and w u2032 represents the remaining wavelet coefficients the algorithm seeks adjusted wavelet coefficients that solve equation 3 has solutions if the rank of the rectangular matrix b is equal to the number of remaining non canceled wavelet coefficients the rank condition however implies that the number of canceled wavelet coefficients must be smaller than the number of masked pixels given a mask and a wavelet decomposition canceled wavelet coefficients called the u201c masked coefficients u201d must be chosen the choice of the masked coefficients impacts the resulting file size canceling a wavelet whose energy is significantly located outside the mask requires a lot of adjustments on the remaining coefficients these adjustments are likely to use coefficients that would be null otherwise empirically good results are achieved by canceling wavelet coefficients when at least half of the wavelets energy is located below the mask once the set of masked coefficients is selected equation 3 may be solved there are many techniques for solving sparse linear systems there is however a method which relies only on the efficient wavelet decomposition and reconstruction algorithms every image can be represented in pixel coordinates i e a collection of pixel values or in wavelet coordinates i e a collection of wavelet coefficients the coordinate transformation is described by matrix a the solutions belong to the intersection of the following sets of images the set p of all images whose pixels located outside the mask are equal to the corresponding pixels in the image being compressed this set is a closed convex affine subspace of the image space the set q of all images whose wavelet representation contains zeroes for all masked coefficients this set also is a closed convex affine subspace of the image space let p respectively q be the projection operator on.