IntroductionCompressionCompressible Image TypesJPEG TechniqueThe Discrete Cosine TransformQuantizationReorderingHuffman CodingComplete Compression ProcessConclusionIntroductionCompressionCompressible Image TypesJPEG TechniqueThe Discrete Cosine . . .QuantizationReorderingHuffman CodingComplete Compression . . .ConclusionHome PageTitle PageJJ IIJ IPage 1 of 17Go BackFull ScreenCloseQuitJPEG Image CompressionIan SnyderDecember 14, 2009AbstractThis paper will outline the process of JPEG image compression and the useof linear algebra as part of this process. It will introduce the reasons for imagecompression and clearly demonstrate each step of the process used by JPEG as asample image is compressed.IntroductionIn 1986 a committee known as the Joint Photographic Experts Group (JPEG) metto discuss and form a standard for image compression since most computers simplyweren’t capable of handling the large files required for storing images. A universalstandard was needed as well since products from different manufacturers of electronicsneeded to be interoperable [11]. A paper by the chairmain of the group, Gregory Wal-lace, on the JPEG’s compression process was subsequently published in 1991, outliningtheir proposed standard of image compression. Adopted in 1994, JPEG’s standard filecompression format has become so widespread that it is now one of the most commonimage file types on the web.IntroductionCompressionCompressible Image TypesJPEG TechniqueThe Discrete Cosine . . .QuantizationReorderingHuffman CodingComplete Compression . . .ConclusionHome PageTitle PageJJ IIJ IPage 2 of 17Go BackFull ScreenCloseQuitFigure 1: Generalized CompressionThis begs the question, why is JPEG’s standard so successful? The answer is sim-ple. The compression technique employed by JPEG allows a large image file to becompressed down to a much smaller size while retaining a substantial amount of theintegrity and quality of the image. The degree of compression in JPEG’s process canbe modified to suit the needs of the individual who is compressing the image, and itis possible to compress an image to one eighth or one ninth of its original size whileretaining enough quality for a decent image [1].CompressionFile compression basically has one goal: take a large file and condense it into a morecompact form for easier storage and/or transport. Image files are much larger thantext files on average, and therefore have a much greater need for compression. Figure 1briefly illustrates the idea behind compression of data for storage or transport and thenthe resulting decompression.Many image compression techniques have been developed along two main lines:‘Lossless’ compression and ‘Lossy’ compression. ‘Lossless’ image compression takesIntroductionCompressionCompressible Image TypesJPEG TechniqueThe Discrete Cosine . . .QuantizationReorderingHuffman CodingComplete Compression . . .ConclusionHome PageTitle PageJJ IIJ IPage 3 of 17Go BackFull ScreenCloseQuitimage data and compacts it as best it can while retaining all of the information. ‘Lossy’image compression, on the other hand, is just what it sounds like: information is lostduring compression. This technique generally results in better compression since someinformation deemed uneccessary is discarded, resulting in an image with a drasticallyreduced file size. Since information is lost, this compression method would seem toresult in an image of poorer quality, but the beauty of JPEG compression is that muchof the visual information that is discarded is imperceptible to the human eye, so theresulting image can virtually be of the same quality.Compressible Image TypesJPEG image compression is suitable for any type of bitmap image. Bitmap images arebasically m × n matrices (or layers of m × n matrices) with each entry in the matrixdetermining the color of one pixel in the image. Each entry, in turn, is represented bya given number of bits. If each pixel was represented by one bit, only two differentcolors could be represented since a computer working in binary could either assign thepixel a value of 0 or 1. If two bits were used to represent each pixel, four colors couldbe represented. The number of possible colors per pixel has 2npossibilities, with nrepresenting the number of bits per pixel. Images with either eight bits per pixel (onebyte) or twenty-four bits per pixel (three bytes) are most common [10].Three different types of bitmap images are generally used. The first type are in-tensity (grayscale) images, where each entry in the matrix has a value between zeroand one, with zero being pure white and one being completely black. Next, there are256-color images, where each entry in the matrix is a digit between 0 and 255 thatcorresponds to a distinct color. In this case each pixel will require eight bits, or onebyte, of storage. The third image type is called truecolor. In truecolor images threematrices are used: one for the shade of red, another for the shade of green, and a thirdfor the shade of blue (See Figure 2). Truecolor images are also known as RGB (Red,IntroductionCompressionCompressible Image TypesJPEG TechniqueThe Discrete Cosine . . .QuantizationReorderingHuffman CodingComplete Compression . . .ConclusionHome PageTitle PageJJ IIJ IPage 4 of 17Go BackFull ScreenCloseQuitGreen, Blue) images because of this. A truecolor image is made by layering these threem × n matrices on top of each other, producing the required color for each pixel [9].JPEG TechniqueThe method employed by JPEG exploits the limitations of the human eye to detectcolor and brightness changes in an image. When dealing with very bright or very dimcolors, the human eye cannot easily detect changes in the color from pixel to pixel,especially if the change is rather large. JPEG’s process takes advantage of this factand discards imperceptible changes in the color (or chrominance) of an image as well assome changes in the brightness (or luminance), which the human eye is a little betterat detecting. This basis for compression makes JPEG’s method ideal for compressingphotographs, which often have large variations in color and brightness from pixel topixel.There are four main steps in the process of JPEG compression: the Discrete CosineTransform (DCT), quantization, reordering, and Huffman coding. The purpose of theDiscrete Cosine Transform is to change the matrix from its pixel color values into amatrix of values that describe the change of color from pixel to