Bilateral filter: Graph spectral interpretation and extensions

In this paper we study the bilateral filter proposed by Tomasi and Manduchi and show that it can be viewed as a spectral domain transform defined on a weighted graph. The nodes of this graph represent the pixels in the image and a graph signal defined on the nodes represents the intensity values. Edge weights in the graph correspond to the bilateral filter coefficients and hence are data adaptive. The graph spectrum is defined in terms of the eigenvalues and eigenvectors of the graph Laplacian matrix. We use this spectral interpretation to generalize the bilateral filter and propose new spectral designs of “bilateral-like” filters. We show that these spectral filters can be implemented with k-iterative bilateral filtering operations and do not require expensive diagonalization of the Laplacian matrix.

1. Introduction

The bilateral filter (BF) proposed by Tomasi and Manduchi [1] has emerged as a powerful tool for image processing. Bilateral filtering smooths images while preserving edges, by taking the weighted average of the nearby pixels. The weights depend on both the spatial distance and photometric distance which provides local adaptivity to the given data. The bilateral filter and its variants are widely used in different applications such as denoising, edge preserving multi-scale decomposition, detail enhancement or reduction and segmentation etc. [2]–[6]. Bilateral filtering was initially developed as an intuitive tool without theoretical justification. Since then, connections between the BF and other well known filtering frameworks such as anisotropic diffusion, weighted least squares, Bayesian methods, kernel regression and non-local means have been explored [7]–[11].