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In signal processing, a lapped transform is a type of linear discrete block transformation where the basis functions of the transformation overlap the block boundaries, yet the number of coefficients overall resulting from a series of overlapping block transforms remains the same as if a non-overlapping block transform had been used.^[1]^[2]^[3]^[4]

Lapped transforms substantially reduce the blocking artifacts that otherwise occur with block transform coding techniques, in particular those using the discrete cosine transform. The best known example is the modified discrete cosine transform used in the MP3, Vorbis, AAC, and Opus audio codecs.^[5]

Although the best-known application of lapped transforms has been for audio coding, they have also been used for video and image coding and various other applications. They are used in video coding for coding I-frames in VC-1 and for image coding in the JPEG XR format. More recently, a form of lapped transform has also been used in the development of the Daala video coding format.^[5]

References

^ Malvar, H. S. (1992). "Signal Processing with Lapped Transforms" (Document). Artech House.
^ de Queiroz, Ricardo L. "On Lapped Transforms". CiteSeerX 10.1.1.91.7148. Retrieved August 20, 2023.
^ Malvar, H. S. (November 1992). "Extended Lapped Transforms: Properties, Applications, and Fast Algorithms" (PDF). IEEE Transactions on Signal Processing. 40 (11): 2703–2714. Bibcode:1992ITSP...40.2703M. doi:10.1109/78.165657.
^ Tran, Trac D.; Liang, Jie; Tu, Chengjie (June 2003). "Lapped Transform via Time-Domain Pre- and Post-Filtering" (PDF). IEEE Transactions on Signal Processing. 51 (6): 1557. Bibcode:2003ITSP...51.1557T. doi:10.1109/TSP.2003.811222. Archived from the original (PDF) on 2016-03-04. Retrieved 2013-06-22.
^ ^a ^b "Next generation video: Introducing Daala". xiph.org. June 20, 2013.

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[1] Malvar, H. S. (1992). "Signal Processing with Lapped Transforms" (Document). Artech House.

[2] Queiroz, Ricardo L. "On Lapped Transforms". CiteSeerX 10.1.1.91.7148. Retrieved August 20, 2023.

[3] Malvar, H. S. (November 1992). "Extended Lapped Transforms: Properties, Applications, and Fast Algorithms" (PDF). IEEE Transactions on Signal Processing. 40 (11): 2703–2714. Bibcode:1992ITSP...40.2703M. doi:10.1109/78.165657.

[4] Tran, Trac D.; Liang, Jie; Tu, Chengjie (June 2003). "Lapped Transform via Time-Domain Pre- and Post-Filtering" (PDF). IEEE Transactions on Signal Processing. 51 (6): 1557. Bibcode:2003ITSP...51.1557T. doi:10.1109/TSP.2003.811222. Archived from the original (PDF) on 2016-03-04. Retrieved 2013-06-22.

[introducingdaala-5] "Next generation video: Introducing Daala". xiph.org. June 20, 2013.

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-In [[signal processing]], a '''lapped transform''' is a type of [[linear transformation|linear]] [[discrete transform|discrete block transformation]] where the [[basis function]]s of the transformation overlap the block boundaries, yet the number of coefficients overall resulting from a series of overlapping block transforms remains the same as if a non-overlapping block transform had been used.<ref>H. S. Malvar. <I>Signal Processing with Lapped Transforms</I>. Artech House, 1992.</ref><ref>{{cite web|url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.91.7148&rep=rep1&type=pdf|title=Lapped Transforms|author=Ricardo L. de Queiroz}}</ref><ref>H. S. Malvar, "[http://research.microsoft.com/pubs/102075/malvar_elt_tsp1192.pdf Extended Lapped Transforms: Properties, Applications, and Fast Algorithms]", ''IEEE Trans. on Signal Processing'', Vol. 40, No. 11, pp. 2703–2714, Nov. 1992.</ref><ref>Trac D. Tran, Jie Liang, Chengjie Tu, "[http://thanglong.ece.jhu.edu/Tran/Pub/prepost.pdf Lapped Transform via Time-Domain Pre- and Post-Filtering]", ''IEEE Trans. on Signal Processing'', Vol. 51, No. 6, June 2003, accessed 2013-06-22.</ref>
+In [[signal processing]], a '''lapped transform''' is a type of [[linear transformation|linear]] [[discrete transform|discrete block transformation]] where the [[basis function]]s of the transformation overlap the block boundaries, yet the number of coefficients overall resulting from a series of overlapping block transforms remains the same as if a non-overlapping block transform had been used.<ref>{{cite document | first = H. S. | last = Malvar | title = Signal Processing with Lapped Transforms | publisher = Artech House | date = 1992 }}</ref><ref>{{cite web | citeseerx = 10.1.1.91.7148 | title = On Lapped Transforms | first = Ricardo L. | last = de Queiroz |url = https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=15fdeb1770347c7e3f4c12407ee00061635d5881 | access-date = August 20, 2023}}</ref><ref>{{cite journal | first = H. S. | last = Malvar | url = http://research.microsoft.com/pubs/102075/malvar_elt_tsp1192.pdf | title = Extended Lapped Transforms: Properties, Applications, and Fast Algorithms | journal = IEEE Transactions on Signal Processing | volume = 40 | issue = 11 | pages = 2703–2714 | date = November 1992 | doi=10.1109/78.165657| bibcode = 1992ITSP...40.2703M }}</ref><ref>{{cite journal | first1 = Trac D. | last1 = Tran | first2 = Jie | last2 = Liang | first3 = Chengjie | last3 = Tu | url = http://thanglong.ece.jhu.edu/Tran/Pub/prepost.pdf | title = Lapped Transform via Time-Domain Pre- and Post-Filtering | journal = IEEE Transactions on Signal Processing | volume = 51 | issue = 6 | date = June 2003 | pages = 1557 | doi = 10.1109/TSP.2003.811222 | bibcode = 2003ITSP...51.1557T | accessdate = 2013-06-22 | url-status = dead | archiveurl = https://web.archive.org/web/20160304123116/http://thanglong.ece.jhu.edu/Tran/Pub/prepost.pdf | archivedate = 2016-03-04 }}</ref>
 Lapped transforms substantially reduce the blocking artifacts that otherwise occur with [[block transform coding]] techniques, in particular those using the [[discrete cosine transform]]. The best known example is the [[modified discrete cosine transform]] used in the [[MP3]], [[Vorbis]], [[Advanced Audio Coding|AAC]], and [[Opus (codec)|Opus]] [[audio codec]]s.<ref name=introducingdaala>{{cite web|url=http://people.xiph.org/~xiphmont/demo/daala/demo1.shtml|title=Next generation video: Introducing Daala|publisher=xiph.org|date=June 20, 2013}}</ref>
-Although the most well-known application of lapped transforms has been for audio coding, they have also been used for video and image coding and various other applications. They are used in video coding for coding [[I-frames]] in [[VC-1]] and for image coding in the [[JPEG XR]] format. More recently, a form of lapped transform has also been used in the development of the [[Daala (video codec)|Daala]] [[video codec]].<ref name=introducingdaala/>
+Although the best-known application of lapped transforms has been for audio coding, they have also been used for video and image coding and various other applications. They are used in video coding for coding [[I-frames]] in [[VC-1]] and for image coding in the [[JPEG XR]] format. More recently, a form of lapped transform has also been used in the development of the [[Daala (video codec)|Daala]] [[video coding format]].<ref name=introducingdaala/>
 == References ==
@@ Line 11: / Line 11: @@
 [[Category:Linear algebra]]
 [[Category:Discrete transforms]]
+{{Compression methods}}