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Vector quantization is a classical quantization technique from signal processing which allows the modeling of probability density functions by the distribution of prototype vectors. Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as It was originally used for data compression. It works by dividing a large set of points (vectors) into groups having approximately the same number of points closest to them. In Linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or equivalently as an Each group is represented by its centroid point, as in k-means and some other clustering algorithms. In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes The k-means algorithm is an algorithm to cluster n objects based on attributes into k partitions, k.

The density matching property of vector quantization is powerful, especially for identifying the density of large and high-dimensioned data. Since data points are represented by the index of their closest centroid, commonly occurring data have low error, and rare data high error. This is why VQ is suitable for lossy data compression. A lossy compression method is one where compressing data and then decompressing it retrieves data that may well be different from the original but is close enough to be useful It can also be used for lossy data correction and density estimation. In Probability and Statistics, density estimation is the construction of an estimate based on observed Data, of an unobservable underlying Probability

Vector quantization is based on the competitive learning paradigm, so it is closely related to the self-organizing map model. A self-organizing map (SOM is a type of Artificial neural network that is trained using Unsupervised learning to produce a low-dimensional (typically two dimensional

Contents

Training

A simple training algorithm for vector quantization is:

  1. Pick a sample point at random
  2. Move the nearest quantization vector centroid towards this sample point, by a small fraction of the distance
  3. Repeat

A more sophisticated algorithm reduces the bias in the density matching estimation, and ensures that all points are used, by including an extra sensitivity parameter:

  1. Increase each centroid's sensitivity by a small amount
  2. Pick a sample point at random
  3. Find the quantization vector centroid with the smallest <distance-sensitivity>
    1. Move the chosen centroid toward the sample point by a small fraction of the distance
    2. Set the chosen centroid's sensitivity to zero
  4. Repeat

It is desirable to use a cooling schedule to produce convergence: see Simulated annealing. Simulated annealing (SA is a generic probabilistic Meta-algorithm for the Global optimization problem namely locating a good approximation to the

The algorithm can be iteratively updated with 'live' data, rather than by picking random points from a data set, but this will introduce some bias if the data is temporally correlated over many samples.

Applications

Vector quantization is used for lossy data compression, lossy data correction and density estimation.

Lossy data correction, or prediction, is used to recover data missing from some dimensions. It is done by finding the nearest group with the data dimensions available, then predicting the result based on the values for the missing dimensions, assuming that they will have the same value as the group's centroid.

For density estimation, the area/volume that is closer to a particular centroid than to any other is inversely proportional to the density (due to the density matching property of the algorithm). In Probability and Statistics, density estimation is the construction of an estimate based on observed Data, of an unobservable underlying Probability

Use in data compression

Vector quantization, also called "block quantization" or "pattern matching quantization" is often used in lossy data compression. A lossy compression method is one where compressing data and then decompressing it retrieves data that may well be different from the original but is close enough to be useful It works by encoding values from a multidimensional vector space into a finite set of values from a discrete subspace of lower dimension. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Subspace may refer to;Mathematics Euclidean subspace, in linear algebra a set of vectors in n -dimensional Euclidean space that is closed under addition A lower-space vector requires less storage space, so the data is compressed. Thanks to the density matching property of vector quantization, the compressed data have errors that are inversely proportional to their density.

The transformation is usually done by projection or by using a codebook. In Cryptography, a codebook is a document used for implementing a code. In some cases, a codebook can be also used to entropy code the discrete value in the same step, by generating a prefix coded variable-length encoded value as its output. In Information theory an entropy encoding is a lossless Data compression scheme that is independent of the specific characteristics of the medium A prefix code is a Code, typically a Variable-length code, with the "prefix property" no Code word is a prefix of any other code word

The set of discrete amplitude levels is quantized jointly rather than each sample being quantized separately. Consider a K-dimensional vector [x1,x2,. . . ,xk] of amplitude levels. It is compressed by choosing the nearest matching vector from a set of N-dimensional vectors [y1,y2,. . . ,yn].

All possible combinations of the N-dimensional vector [y1,y2,. . . ,yn] form the codebook.

Block Diagram: A simple vector quantizer is shown below


Only the index of the codeword in the codebook is sent instead of the quantized values. This conserves space and achieves more compression.

Twin vector quantization (VQF) is part of the MPEG-4 standard dealing with time domain weighted interleaved vector quantization. In Data compression, twin vector quantization is related to Vector quantization, but the speed of the quantization is doubled by the secondary vector analyzer MPEG-4 is a collection of methods defining compression of audio and visual (AV digital data

Video codecs based on vector quantization

and old versions of its spiritual successors:

All of which are superseded by the MPEG family. Cinepak is a Video codec developed by SuperMatch a division of SuperMac Technologies, and released in 1992 as part of Apple Computer's The Sorenson codec (also known as Sorenson Video Codec, Sorenson Video Quantizer or SVQ) is a Digital video Codec devised by the company Indeo Video (commonly known now simply as "Indeo" is a Video codec developed by Intel in 1992

Audio codecs based on vector quantization

See also

Part of this article was originally based on material from the Free On-line Dictionary of Computing and is used with permission under the GFDL. Code excited linear prediction ( CELP) is a Speech coding algorithm originally proposed by M G729 is an Audio data compression algorithm for voice that compresses voice audio in chunks of 10 milliseconds TwinVQ (transform-domain weighted interleaved Vector quantization) is an audio compression technique developed by Nippon Telegraph and Telephone Corporation (NTT Vorbis is a free and open source, lossy audio Codec project headed by the Xiph Adaptive Multi Rate – WideBand plus ( AMR-WB+) is an audio Codec that extends AMR-WB. DTS Coherent Acoustics is the full name for the audio format standard usually known as just DTS. Speech coding is the application of Data compression of Digital audio signals containing Speech. Vorbis is a free and open source, lossy audio Codec project headed by the Xiph In Mathematics, a Voronoi diagram, named after Georgy Voronoi, also called a Voronoi Tessellation, a Voronoi decomposition, or Rate–distortion theory is a major branch of Information theory which provides the theoretical foundations for Lossy data compression; it addresses the problem of Clustering is the classification of objects into different groups or more precisely the partitioning of a Data set into Subsets (clusters LVQ, or Learning Vector Quantization, is a prototype-based supervised classification Algorithm. In Geometry, a centroidal Voronoi tessellation (CVT is a special type of Voronoi tessellation or Voronoi diagrams. The Free On-line Dictionary of Computing ( FOLDOC) is an online searchable encyclopedic Dictionary of Computing subjects

References

  1. ^ Vorbis I Specification. Xiph. org (2007-03-09). Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 590 - Bahram Chobin is crowned as king Barham VI of Persia. Retrieved on 2007-03-09. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 590 - Bahram Chobin is crowned as king Barham VI of Persia.

External links

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