The Process of Quantization

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Quantization refers to the process of approximating the continuous set of values in the image data with a finite (preferably small) set of values. The input to a quantizer is the original data, and the output is always one among a finite number of levels. The quantizer is a function whose set of output values are discrete, and usually finite. Obviously, this is a process of approximation, and a good quantizer is one which represents the original signal with minimum loss or distortion.
A quantizer simply reduces the number of bits needed to store the transformed coefficients by reducing the precision of those values. Since this is a many-to-one mapping, it is a lossy process and is the main source of compression in an encoder. Quantization can be performed on each individual coefficient, which is known as Scalar Quantization (SQ).
There are two types of quantization - Scalar Quantization and Vector Quantization. In scalar quantization, each input symbol is treated separately in producing the output, while in vector quantization the input symbols are clubbed together in groups called vectors, and processed to give the output. This clubbing of data and treating them as a single unit increases the optimality of the vector quantizer, but at the cost of increased computational complexity.
Coefficients that corresponds to smooth parts of data become small. (Indeed, their difference, and therefore their associated wavelet coefficient, will be zero, or very close to it). So we can throw away these coefficients without significantly distorting the image. We can then encode the remaining coefficients and transmit them along with the overall average value.
Discrete Wavelet Transform (DWT):
A discrete wavelet transformation apply on an image consists of four frequency bands as given in figure (4.1). The top- left corner of the transformed image is the LL band of original image, low-frequency coefficients. The bottom-right corner "HH" contains residual diagonal frequencies. The main reason for use DWT, for reduce an image dimensions, and the high-frequencies coefficients are ignored (i.e. not used in this work), this will effects on the image quality, while this process increasing compression ratio.

Figure (4.1): Original Image After Discrete Wavelet Transform Application
JPEG Technique:
JPEG is a lossy image compression method for color or grayscale images. An important feature of JPEG is its use the quality parameter "Quality", allowing the user to adjust the amount of the data lost over a very wide range. In this work we apply this technique on the sub-image LL, and the image quality problem are solved in the (4.

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MLA Citation:
"The Process of Quantization." 19 Jan 2018
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1). The JPEG technique explained in the following steps:
1. The pixels of an image are organized in groups of 8×8 pixels called data units, and each data unit is compressed separately. If the number of image rows or columns is not a multiple of 8, the bottom row and the rightmost column are padded with zeros as many times as necessary.
2. The discrete cosine transform (DCT) is then applied on the each group of 8×8 pixels to create an 8×8 map of frequency domain. The frequency domain consists of; DC coefficient at the first location in 8×8 frequency component, which is represents average pixels value.
3. Each of the 64 frequency components in an 8×8 map are divided by a separate numbers its called Quantization Coefficient (QC), and then rounded to an integer.
A simple quantization table Q is computed, based on one parameter R specified by the user. A simple expression such as Qij = 1+(i + j) × R guarantees that QCs start small at the upper-left corner and get bigger toward the lower-right corner [6].
4. Each 8×8 map frequency component scanned by using zigzag scan, to create a one dimensional array contains 64 quantized frequency coefficients. The one-dimensional array encoded using Huffman coding.
The figure (4.2) shows that compression by JPEG technique The parameter R in QC computed by selecting maximum coefficient from an 8x8 frequency components. But in the decompression part, we obtain low quality image (degraded image).

Figure (4.2): JPEG Technique Applied on Subimage
Image Quality and Coding:
Before apply compression on an image, we want to improve image quality. The image quality obtains from the difference between original sub-image (LL) and decoded sub-image (LLDecode ), as shown in the following figure (4.3):
The difference matrix in the above figure called D-Matrix. To compress D-Matrix, at first we minimize the size for the matrix and then compress each row by Huffman encoding and RLE, the following figure shows compression process for D-Matrix:

Figure (4.3): Difference matrix obtained from subtraction between original LL and LLDecode.

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