Aim

Gain experience of designing systems for image approximation using signal transforms

Task

You are designing an image compression system as a component of a new photography transmission system for satellite imaging. Your task is to design the image compression algorithm using the wavelet transform. Write a detailed individual report following the points below. For each task below, you need to provide a detailed explanation with thorough justification for each choice you make.

Using only the operator symbols for filtering and downsampling, show and explain a detailed schematic diagram of the forward wavelet transform applied to images with only 1 decomposition level. Recall that the wavelet transform needs to be applied along horizontal and vertical directions when applied to images.
Using only the operator symbols for filtering and upsampling, show and explain a detailed schematic diagram of the inverse wavelet transform used for reconstruction of images with only 1 synthesis level.
Using the schematic diagram derived in A. as a building block, show and explain in detail the schematic diagram for deploying the wavelet transform with 4 iterations (decomposition levels) to images. Use the schematic diagram from A. as a box with input(s) and output(s) – there is no need to show again the content of the box.
Using the schematic diagram derived in B. as a building block, show and explain in detail the schematic diagram for reconstructing images using 4 iterations (decomposition levels).
Show a schematic diagram that combines the ones derived in C. and D. to apply image approximation. Assume that in the approximation process, all transform coefficients that do not belong to the lowest pass subband are set to zero.
Calculate how many operations (additions and multiplications) are required to be carried out for approximation (forward, then inverse wavelet transform with the modification as in E., that is zeroing out all the coefficients except the ones from the lowest pass subband) of an image of the size 1000 x 1000 pixels. Assume that the wavelet filters used in the process consist of 4 coefficients. Show and explain every step of the calculations. Show how the downsampling operator can reduce the number of operations.