Multi resolution approximation (ECE 592 Module 43)
Module 43 looks at multi resolution approximation, and connects ideas between wavelets and approximation of spaces. We have a sequence of subspaces of L2(R), which is comprised of finite energy continuous time functions. This sequence of subspaces is called a multi resolution approximation (MRA) if the subspaces adhere to some technical conditions, for example that they are nested, i.e., V_{j+1} is a subset of V_j. In the limit of negative j, the space is L2(R), and for large positive j, V_j becomes the zero function. We can express V_{j-1}, a larger space, as a direct sum of V_j and W_j, the orthogonal complement of V_j in V_{j-1}; the W_j at different scales are actually wavelet components. This direct sum allows us to decompose a function into its components at different scales. Finally, from the computational angle, we can implement an MRA as a combination of wavelets at different scales using filter banks.
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