The traditional space-invariant isotropic kernel utilized by a bilateral filter (BF) frequently leads to blurry edges and gradient reversal artifacts due to the existence of a large amount of outliers in the local averaging window. However, the efficient and accurate estimation of space-variant kernels which adapt to image structures, and the fast realization of the corresponding space-variant bilateral filtering are challenging problems. To address these problems, we present a space-variant BF (SVBF), and its linear time and error-bounded acceleration method. First, we accurately estimate spacevariant anisotropic kernels that vary with image structures in linear time through structure tensor and minimum spanning tree. Second, we perform SVBF in linear time using two error-bounded approximation methods, namely, low-rank tensor approximation via higher-order singular value decomposition and exponential sum approximation. Therefore, the proposed SVBF can efficiently achieve good edge-preserving results. We validate the advantages of the proposed filter in applications including: image denoising, image enhancement, and image focus editing. Experimental results demonstrate that our fast and error-bounded SVBF is superior to state-of-the-art methods.