A Method for Enforcing Integrability in Shape from Shading Algorithms
IEEE Transactions on Pattern Analysis and Machine Intelligence
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In the past, several methods based on iterative solution of pressure-Poisson equation have been developed for measurement of pressure from phase-contrast magnetic resonance (PC-MR) data. In this paper, a non-iterative harmonics-based orthogonal projection method is discussed which can keep the pressures measured based on the Navier-Stokes equation independent of the path of integration. The gradient of pressure calculated with Navier-Stokes equation is expanded with a series of orthogonal basis functions, and is subsequently projected onto an integrable subspace. Before the projection step however, a scheme is devised to eliminate the discontinuity at the vessel boundaries. The approach was applied to velocities obtained from computational fluid dynamics (CFD) simulations of stenotic flow and compared with pressures independently obtained by CFD. Additionally, MR velocity data measured in in-vitro phantom models with different degree of stenoses and different flow rates were used to test the algorithm and results were compared with CFD simulations. The pressure results obtained from the new method were also compared with pressures calculated by an iterative solution to the pressure-Poisson equation. Experiments have shown that the proposed approach is faster and is less sensitive to noise.