In some imaging modalities based on coherent radiation, the noise contaminating an image may contain useful information, thereby necessitating the separation of the noise field rather than just denoising. When the algebraic operation that relates the image and noise is known, the noise component can be estimated in a straightforward manner after denoising. However, for some statistical models such as Poisson noise, this algebraic relation is not known. In this paper, we propose a method for simultaneously estimating the image and separating the noise field, when we do not know the algebraic relation between them. It is assumed that the image is sparse and the noise field is not, and appropriate regularizers are used on them. We use a polynomial representation to relate the image and noise with the observed image, and iteratively estimate the polynomial coefficients, the image, and noise component. Experimental results show that the method correctly estimates the model coefficients and the estimated noise components follow their respective statistical distributions.