Kolmogorov's structure function (KSF) is used in the algorithmic theory of complexity for describing the structure of a string by use of models (programs) of increasing complexity. Recently, inspired by the structure function, an extension of the minimum description length theory was introduced for achieving a decomposition of the
total description of the data into a noise part and a model part, where the models are parametric distributions instead of programs, the code length necessary for the model part being restricted by a parameter. In this way a new "rate-distortion" type of curve is obtained, which may be further used as a general model of the data,
quantifying the amount of noise left "unexplained" by models of increasing complexity. In this paper we present a complexity-noise function for a class of hierarchical image models in the wavelet
transform domain, in the spirit of the Kolmogorov structure function. The minimization of the model description can be shown to have a form similar to one resulting from the minimization in the rate-distortion sense, and thus it will be achieved as in lossy image compression. As an application of the complexity-noise function introduced we study the image denoising problem and analyze the conditions under which the best reconstruction along the complexity-noise function is obtained.