1 4/8/2017ENEE631 Spring’09 Lecture 10 (2/25/2009)Unitary Transformxxx Lecture:xxx. – xx slides, about xx minSpring ’09 Instructor: Min Wu Electrical and Computer Engineering Department,University of Maryland, College Parkbb.eng.umd.edu (select ENEE631 S’09)
3 Recap: Scalar Quantizer 4/8/2017…Recap: Scalar QuantizerQuantize one sample at a timeQuantizer…Input/Output responseInput xOutput Q(x)…quantization errorM. Wu: ENEE631 Digital Image Processing (Spring'09)
4 Recap: MMSE Quantizer Example – Gaussian Source 4/8/2017Recap: MMSE Quantizer Example – Gaussian SourceTruncated Gaussian N(0,1) with L=16 quantizerStart with uniform quantizer Use iterative algorithm.optimum thresholds (red) and reconstruction values (blue)From B. Liu PU EE488 F’06M. Wu: ENEE631 Digital Image Processing (Spring'09)
36 Visualizing Fourier Basis Images 4/8/2017Visualizing Fourier Basis ImagesFourier basis uses complex exponentialsTheir real and imaginary parts give smoothly varying sinusoidal patterns in different frequencies and orientationsexp[ j 2 (ux + vy) ] = cos[2 (ux + vy)] + j sin[2 (ux + vy)]vReal(cos) part(u, v)(1, 0)(1, 1)(0, 5)Imaginary(sin) partFigures from Mani Thomas U.Del CISC489/689 2D FTM. Wu: ENEE631 Digital Image Processing (Spring'09)
37 8x8 DFT Basis Images Figures from John Woods’ book. 4/8/20178x8 DFT Basis ImagesNote at multiple of 4, the basis is real valued.Figures from John Woods’ book.M. Wu: ENEE631 Digital Image Processing (Spring'09)
41 Clarifications “Dimension” Eigenvalues of unitary transform 4/8/2017Clarifications“Dimension”Dimension of a signal ~ # of index variablesaudio and speech is 1-D signal, image is 2-D, video is 3-DDimension of a vector space ~ # of basis vectors in itEigenvalues of unitary transformAll eigenvalues have unit magnitude (could be complex valued)By definition of eigenvalues ~ A x = xBy energy preservation of unitary ~ || A x || = ||x||Eigenvalues here are different from the eigenvalues in K-L transformK-L concerns the eigen of covariance matrix of random vectorEigenvectors ~ we generally consider the orthonormalized onesM. Wu: ENEE631 Digital Image Processing (Spring'09)
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