Transform coding

Last updated June 20, 2023

Transform coding is a type of data compression for "natural" data like audio signals or photographic images. The transformation is typically lossless (perfectly reversible) on its own but is used to enable better (more targeted) quantization, which then results in a lower quality copy of the original input (lossy compression).

In transform coding, knowledge of the application is used to choose information to discard, thereby lowering its bandwidth. The remaining information can then be compressed via a variety of methods. When the output is decoded, the result may not be identical to the original input, but is expected to be close enough for the purpose of the application.

Colour television

NTSC

One of the most successful transform encoding system is typically not referred to as such—the example being NTSC color television. After an extensive series of studies in the 1950s, Alda Bedford showed that the human eye has high resolution only for black and white, somewhat less for "mid-range" colors like yellows and greens, and much less for colors on the end of the spectrum, reds and blues.

Using this knowledge allowed RCA to develop a system in which they discarded most of the blue signal after it comes from the camera, keeping most of the green and only some of the red; this is chroma subsampling in the YIQ color space.

The result is a signal with considerably less content, one that would fit within existing 6 MHz black-and-white signals as a phase modulated differential signal. The average TV displays the equivalent of 350 pixels on a line, but the TV signal contains enough information for only about 50 pixels of blue and perhaps 150 of red. This is not apparent to the viewer in most cases, as the eye makes little use of the "missing" information anyway.

PAL and SECAM

The PAL and SECAM systems use nearly identical or very similar methods to transmit colour. In any case both systems are subsampled.

Digital

The term is much more commonly used in digital media and digital signal processing. The most widely used transform coding technique in this regard is the discrete cosine transform (DCT),^[1]^[2] proposed by Nasir Ahmed in 1972,^[3]^[4] and presented by Ahmed with T. Natarajan and K. R. Rao in 1974.^[5] This DCT, in the context of the family of discrete cosine transforms, is the DCT-II. It is the basis for the common JPEG image compression standard,^[6] which examines small blocks of the image and transforms them to the frequency domain for more efficient quantization (lossy) and data compression. In video coding, the H.26x and MPEG standards modify this DCT image compression technique across frames in a motion image using motion compensation, further reducing the size compared to a series of JPEGs.

In audio coding, MPEG audio compression analyzes the transformed data according to a psychoacoustic model that describes the human ear's sensitivity to parts of the signal, similar to the TV model. MP3 uses a hybrid coding algorithm, combining the modified discrete cosine transform (MDCT) and fast Fourier transform (FFT).^[7] It was succeeded by Advanced Audio Coding (AAC), which uses a pure MDCT algorithm to significantly improve compression efficiency.^[8]

The basic process of digitizing an analog signal is a kind of transform coding that uses sampling in one or more domains as its transform.

Related Research Articles

Audio signal processing is a subfield of signal processing that is concerned with the electronic manipulation of audio signals. Audio signals are electronic representations of sound waves—longitudinal waves which travel through air, consisting of compressions and rarefactions. The energy contained in audio signals is typically measured in decibels. As audio signals may be represented in either digital or analog format, processing may occur in either domain. Analog processors operate directly on the electrical signal, while digital processors operate mathematically on its digital representation.

In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information. Typically, a device that performs data compression is referred to as an encoder, and one that performs the reversal of the process (decompression) as a decoder.

<span class="mw-page-title-main">Lossy compression</span> Data compression approach that reduces data size while discarding or changing some of it

In information technology, lossy compression or irreversible compression is the class of data compression methods that uses inexact approximations and partial data discarding to represent the content. These techniques are used to reduce data size for storing, handling, and transmitting content. The different versions of the photo of the cat on this page show how higher degrees of approximation create coarser images as more details are removed. This is opposed to lossless data compression which does not degrade the data. The amount of data reduction possible using lossy compression is much higher than using lossless techniques.

<span class="mw-page-title-main">MP3</span> Digital audio format

MP3 is a coding format for digital audio developed largely by the Fraunhofer Society in Germany under the lead of Karlheinz Brandenburg, with support from other digital scientists in the United States and elsewhere. Originally defined as the third audio format of the MPEG-1 standard, it was retained and further extended — defining additional bit-rates and support for more audio channels — as the third audio format of the subsequent MPEG-2 standard. A third version, known as MPEG 2.5 — extended to better support lower bit rates — is commonly implemented, but is not a recognized standard.

Speech coding is an application of data compression to digital audio signals containing speech. Speech coding uses speech-specific parameter estimation using audio signal processing techniques to model the speech signal, combined with generic data compression algorithms to represent the resulting modeled parameters in a compact bitstream.

Image compression is a type of data compression applied to digital images, to reduce their cost for storage or transmission. Algorithms may take advantage of visual perception and the statistical properties of image data to provide superior results compared with generic data compression methods which are used for other digital data.

Motion compensation in computing, is an algorithmic technique used to predict a frame in a video, given the previous and/or future frames by accounting for motion of the camera and/or objects in the video. It is employed in the encoding of video data for video compression, for example in the generation of MPEG-2 files. Motion compensation describes a picture in terms of the transformation of a reference picture to the current picture. The reference picture may be previous in time or even from the future. When images can be accurately synthesized from previously transmitted/stored images, the compression efficiency can be improved.

Digital audio is a representation of sound recorded in, or converted into, digital form. In digital audio, the sound wave of the audio signal is typically encoded as numerical samples in a continuous sequence. For example, in CD audio, samples are taken 44,100 times per second, each with 16-bit sample depth. Digital audio is also the name for the entire technology of sound recording and reproduction using audio signals that have been encoded in digital form. Following significant advances in digital audio technology during the 1970s and 1980s, it gradually replaced analog audio technology in many areas of audio engineering, record production and telecommunications in the 1990s and 2000s.

A video codec is software or hardware that compresses and decompresses digital video. In the context of video compression, codec is a portmanteau of encoder and decoder, while a device that only compresses is typically called an encoder, and one that only decompresses is a decoder.

A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images, digital video, digital audio, digital television, digital radio, and speech coding. DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.

<span class="mw-page-title-main">Compression artifact</span> Distortion of media caused by lossy data compression

A compression artifact is a noticeable distortion of media caused by the application of lossy compression. Lossy data compression involves discarding some of the media's data so that it becomes small enough to be stored within the desired disk space or transmitted (streamed) within the available bandwidth. If the compressor cannot store enough data in the compressed version, the result is a loss of quality, or introduction of artifacts. The compression algorithm may not be intelligent enough to discriminate between distortions of little subjective importance and those objectionable to the user.

Advanced Audio Coding (AAC) is an audio coding standard for lossy digital audio compression. Designed to be the successor of the MP3 format, AAC generally achieves higher sound quality than MP3 encoders at the same bit rate.

The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset, where subsequent blocks are overlapped so that the last half of one block coincides with the first half of the next block. This overlapping, in addition to the energy-compaction qualities of the DCT, makes the MDCT especially attractive for signal compression applications, since it helps to avoid artifacts stemming from the block boundaries. As a result of these advantages, the MDCT is the most widely used lossy compression technique in audio data compression. It is employed in most modern audio coding standards, including MP3, Dolby Digital (AC-3), Vorbis (Ogg), Windows Media Audio (WMA), ATRAC, Cook, Advanced Audio Coding (AAC), High-Definition Coding (HDC), LDAC, Dolby AC-4, and MPEG-H 3D Audio, as well as speech coding standards such as AAC-LD (LD-MDCT), G.722.1, G.729.1, CELT, and Opus.

In audio signal processing, pre-echo, sometimes called a forward echo, is a digital audio compression artifact where a sound is heard before it occurs. It is most noticeable in impulsive sounds from percussion instruments such as castanets or cymbals.

A timeline of events related to information theory, quantum information theory and statistical physics, data compression, error correcting codes and related subjects.

<span class="mw-page-title-main">K. R. Rao</span> Indian-American electrical engineer

Kamisetty Ramamohan Rao was an Indian-American electrical engineer. He was a professor of Electrical Engineering at the University of Texas at Arlington. Academically known as K. R. Rao, he is credited with the co-invention of discrete cosine transform (DCT), along with Nasir Ahmed and T. Natarajan due to their landmark publication, Discrete Cosine Transform.

A video coding format is a content representation format for storage or transmission of digital video content. It typically uses a standardized video compression algorithm, most commonly based on discrete cosine transform (DCT) coding and motion compensation. A specific software, firmware, or hardware implementation capable of compression or decompression to/from a specific video coding format is called a video codec.

Nasir Ahmed is an Indian-American electrical engineer and computer scientist. He is Professor Emeritus of Electrical and Computer Engineering at University of New Mexico (UNM). He is best known for inventing the discrete cosine transform (DCT) in the early 1970s. The DCT is the most widely used data compression transformation, the basis for most digital media standards and commonly used in digital signal processing. He also described the discrete sine transform (DST), which is related to the DCT.

An online video platform (OVP), provided by a video hosting service, enables users to upload, convert, store and play back video content on the Internet, often via a private server structured, large-scale system that may generate revenue. Users will generally upload video content via the hosting service's website, mobile or desktop application, or other interfaces (API). An example of an OVP is YouTube. The type of video content uploaded might be anything from shorts to full-length TV shows and movies. The video host stores the video on its server and offers users the ability to enable different types of embed codes or links that allow others to view the video content. The website, mainly used as the video hosting website, is usually called the video-sharing website.

An audio coding format is a content representation format for storage or transmission of digital audio. Examples of audio coding formats include MP3, AAC, Vorbis, FLAC, and Opus. A specific software or hardware implementation capable of audio compression and decompression to/from a specific audio coding format is called an audio codec; an example of an audio codec is LAME, which is one of several different codecs which implements encoding and decoding audio in the MP3 audio coding format in software.

References

↑ Muchahary, D.; Mondal, A. J.; Parmar, R. S.; Borah, A. D.; Majumder, A. (2015). "A Simplified Design Approach for Efficient Computation of DCT". 2015 Fifth International Conference on Communication Systems and Network Technologies: 483–487. doi:10.1109/CSNT.2015.134. ISBN 978-1-4799-1797-6. S2CID 16411333.
↑ Chen, Wai Kai (2004). The Electrical Engineering Handbook. Elsevier. p. 906. ISBN 9780080477480.
↑ Ahmed, Nasir (January 1991). "How I Came Up With the Discrete Cosine Transform". Digital Signal Processing . 1 (1): 4–5. doi:10.1016/1051-2004(91)90086-Z.
↑ Stanković, Radomir S.; Astola, Jaakko T. (2012). "Reminiscences of the Early Work in DCT: Interview with K.R. Rao" (PDF). Reprints from the Early Days of Information Sciences. 60. Retrieved 13 October 2019.
↑ Ahmed, Nasir; Natarajan, T.; Rao, K. R. (January 1974), "Discrete Cosine Transform", IEEE Transactions on Computers, C-23 (1): 90–93, doi:10.1109/T-C.1974.223784, S2CID 149806273
↑ "T.81 – Digital compression and coding of continuous-tone still images – Requirements and guidelines" (PDF). CCITT. September 1992. Retrieved 12 July 2019.
↑ Guckert, John (Spring 2012). "The Use of FFT and MDCT in MP3 Audio Compression" (PDF). University of Utah . Retrieved 14 July 2019.
↑ Brandenburg, Karlheinz (1999). "MP3 and AAC Explained" (PDF). Archived (PDF) from the original on 2017-02-13.

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[Muchahary-1] Muchahary, D.; Mondal, A. J.; Parmar, R. S.; Borah, A. D.; Majumder, A. (2015). "A Simplified Design Approach for Efficient Computation of DCT". 2015 Fifth International Conference on Communication Systems and Network Technologies: 483–487. doi:10.1109/CSNT.2015.134. ISBN 978-1-4799-1797-6. S2CID 16411333.

[2] Chen, Wai Kai (2004). The Electrical Engineering Handbook. Elsevier. p. 906. ISBN 9780080477480.

[Ahmed-3] Ahmed, Nasir (January 1991). "How I Came Up With the Discrete Cosine Transform". Digital Signal Processing . 1 (1): 4–5. doi:10.1016/1051-2004(91)90086-Z.

[Stankovic-4] Stanković, Radomir S.; Astola, Jaakko T. (2012). "Reminiscences of the Early Work in DCT: Interview with K.R. Rao" (PDF). Reprints from the Early Days of Information Sciences. 60. Retrieved 13 October 2019.

[pubDCT-5] Ahmed, Nasir; Natarajan, T.; Rao, K. R. (January 1974), "Discrete Cosine Transform", IEEE Transactions on Computers, C-23 (1): 90–93, doi:10.1109/T-C.1974.223784, S2CID 149806273

[t81-6] "T.81 – Digital compression and coding of continuous-tone still images – Requirements and guidelines" (PDF). CCITT. September 1992. Retrieved 12 July 2019.

[Guckert-7] Guckert, John (Spring 2012). "The Use of FFT and MDCT in MP3 Audio Compression" (PDF). University of Utah . Retrieved 14 July 2019.

[brandenburg-8] Brandenburg, Karlheinz (1999). "MP3 and AAC Explained" (PDF). Archived (PDF) from the original on 2017-02-13.

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