The traditional space-invariant isotropic kel 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 kels 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 space-variant anisotropic kels 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.