Multiresolution and Astronomical Image Processing 1. the Discrete Wavelet Transform 1.1. Mallat's Transform

We present several wavelet transform algorithms and their applications in astronomical image processing (restoration, object detection , compression, etc.). Extensive literature exists on the wavelet transform and its application (Chui 1992; Daubechies 1992; Meyer 1989). A discrete wavelet transform approach can be obtained from multiresolution analysis (Mallat 1989). Multiresolution analysis results from the embedded subsets generated by interpolations at different scales. A function f(x) is projected at each step j onto the subset V j. This projection is deened by the scalar product c j (k) of f(x) with the scaling function (x) which is dilated and translated: c j (k) =< f(x); 2 ?j (2 ?j x ? k) > : (1) (x) is a scaling function which has the property 1 2 x 2 = X n h(n)(x ? n): (2) Equation 2 permits the set c j+1 (k) to be computed directly from c j (k). If we start from the set c 0 (k), we compute all the sets c j (k), with j > 0, without directly computing any other scalar product: c j+1 (k) = X n h(n ? 2k)c j (n):