Editing JPEG 2000 (section)

===Wavelet transform===
[[File:Wavelet Bior2.2.svg|thumb|[[Cohen-Daubechies-Feauveau wavelet|CDF]] 5/3 wavelet used for lossless compression]]
[[File:Jpeg2000 2-level wavelet transform-lichtenstein.png|thumb|256px|An example of the wavelet transform that is used in JPEG&nbsp;2000. This is a 2nd-level CDF 9/7 [[wavelet transform]].]]

These tiles are then [[wavelet transform|wavelet-transformed]] to an arbitrary depth, in contrast to JPEG&nbsp;1992 which uses an 8×8 block-size [[discrete cosine transform]]. JPEG&nbsp;2000 uses two different [[wavelet]] transforms:

# ''irreversible'': the [[Cohen-Daubechies-Feauveau wavelet|CDF]] 9/7 wavelet transform (developed by [[Ingrid Daubechies]]).<ref name="Unser">{{cite journal |last1 = Unser |first1 = M. |last2=Blu |first2=T. |title = Mathematical properties of the JPEG2000 wavelet filters |journal=IEEE Transactions on Image Processing |year = 2003 |volume=12 |issue=9 |pages=1080–1090 |doi=10.1109/TIP.2003.812329 |pmid=18237979 |bibcode=2003ITIP...12.1080U |s2cid=2765169 |url=https://pdfs.semanticscholar.org/6ed4/dece8b364416d9c390ba53df913bca7fb9a6.pdf |archive-url=https://web.archive.org/web/20191013222932/https://pdfs.semanticscholar.org/6ed4/dece8b364416d9c390ba53df913bca7fb9a6.pdf |archive-date=2019-10-13 }}</ref> It is said to be "irreversible" because it introduces quantization noise that depends on the precision of the decoder.
# ''reversible'': a rounded version of the biorthogonal Le Gall–Tabatabai (LGT) 5/3 wavelet transform<ref>{{cite web |last=Sullivan |first=Gary |title=General characteristics and design considerations for temporal subband video coding |publisher=[[Video Coding Experts Group]] |website=[[ITU-T]] |date=8–12 December 2003 |url=https://www.itu.int/wftp3/av-arch/video-site/0312_Wai/VCEG-U06.doc |access-date=13 September 2019}}</ref><ref name="Unser" /><ref>{{cite book |last=Bovik |first=Alan C. |title=The Essential Guide to Video Processing |year=2009 |publisher=[[Academic Press]] |isbn=9780080922508 |page=355 |url=https://books.google.com/books?id=wXmSPPB_c_0C&pg=PA355 }}</ref> (developed by Didier Le Gall and Ali J. Tabatabai).<ref>{{cite conference |last1=Le Gall |first1=Didier |last2=Tabatabai |first2=Ali J. |title=Sub-band coding of digital images using symmetric short kernel filters and arithmetic coding techniques |conference=ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing |date=1988 |pages=761–764 |volume=2 |doi=10.1109/ICASSP.1988.196696 |s2cid=109186495 }}</ref> It uses only integer coefficients, so the output does not require rounding (quantization) and so it does not introduce any quantization noise. It is used in lossless coding.

The wavelet transforms are implemented by the [[lifting scheme]] or by [[convolution]].