1 / 24

Scalar Quantization

Scalar Quantization. CAP5015 Fall 2004. Quantization. Definition: Quantization: a process of representing a large – possibly infinite – set of values with a much smaller set. Scalar quantization: a mapping of an input value x into a finite number of output values, y : Q: x ® y

Ava
Télécharger la présentation

Scalar Quantization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Scalar Quantization CAP5015 Fall 2004

  2. Quantization • Definition: • Quantization: a process of representing a large – possibly infinite – set of values with a much smaller set. • Scalar quantization: a mapping of an input value x into a finite number of output values, y: Q: x ® y • One of the simplest and most general idea in lossy compression.

  3. Scalar Quantization • Many of the fundamental ideas of quantization and compression are easily introduced in the simple context of scalar quantization. • An example: any real number x can be rounded off to the nearest integer, say q(x) = round(x) • Maps the real line R(a continuous space) into a discrete space.

  4. Quantizer • The design of the quantizer has a significant impact on the amount of compression obtained and loss incurred in a lossy compression scheme. • Quantizer: encoder mapping and decode mapping. • Encoder mapping • – The encoder divides the range of source into a number of intervals • – Each interval is represented by a distinct codeword • Decoder mapping • – For each received codeword, the decoder generates a reconstruct value

  5. Components of a Quantizer • Encoder mapping: Divides the range of values that the source generates into a number of intervals. Each interval is then mapped to a codeword. It is a many-to-one irreversible mapping. The code word only identifies the interval, not the original value. If the source or sample value comes from a analog source, it is called a A/D converter.

  6. Mapping of a 3-bit Encoder Codes … 000 001 010 011 100 101 110 111 -3.0 -2.0 -1.0 0 1.0 2.0 3.0 input

  7. Mapping of a 3-bit D/A Converter

  8. Components of a Quantizer 2. Decoder: Given the code word, the decoder gives a an estimated value that the source might have generated. Usually, it is the midpoint of the interval but a more accurate estimate will depend on the distribution of the values in the interval. In estimating the value, the decoder might generate some errors. (Give Table 8.1 and explain)

  9. Digitizing a Sine Wave

  10. Step Encoder

  11. resulting quantization error (‘noise’) so that

  12. Probability Density Function • A probability density function f(x) of the random variable x is said to meet the following criterion : • Probability associated with a value of x in its domain X is given by Pr( X<= x ). • The corresponding cumulative distribution function CDF or F(x) requires that F(x) is non-decreasing for x[1] <= x[2]. When sampling occurs at discrete intervals then F(x) is said to be monotonically increasing. • F(x) is said to be continuous from the right or that the limit of f(x + e) exists when evaluated as e-> 0 from the right positive abscissa. • In the discrete case the point probabilities of particular values of x[i] have a probability that is always greater or equal to 0, p[i] == Pr( X = x[i] ) >= 0. • CDF may be expressed as  • In the continuous case, the CDF may be expressed as the following relationship: 

  13. Quantization operation: • – Let M be the number of reconstruction levels where the decision boundaries are and the reconstruction levels are

  14. Quantization Problem • MSQE (mean squared quantization error) • If the quantization operation is Q • Suppose the input is modeled by a random variable X with pdf fX(x). The MSQE is

  15. Quantization Problem • Rate of the quantizer • The average number of bits required to represent a single quantizer output • –For fixed-length coding, the rate R is: • For variable-length coding, the rate will depend on the probability of occurrence of the outputs

  16. Quantization Problem • Quantizer design problem • Fixed -length coding • Variable-length coding If li is the length of the codeword corresponding to the output yi, and the probability of occurrence of yi is: The rate is given by:

  17. Uniform Quantization

  18. Uniform Quantizer Zero is one of the output levels M is odd Zero is not one of the output levels M is even

  19. Uniform Quantization of A Uniformly Distributed Source

  20. Uniform Quantization of A Uniformly Distributed Source

  21. Uniform Quantization of A Non-uniformly Distributed Source

  22. Image Compression Original 8bits/pixel 3bits/pixel

  23. Image Compression 2bits/pixel 1bit/pixel

More Related