The task of compression consists of two components, an encoding algorithm that takes a file and generates a “compressed” representation (hopefully with fewer bits), and a decoding algorithm that reconstructs the original file or some approximation of it from the compressed representation. These two components are typically intricately tied together since they both have to understand the shared compressed representation. We distinguish between lossless algorithms, which can reconstruct the original file exactly from the compressed file, and lossy algorithms, which can only reconstruct an approximation of the original file.

The discrete cosine transforms is a technique for converting a data (image pixels) into sets frequency. Therefore, The key to the compression process discussed here is a mathematical transformation known as the Discrete Cosine Transform (DCT).Among the emerging of compression techniques standards are JPEG, for compression of still images.

Before beginning, it should be noted that the pixel values of a black-and-white image rang from 0 to 255, where pure black is represented by 0, and pure white by 255. Since an image comprises hundreds or even thousands of blocks of pixels, the following description of what happens to one block is a microcosm of the JPEG process; what is done to one block of image pixels is done to all of them, in the order earlier specified.

For Example Original Lena Image 256x256

Now, let's start with a block of image-pixel values.

Now, the DCT is applied on the above matrix by using the equation : This yields the Following Matrix. This block matrix now consists of 64 DCT coefficients, The top-left coefficient, D00, correlates to the low frequencies of the original image block.

Quantization The 8x8 block of DCT coefficients is now ready for compression by quantization. A remarkable and highly useful feature of JPEG process is that in this step.

Quantization is achieved by dividing each element in the transformed image matrix D by the corresponding element in the quantization matrix, and then rounded to the nearest integer.

Recall that the coefficients situated near the upper-left corner correspond to the lower frequencies-to which the human eye is most sensitive-of the image block. In addition, the zeros represent the less important, higher frequencies that have been discarded, giving rise to the lossy part of compression.

Encoding After quantization, it is not unusual for more than half of the DCT coefficients to equal zero. JPEG incorporates run-length coding to take advantage of this. For each non- zero DCT coefficient, JPEG records the number of zeros that preceded the number, the number of bits needed to represent the number's amplitude, and the amplitude itself. To consolidate the runs of zeros, JPEG processes DCT coefficients in the zigzag pattern shown in figure.

Zigzag sequence

Decompression Reconstruction of image begins by decoding the bit stream representing the compressed matrix C. Each element of matrix C is then multiplied by the corresponding element of the quantization matrix originally used. The IDCT equation is : Where

is next applied to matrix R, which is rounded to the nearest integer, giving us the decompressed of original 8*8 image block

OriginalDecompression

Original Lena ImageReconstruct Lena Image