University of Maryland at College ParkDepartment of Electrical and Computer EngineeringENEE 624 Advanced Digital Signal ProcessingProject 1Spring 2003Issued: Wednesday, March 5, 2003 Due: Monday, March 31, 2003Problem 1.1 Perfect Reconstruction FIR QMF Bank Designx[n]- -?--H0(z)H1(z)--↓ 2↓ 2--↑ 2↑ 2--F0(z)F1(z)-- -?ˆx[n]Figure 1.1-1 Two-channel QMF bank.(a) Design a two-channel QMF bank (see Fig. 1.1-1) with the following properties:(i) all four filters are FIR filters;(ii) H0(z) is a lowpass filter with ωs= 0.6 π and δs= 0.01;(iii) the overall system is a perfect reconstruction (PR) system.Check the PR property by examining whether the bank indeed perfectly reconstructs thesupplied signal xa in file testsigs.mat.(b) Repeat part (a) for δs= 0.001; again check the PR property.(c) Next consider compressing each of the signals x1[n] and x2[n] (given by the Matlab vec-tors x1 and x2 in testsigs.mat, respectively) by quantizing the outputs of the decimatedanalysis filter bank in Fig. 1.1-1. In each case, design the QMF bank (or, equivalently,the prototype filter H0(z)) and choose a quantization/compression strategy so that:(i)Pn(ˆxi[n + N] − xi[n])2Pnx2i[n]< 0.01 ,where N corresponds to the delay induced by the PR system;(ii) the total number of bits needed to describe the (quantized) outputs of the decimatedanalysis filter bank is minimized. Discuss your findings.Project 1 2Problem 1.2 Subband Coding for Image CompressionThis part of the project is open-ended; its purpose is the exploration of the effectiveness ofsubband coding for lossy image compression. All the compression systems you are asked todesign are in the form of a decimated analysis filter bank followed by an appropriately designedmethod of your choice for compressing the filter bank outputs into bit streams. The resultingdescription denotes (hopefully) an efficiently compressed version of the original image. The testimages that you will be using in this project are provided in the files lena.pgm, houses.pgm, andlighthouse.pgm. Throughout the project you are free to select the compression strategy of yourchoice (clearly, one such strategy pertains to simply quantizing the analysis bank outputs). Ineach case, you also have to design the associated synthesis bank for signal reconstruction. Notethat certain parameters from your subband coder compression method may have to be madeknown to the subband decoder so as to allow the decoder to accurately map the bits producedby the subband coder into quantized signal levels and consequently construct the signals thatare the inputs of the synthesis bank. Clearly, any such parameters have to be considered aspart of the compressed description, as accurate image reconstruction is not possible withoutthem.You are asked to explore four different types of filter banks. Your task in each case is tominimize the number of bits in the compressed description so that the reconstructed imageis still of “acceptable quality.” Clearly, in this context “acceptable quality” is a subjectiveattribute; you have to use your own judgment as to what you deem “perceptually acceptable.”In the process you may also want to consider whether you can develop what you feel is ameaningful metric for quantifying the perceptual image distortion.(a) Design a PR FIR QMF bank of the type you designed in Problem 1.1 of the project.First check whether your bank is indeed PR (in the absence of compression) by usingthe supplied images as input and comparing it to the output. Next explore strategies forcompression of the decimated analysis bank outputs; your goal is to minimize the size ofthe image description produced by the subband coder subject to what you still deem as anacceptably accurate reconstruction (in a perceptual sense) of the original image. Discussyour observations and findings, and in particular the effectiveness of your compressionalgorithm for each image and the trade-offs it provides between compression gain andperceptual quality of the reconstruction.(b) Repeat part (a) for a two-channel decimated uniform DFT bank, where the prototypeFIR filter H0(z) (based on which the other three FIR filters are selected) is lowpass andlinear phase. Design H0(z) so as to make the amplitude distortion of the overall system assmall as possible while preserving the lowpass and linear-phase characteristics of H0(z).You may find such methods for designing optimized filter banks of this type (as wellas tables of sample prototype filters) in [1, 2]. First determine whether the designedbank accurately reproduces the images in the absence of compression. Then, explorecompression of the analysis bank outputs and, in particular, for each image minimize thesubband coder image description size subject to the constraint that the reconstructedimage is of acceptable quality. Discuss your findings.(c) In this part you are to develop a nonuniform three-channel PR filter bank, by splittingthe lowpass signal output of a two-channel analysis bank via another two-channel analysisProject 1 3filter bank. For both banks use a two-channel PR bank of the type you developed in part(a). First, check whether the resulting three-channel bank perfectly reconstructs theoriginal images in the absence of compression. Then, explore compression strategies andminimize the description size subject to what you still deem as acceptably accurate imagereconstruction. Compare your results to those of parts (a) and (b).(d) The concept of 1-D subband coding can be easily extended to 2-D: each row of the imageto be compressed can be first separated via 1-D two-channel subband coding to its lowand high frequency components. Then, by viewing the two collections of the resultinglowpass and highpass row signals as two separate collections of column signals, we canapply 1-D subband coding of each column via a two-channel QMF bank into highpass andlowpass column-wise signals. The operation is graphically demonstrated in Fig. 1.2-1.Image-1-D row-wisesubband codingL R-1-D column-wisesubband codingLHLLHHHLFigure 1.2-1 2-D subband coding of images.Verify that such a 2-D subband coding method yields perfect reconstruction, if each 1-D subband coding operation has the perfect reconstruction property. Again determinethe minimum subband coder description size (resulting from compressing the resulting2-D bank coefficients) subject to acceptable reconstructed image quality for each of thesupplied images. Compare your results against the preceding