quantization4up.220.06 - An Introduction to Quantization...

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S. S. Hemami-ECE220 -1-10/22/06 An Introduction to Quantization Simple examples to get started. Formal definition. The uniform quantizer. Analysis of quantization error for uniform quantizer. Signal-to-noise ratio and 6 dB/bit. In this material, we will begin to see why signal processing requires probability and statistics (310 and 411). Prof. Hemami ECE 220 Fall 2006 S. S. Hemami-ECE220 -2-10/22/06 Outline 2 definitions A process by which a signal that takes on continuous or discrete values is represented using a lesser number of values from a discrete set. Example — your grade. A process by which we decrease the number of bits required to store a set of values by reducing the precision in those values. Example — use “thousands of colors” instead of “millions of colors” for your computer display. S. S. Hemami-ECE220 -3-10/22/06 A Conceptual Example — Your Grade Averaging all your scores as weighted according to the course information sheet, your raw grade g is... , and Your assigned grade is one of 5 letter grades: , The quantizer has an codebook of size 5. Once your raw grade has been quantized to a letter grade, you cannot recover your raw grade. Quantization is irreversible. g 0 g 100 L Q g ( ) = Q g ( ) A B C D F = S. S. Hemami-ECE220 -4-10/22/06 One Potential Grade Quantizer Observe that all the bins are the same size (20 points). This is a uniform quantizer . 20 40 60 80 100 F D C B A Input g (raw grade) Output L=Q(g) (letter grade)
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S. S. Hemami-ECE220 -5-10/22/06 A Better Grade Quantizer (from my perspective) Now we have bin sizes of 52, 8, 8, 12, and 20. This is a nonuniform quantizer . 52 60 80 100 F D C B A Input g (raw grade) Output L=Q(g) (letter grade) 68 S. S. Hemami-ECE220 -6-10/22/06 Other Quantization Examples Temperature as reported to us by the weatherman on TV, weather.com, etc. We are always given temperatures as integers. The actual measured temperatures are quantized with a step size/bin width of 1. Pixel values in digital images. 8-bit images contain values ranging from 0 to 255 ( ). The sensed luminance values, however, are not integers in the camera CCD. Medical images are often quantized to 12 bits instead of 8. 2 8 1 S. S. Hemami-ECE220 -7-10/22/06 Formal Definition An N-point scalar (1 dimensional) quantizer Q is a mapping where R is the real line and is the output set or codebook with size N . Terminology: The output values are also called o utput levels, output points, reproduction values, reconstruction values/points . The input (real line) is partitioned into cells (also called bins ). Q : R C C y 1 y 2 y N , , , { } R y i S. S. Hemami-ECE220 -8-10/22/06 A 7-point Quantizer Not shown on plot are the left/right endpoints of the leftmost/rightmost bins: Note x 2 x 5 x 3 x 4 x 6 x 1 y 4 y 5 y 6 y 2 y 1 input y 3 output x 0 x 7 , + = = x 0 y 1 x 1 y 2 y N x N < < < < < <
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S. S. Hemami-ECE220 -9-10/22/06 Operation when . In English, “output a value of when the input is in the range .” We call the range a “bin” or a cell.
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