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The bilateral filter represents a wide group of nonlinear filters for edge-preserving image smoothing. In this work, we study the convergence properties of the bilateral filter algorithm. The understanding is established that the bilateral filter is an optimization procedure. We demonstrate that the bilateral filter is equivalent to minimizing a robust cost criterion using iterative reweighting, which is a good approximation to the very fast but unstable Newton's method. Further, the results of the analysis allow us to derive an improved hybrid smoothing scheme with concerns of computational efficiency and edge preservation.