JPEG compression steps in detail:-

• Step1: Converting the RGB image in to Grey scale image:

 The original RGB color image is first converted to black and white image.

 To compress a grey-scale image in JPEG, each pixel value is translated into grey value (0-255).

 A color image is composed of pixels; these pixel values have RGB values, so to compress RGB color image, the work is three times as much because JPEG treats each color component separately.

 The R (red) component is compressed first, the G (green) component and lastly B (blue) component.

 The black and white image allows greater compression without a significant effect on image quality.

• Step2: Apply DCT (Discrete Cosine Transform):

 In this step, grey scale image with 8*8 block image is divided into 8-pixel by 8-pixel block shown in Fig1 below.

Fig1: Input Matrix of DCT coefficient

 This image is given as an input to DCT formula.

DCT Calculation:

• The formula for discrete cosine transfer (creating DCT coefficient) is as follows:

  $DCT (i,j) = \frac{1}{\sqrt{2N}} C(i) C(j) \sum \sum pixel (x,y) cos[\frac{(2x + 1) i \pi}{2N}] cos[\frac{(2y + 1) j \pi}{2N}]$ ---- (i)

• The formula for inverse discrete cosine transform (restoring original pixel information from a DCT coefficient) is:

  $Pixel (x,y) = \frac{1}{\sqrt{2N}} \sum \sum C(i) C(j) DCT (i,j) cos[\frac{(2x + 1) i \pi}{2N}] cos[\frac{(2x + 1) j \pi}{2N}]$ ---- (ii)

 Then DCT coefficients are generated by applying DCT formula on 8*8 block using Equation 1 shown above.

 After applying DCT formula, we will get 8*8 output matrix showing DCT coefficients as shown in Fig2 below.

Fig2: Output Matrix of DCT coefficient

• Step3: Quantization:

 It is the process of reducing precision of an integer, thereby reducing the number of bits required to store the integer.

 Quantization is the process to determine what information can be safely discarded with significant loss in visual fidelity.

 The quantized coefficient is described by the following equation: Quantized coefficient (i, j) = DCT (i, j)/Quantum (i, j)

 Quantized coefficient is result of dividing a DCT output matrix by a quantum matrix to generate quantized DCT values.

 Typically, JPEG quantum matrix has large values in lower half of quantum matrix which in turn results in zero quantized coefficients in lower half of quantized matrix.

 Quantized matrix is shown below in Fig3.

 Therefore, matrix ends up with most DCT values as zeros and some non-zero DCT values.

• Step4: Zig-Zag sequence:

 JPEG came up with ordering quantized DCT coefficient in zigzag sequence as shown in figure below.

 After the quantization step, most of the coefficients towards the lower right corner are zero.

 The zigzag ordering rearranges the coefficients in one dimensional order, in result most of the zeros will be placed at the end of the stream.

• Step5: Entropy encoding:

 Entropy encoding is done using Huffman coding.

 In Huffman coding, the most frequent character gets the smallest code and the least frequent character gets the largest code.