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We present a fast convolution-based technique for computing an approximate,signed Euclidean distance function $S$ on a set of 2D and 3D grid locations.Instead of solving the non-linear, static Hamilton-Jacobi equation ($\|\nablaS\|=1$), our solution stems from first solving for a scalar field $\phi$ in alinear differential equation and then deriving the solution for $S$ by takingthe negative logarithm. In other words, when $S$ and $\phi$ are related by$\phi = \exp \left(-\frac{S}{\tau} \right)$ and $\phi$ satisfies a specificlinear differential equation corresponding to the extremum of a variationalproblem, we obtain the approximate Euclidean distance function $S = -\tau\log(\phi)$ which converges to the true solution in the limit as $\tau\rightarrow 0$. This is in sharp contrast to techniques like the fast marchingand fast sweeping methods which directly solve the Hamilton-Jacobi equation bythe Godunov upwind discretization scheme. Our linear formulation results in aclosed-form solution to the approximate Euclidean distance function expressibleas a discrete convolution, and hence efficiently computable using the fastFourier transform (FFT). Our solution also circumvents the need for spatialdiscretization of the derivative operator. As $\tau\rightarrow0$ we show theconvergence of our results to the true solution and also bound the error for agiven value of $\tau$. The differentiability of our solution allows us tocompute---using a set of convolutions---the first and second derivatives of theapproximate distance function. In order to determine the sign of the distancefunction (defined to be positive inside a closed region and negative outside),we compute the winding number in 2D and the topological degree in 3D, whosecomputations can also be performed via fast convolutions. We demonstrate theefficacy of our method through a set of experimental results.

A new variational principle for the Euclidean distance function: Linear approach to the non-linear eikonal problem