Final Exam 2 Solutions

# Find the optimal values for predictor coefficients a

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Unformatted text preview: ficients a and d that will minimize the mean square prediction error, and determine the corresponding minimal prediction error. What is the coding gain compared to code each sample directly? Hint: you can treat C=(A+B)/2 as a sample and make use of correlations among F,C,D. B D A F Figure 2 The predictor coefficients should satisfy the following equation: E(CC)=E((A+B)*(A+B))/4=(E(A^2)+E(B^2)+2E(AB))/4=^2 / 2 E(CD)=E((A+B)D)/2=0 E(FC)=E(F(A+B))/2= \rho_s \sigma^2 E(FD)=\rho_t \sigma^2 The solution is a=2 \rho_s, d=2\rho_t. You can work out the rest. 3. (15 pt) Consider applying transform coding to every three pixels A, B, C in an image. The transform is 3-point DCT, whose basis vectors are given below. Suppose every pixel has the same variance 2 and the correlation between samples are EA, B 2 , EA, C 2 , EB, C 0. (a) Determine the covariance matrix of the original three pixels. (b) Determine the covariance matrix of the transformed coefficients. (c) Determine the variances of the transformed coefficients. (c) If the total number of bits for the three coefficients is 3R, what is the optimal bit allocation to each coefficient? (d) What is the coding gain over direct coding of individual pixels? Express your results in terms of given variables. 1 1 1 1 1 1 1, u 1 0 , u 2 2 . The DCT basis vectors are: u 0 3 2 6 1 1 1 Solution: This is straight forward. 4. (15 pt) Consider stereo imaging using parallel camera configuration with a baseline distance of B and focal length F. Assume a 3D point with coordinate (X,Y,Z) is projected into left and right image planes with coordinates ( xl , yl ) and ( xr , y r ) . These coordinates are related by X B/2 X B/2 Y xl F , xr F , yl y r F . Z Z Z a. Suppose the 3D points being imaged fall on a planar surface described by the plane equation Z aX bY cB . Show that the horizontal disparity d xl x r at the image point ( xr , y r ) ca...
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## This document was uploaded on 03/12/2014 for the course EL 6123 at NYU Poly.

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