A closed-form solution for natural image matting is presented, which allows for the extraction of a foreground object from an image using minimal user input. The method derives a cost function based on local smoothness assumptions on foreground and background colors, enabling the analytical elimination of these colors to obtain a quadratic cost function in the alpha matte. This allows for the globally optimal alpha matte to be found by solving a sparse linear system. The closed-form solution also enables the prediction of solution properties by analyzing the eigenvectors of a sparse matrix related to spectral image segmentation algorithms. The method is shown to produce high-quality mattes from a small amount of user input, such as scribbles, and is effective on natural images with mixed pixels or complex foreground/background boundaries. The approach is compared to existing methods, including Bayesian matting, Poisson matting, and random walk matting, and is shown to produce results comparable to or better than these methods in terms of visual quality. The method is also effective in handling ambiguous color situations and can be used for tasks such as shadow and smoke matting. The algorithm is implemented using a sparse linear system solver and is shown to be efficient and effective on both small and large images. The method is also shown to be robust to noise and can be used for tasks such as foreground and background reconstruction. The results demonstrate that the closed-form solution provides a solid theoretical foundation for natural image matting and is effective in a wide range of applications.A closed-form solution for natural image matting is presented, which allows for the extraction of a foreground object from an image using minimal user input. The method derives a cost function based on local smoothness assumptions on foreground and background colors, enabling the analytical elimination of these colors to obtain a quadratic cost function in the alpha matte. This allows for the globally optimal alpha matte to be found by solving a sparse linear system. The closed-form solution also enables the prediction of solution properties by analyzing the eigenvectors of a sparse matrix related to spectral image segmentation algorithms. The method is shown to produce high-quality mattes from a small amount of user input, such as scribbles, and is effective on natural images with mixed pixels or complex foreground/background boundaries. The approach is compared to existing methods, including Bayesian matting, Poisson matting, and random walk matting, and is shown to produce results comparable to or better than these methods in terms of visual quality. The method is also effective in handling ambiguous color situations and can be used for tasks such as shadow and smoke matting. The algorithm is implemented using a sparse linear system solver and is shown to be efficient and effective on both small and large images. The method is also shown to be robust to noise and can be used for tasks such as foreground and background reconstruction. The results demonstrate that the closed-form solution provides a solid theoretical foundation for natural image matting and is effective in a wide range of applications.