Gaussian Lifting for Fast Bilateral and Nonlocal Means Filtering

Recently, many fast implementations of the bilateral and the nonlocal filters were proposed based on lattice and vector quantization, e.g. clustering, in higher dimensions. However, these approaches can still be inefficient owing to the complexities in the resampling process or in filtering the high-dimensional resampled signal. In contrast, simply scalar resampling the high-dimensional signal after decorrelation presents the opportunity to filter signals using multi-rate signal processing techniques. This work proposes the Gaussian lifting framework for efficient and accurate bilateral and nonlocal means filtering, appealing to the similarities between separable wavelet transforms and Gaussian pyramids. Accurately implementing the filter is important not only for image processing applications, but also for a number of recently proposed bilateral-regularized inverse problems, where the accuracy of the solutions depends ultimately on accurate filter implementations. We show that our Gaussian lifting approach filters images more accurately and efficiently across many filter scales. Adaptive lifting schemes for bilateral and nonlocal means filtering are also explored.