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# Image Compression:

## How Math Led to the JPEG2000 Standard

### Issues with JPEG

The JPEG standard is still the most popular standard for storing images on web servers and the resolution at which images are compressed using the standard is acceptable for applications such as web browsing. As the Digital Age continues, people who work with digital images identified some problems with JPEG as well as some amenities that would further improve the standard.

In no particular order, here are the issues with the JPEG Image Compression Standard:

• No Lossless Compression Option - For many applications (magazine advertisements, for example), loss of resolution is not acceptable.
• Decoupling of Images - Partitioning the image into 8 x 8 blocks means that blocks are viewed independently - they are transformed and quantized independent of each other. This is a disadvantage when trying to exploit the homogeneity of a region that might be larger than an 8 x 8 block.
• Block Effects - Since the image is partitioned into blocks, there is often not a smooth transition between blocks in the compressed image.
• Edge Effects - The DCT is built from samples of the cosine function and thus works best when input data are periodic. It is not typically the case that the intensities in rows or columns of a digital image are periodic.
• Global Transformation - The DCT is a global transformation. For example, consider the application of the 8 x 8 DCT matrix U to the 8-vector \bf{v} to obtain \bf{y} = U\bf{v}. Each element of y is constructed using every element of \bf{v}. So if there a perturbance in v, then it affects every element in y.

### Features and Enhancements

Most of the problems encountered by the DCT in JPEG were solved with the incorporation of the Discrete Wavelet Transformation in JPEG2000. The JPEG2000 standard uses two wavelet transformations - one is used for lossless compression and designed so that it maps integers to integers and the other is used lossy compression. Both are local transformations - the LeGall filter uses only 5 and 3 elements per inner products for the lowpass and highpass portions of the transformation, respectively and the CDF97 filter uses 9 and 7 elements per inner product for the lowpass and highpass portions respectively. Both filters are symmetric and as such, the transformations can be modified to better handle the processing of intensities that appear at the end of a row or column.

Here are some other features and enhancements of the JPEG2000 Image Compression Standard:

• Better CompressionIn many cases, we set the compression rate and then adjust the algorithm accordingly. The JPEG2000 produces higher-quality images for low bit rates (.025bpp and lower) than does JPEG.
• Progressive Signal Transmission JPEG2000 can reconstruct digital images (at an increasing rate of resolution) as they are received by a browser.
• Tiling JPEG2000 allows users to tile an image into non-overlapping blocks and then process each block in a manner similar to the JPEG standard. These tilings need not be 8 x 8 blocks.
• Regions of Interest Users of JPEG2000 can identify Regions of Interest in an image and encode these ROIs so that they have a higher resolution in the compressed version of the image.
• Larger Image Size The JPEG standard can handle images of size up to 64,000 x 64,000. JPEG2000 can handle images 4,294,967,295 x 4,294,967,295. (4,294,967,295 = 2^{32}-1).
• Multiple Channels The JPEG standard can support the compression of three channels (color images). JPEG2000 can support the compression of up to 256 channels. Such large channel numbers are common for satellite data.

 Wavelets and JPEG2000 Home Digital Image Basics Compression in a Nutshell Basic JPEG Wavelet Transformations JPEG2000 Features/Enhancements Quantization References and Links

Images courtesy of Radka Tezaur.