This paper proposes novel methods for detecting and separating smoke from a single image frame. Specifically, an image formation model is derived based on the atmospheric scattering models. The separation of a frame into quasi-smoke and quasi-background components is formulated as convex optimization that solves a sparse representation problem using dual dictionaries for the smoke and background components, respectively. A novel feature is constructed as a concatenation of the respective sparse coefficients for detection. In addition, a method based on the concept of image matting is developed to separate the true smoke and background components from the smoke detection results. Extensive experiments on detection were conducted and the results showed that the proposed feature significantly outperforms existing features for smoke detection. In particular, the proposed method is able to differentiate smoke from other challenging objects (e.g. fog/haze, cloud, and so on) with similar visual appearance in a gray-scale frame. Experiments on smoke separation also demonstrated that the proposed separation method can effectively estimate/separate the true smoke and background components.