the given input covariance/correlation structure are incorporated in the design of optimal filter banks. These filter banks resemble the signal dependent Karhunen–Loève transform (KLT) that is the optimal block transform where the length L of basis functions (filters) and the subspace dimension M are the same. Multidimensional filter banks Multidimensional filtering, downsampling, and upsampling are the main parts of multirate systems and filter banks. A complete filter bank consists of the analysis and synthesis side. The analysis filter bank divides an input signal to different subbands with different frequency spectra. The synthesis part reassembles the different subband signals and generates a reconstructed signal. Two of the basic building blocks are the decimator and expander. For example, the input divides into four directional sub bands that each of them covers one of the wedge-shaped frequency regions. In 1D systems, M-fold decimators keep only those samples that are multiples of M and discard the rest. while in multi-dimensional systems the decimators are D × D nonsingular integer matrix. it considers only those samples that are on the lattice generated by the decimator. Commonly used decimator is the quincunx decimator whose lattice is generated from the Quincunx matrix which is defined by The quincunx lattice generated by quincunx matrix is as shown; the synthesis part is dual to the analysis part. Filter banks can be analyzed from a frequency-domain perspective in terms of subband decomposition and reconstruction. However, equally important is Hilbert-space interpretation of filter banks, which plays a key role in geometrical signal representations. For generic K-channel filter bank, with analysis filters , synthesis filters , and sampling matrices . In the analysis side, we can define vectors in as , each index by two parameters: and . Similarly, for the synthesis filters we can define . Considering the definition of analysis/synthesis sides we can verify that and for reconstruction part: . In other words, the analysis filter bank calculate the inner product of the input signal and the vector from analysis set. Moreover, the reconstructed signal in the combination of the vectors from the synthesis set, and the combination coefficients of the computed inner products, meaning that If there is no loss in the decomposition and the subsequent reconstruction, the filter bank is called perfect reconstruction. (in that case we would have . Figure shows a general multidimensional filter bank with N channels and a common sampling matrix M. The analysis