Editing JPEG 2000 (section)

===Coding===
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The result of the previous process is a collection of ''sub-bands'' which represent several approximation scales. A sub-band is a set of ''coefficients''—[[real numbers]] which represent aspects of the image associated with a certain frequency range as well as a spatial area of the image.

The quantized sub-bands are split further into ''precincts'', rectangular regions in the wavelet domain. They are typically sized so that they provide an efficient way to access only part of the (reconstructed) image, though this is not a requirement.

Precincts are split further into ''code blocks''. Code blocks are in a single sub-band and have equal sizes—except those located at the edges of the image. The encoder has to encode the bits of all quantized coefficients of a code block, starting with the most significant bits and progressing to less significant bits by a process called the ''EBCOT'' scheme. ''EBCOT'' here stands for ''Embedded Block Coding with Optimal Truncation''. In this encoding process, each [[bit plane]] of the code block gets encoded in three so-called ''coding passes'', first encoding bits (and signs) of insignificant coefficients with significant neighbors (i.e., with 1-bits in higher bit planes), then refinement bits of significant coefficients and finally coefficients without significant neighbors. The three passes are called ''Significance Propagation'', ''Magnitude Refinement'' and ''Cleanup'' pass, respectively.

In lossless mode all bit planes have to be encoded by the EBCOT, and no bit planes can be dropped.

The bits selected by these coding passes then get encoded by a context-driven binary [[Arithmetic coding|arithmetic coder]], namely the binary MQ-coder (as also employed by [[JBIG2]]). The context of a coefficient is formed by the state of its eight neighbors in the code block.

The result is a bit-stream that is split into ''packets'' where a ''packet'' groups selected passes of all code blocks from a precinct into one indivisible unit. Packets are the key to quality scalability (i.e., packets containing less significant bits can be discarded to achieve lower bit rates and higher distortion).

Packets from all sub-bands are then collected in so-called ''layers''.
The way the packets are built up from the code-block coding passes, and thus which packets a layer will contain, is not defined by the JPEG&nbsp;2000 standard, but in general a codec will try to build layers in such a way that the image quality will increase monotonically with each layer, and the image distortion will shrink from layer to layer. Thus, layers define the progression by image quality within the codestream.

The problem is now to find the optimal packet length for all code blocks which minimizes the overall distortion in a way that the generated target bitrate equals the demanded bit rate.

While the standard does not define a procedure as to how to perform this form of [[rate–distortion optimization]], the general outline is given in one of its many appendices: For each bit encoded by the EBCOT coder, the improvement in image quality, defined as mean square error, gets measured; this can be implemented by an easy table-lookup algorithm. Furthermore, the length of the resulting codestream gets measured. This forms for each code block a graph in the rate–distortion plane, giving image quality over bitstream length. The optimal selection for the truncation points, thus for the packet-build-up points is then given by defining critical ''slopes'' of these curves, and picking all those coding passes whose curve in the rate–distortion graph is steeper than the given critical slope. This method can be seen as a special application of the method of ''[[Lagrange multiplier]]'' which is used for optimization problems under constraints. The [[Lagrange multiplier]], typically denoted by λ, turns out to be the critical slope, the constraint is the demanded target bitrate, and the value to optimize is the overall distortion.

Packets can be reordered almost arbitrarily in the JPEG&nbsp;2000 bit-stream; this gives the encoder as well as image servers a high degree of freedom.

Already encoded images can be sent over networks with arbitrary bit rates by using a layer-progressive encoding order. On the other hand, color components can be moved back in the bit-stream; lower resolutions (corresponding to low-frequency sub-bands) could be sent first for image previewing. Finally, spatial browsing of large images is possible through appropriate tile or partition selection. All these operations do not require any re-encoding but only byte-wise copy operations.{{Citation needed|date=December 2022}}