, Geodesic Equation

The Courant metric in shape analysis (16) is extended here to classes of non smooth subsets in D. The intrinsic tube analysis which is evoked here is developped in (25), (24). The characteristic function of Q is ζ ∈ L∞(I ×D) verifying ζ = ζ2 and ζ(t) = χΩt where the measurable set Ωt is defined in D up to a zero measure subset. That theory can be extended to boundaries with the approach of (2). In the second part we adopt the eulerian modeling (5; 16; 8; 11) which has been extended to non smooth vector fields in (17; 25; 24; 8)... Making use of the transverse field approach (8; 4; 18) we derive the euler equation for the geodesic-tube which has been presented in several image anlysis conferences (”Shape Space” IMA , march 06, MIA06 Paris, Obergurgl ...) with application developed with L. Blanchard (26). The technic is inspired from (7). Following (17), (23), we consider tubes which are continuous with respect to the the L1(D) topology and with time integrable perimeter, then we introduce the set of characteristic functions