Joint filtering mainly uses an additional guidance image as a prior and transfers its structures to the target image in the filtering process. Different from existing approaches that rely on local linear models or hand-designed objective functions to extract the structural information from the guidance image, we propose a new joint filtering method based on a spatially variant linear representation model (SVLRM), where the target image is linearly represented by the guidance image. However, learning SVLRMs for vision tasks is a highly ill-posed problem. To estimate the spatially variant linear representation coefficients, we develop an effective approach based on a deep convolutional neural network (CNN). As such, the proposed deep CNN (constrained by the SVLRM) is able to model the structural information of both the guidance and input images. We show that the proposed approach can be effectively applied to a variety of applications, including depth/RGB image upsampling and restoration, flash deblurring, natural image denoising, and scale-aware filtering. In addition, we show that the linear representation model can be extended to high-order representation models (e.g., quadratic and cubic polynomial representations). Extensive experimental results demonstrate that the proposed method performs favorably against the state-of-the-art methods that have been specifically designed for each task.