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Final Exam 2 Solutions

Final Exam 2 Solutions - Polytechnic University Dept...

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Polytechnic University, Dept. Electrical and Computer Engineering EL6123 --- Video Processing, S08 (Prof. Yao Wang) Final Exam, 5/1/2008, 3:00-5:50 Two sheest of notes (double sided) allowed, No peak into neighbors. Cheating will result in failing grade! 1. (15 pt) Consider coding a 2-D random vector that is uniformly distributed over the region illustrated in Fig. 1(a). Suppose you want to design a codebook with 2 codewords. One possible codebook construction (codeword locations and region partition) is illustrated in Figure 1(b). a. Determine the value of y* in the upper triangle in Fig. 1(b) that will minimize the mean square error of the quantizer. Also determine the corresponding minimal mean square error. b. Another possible codebook configuration is shown in Fig. 1(c). Is this codebook better or worse than that in Fig. 1(b)? why? Solution: a) For given y*, the MSE can be computed as ܦሺݕ ∗ሻ ൌ 4׬׬ ሺሺݔ െ 0ሻ ൅ሺݕെݕ∗ሻ ሻ ݀ݕ ݀ݔ ଶሺଵି௫ሻ One can find the optimal y* by minimizing D(y*), which can be done by setting డ஽ డ௬ ൌ0 . Detailed solution skipped. Once the optimal y* is found, you can substitute back into the equation for D(y*) to determine the MSE corresponding to this y*. (My rough calculation tells me y*=(\sqrt(21)-3 )/6. (but I cannot guarantee this is correct). b) The coded in Fig.(c) is worser, as it has more points in each partition that are farther away from the codeword for that partition, than in Fig. (b). 2. (15 pt) Consider the following predictive coding method (see Fig. 2). A sample in frame n, ) , ( y x f F n is predicted from its two neighboring pixels in the same frame ) , 1 ( y x f A n and ) 1 , ( y x f B n and a pixel in a previous frame ) , ( 1 y x f D n , using the linear predictor: dD B A a F 2 / ) ( . Suppose all samples have the same variance 2 , and the correlations between these samples are , } { , } { } { 2 2 t s FD E FB E FA E and all other samples are uncorrelated. Find the optimal values for predictor coefficients 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. Figure 1 -1 -2 1 2 (a) -1 -2 1 2 (b) -1 -2 1 2 (c) y* -y* y*
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Figure 2 The predictor coefficients should satisfy the following equation: ܧሺܥܥሻ ܧሺܥܦሻ ܧሺܦܥሻ ܧሺܦܦሻ ൨ቂ ܽ ݀ ቃൌ൤ ܧሺܨܥሻ ܧሺܨܦሻ 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       . 0 , , , , , 2 2 C B E C A E B A E 
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Final Exam 2 Solutions - Polytechnic University Dept...

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