eee508_ScalarQuantization

eee508_ScalarQuantization - Qua t at o Quantization...

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Quantization Quantization: main component in source coders ¾ Problem: Original image is often very large ± Reduce the data 9 Remove redundancy ± no loss ± info content preserved Prediction (DPCM, ADPCM) Entropy coding Transform coding (e.g., transforms used to decorrelate data) subband coding 9 Remove unnecessary, “least important” info ± loss scalar quantization vector quantization EEE 508 - Lecture 7
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Scalar quantization Example : Uniform and non-uniform scalar quantization x Uniform Uniform quantization ¾ Better performance (in terms of quantization distribution) can be t 0 achieved by designing quantizer that match the pdf of the signal x EEE 508 - Lecture 7
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Scalar quantization Example : Uniform and non-uniform scalar quantization Non-uniform x [ n ] Non uniform quantization ¾ Might be better to have lots of steps at low levels and a few at high n 0 levels ¾ Look at pdf of signal or image and match the quantizer to it EEE 508 - Lecture 7
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Scalar quantization General Approach: ¾ Take histogram of image and normalize it (so that area = 1) to obtain a pdf. Usually, it takes a form close to a Gaussian or a Laplacian distribution ¾ Then design quantizer (choose { V j } and { r j } ) to match pdf and minimize distortion ¾ To solve the obtained resulting equations is not impossible but they are non linear So usually the equations have been solved they are non-linear. So, usually, the equations have been solved numerically for different distributions and number of quantization levels and the parameters of optimal quantizers (Lloyd-Max) are tabulated. EEE 508 - Lecture 7
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Scalar quantization Simple example G Q ( x ) = x Two level quantizer p ( x ) x input with pdf x - x ¾ What is the optimal reconstruction level ? ± ˆ ²³ ² ³ ´ f f ± ± dx x p x x e E E x x e e Q 2 2 2 ˆ 2 V EEE 508 - Lecture 7
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Scalar quantization ¾ Use symmetry of the problem: ± ² ± ² ½ ­ ³ x p x p ±² ± ² ¿ ¾ ¯ ® ³ ´ ³ ³ 2 2 )) ( ( )) ( ( x e x e x Q x Q f f 2 2 2 ˆ 2 d p d p E G ± ² ± ² ± ² ± ² µ µ ³ ³ 0 0 2 dx x x dx x x x Q ± ² ± ² ± ² 0 2 4 2 0 Take 2 2 » º « ª ³ ´ f f f dx x p dx x xp dx x p x d dE 0 0 0 ¼ ¬ µ µ µ d d ±² ±² 0 4 4 0 0 ³ ´ µ µ f f dx x xp dx x p 2 1 ±² ±³ ´ µ µ f f ´ ´ 2 4 2 dx x xp dx x xp EEE 508 - Lecture 7 0 0
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Scalar quantization ± Apply same idea for the more general M-level case Note: In previous example, the quantization levels are fixed and the optimzation is performed with respect to the reconstruction levels assuming a uniform-quantizer structure. More generally, we would want to optimize with respect to both quantization levels { V j } and reconstruction levels { r j } ± Lloyd-Max Quantizer optimal in the mean square sense (mse distortion Quantizer optimal in the mean square sense (mse distortion minimized).
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eee508_ScalarQuantization - Qua t at o Quantization...

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