Imaging model at different resolutions

Multiresolution analysis is being extensively used in image processing and computer vision problems. We derive the image formation model at a coarse resolution given the image model at a fine resolution, in particular we give an expression to calculate the blur matrix at a coarse resolution given the blur matrix at fine resolution, In particular, we consider the image model Y^Ω=ℋ^ΩX^Ω+W ^Ω at resolution Ω and show that the image model at a coarser resolution, Ω-1, retains the same structure and can be shown to be Y^Ω-1=ℋ^Ω-1X^Ω-1+W ^Ω-1