1. A Simple Image Compression : JPEG
Overall Process :
The image is broken into 8x8 blocks of pixels
Working from Left to Right and Top to Bottom, Discrete Cosine Transform(DCT) is applied to each of the 8x8 blocks.
Each Block is compressed through Quantization
The array of compressed blocks that constitute the image is stored in a drastically reduced amount of space.
The image is reconstructed by the use if Inverse Discrete Cosine Transform(IDCT).

2. DCT Computation
The DCT Coefficient D(i,j) for the (i,j) pixel is computed as follows:

3. DCT Computation
For a standard (8x8) block as used in jpeg, N=8 and (x,y) range from 0 to 7. Hence, the DCT computation has reduced to:

4. Matrix Representation of the DCT Equation
Let Tij be computed as :

5. Matrix form of Tij
For an (8x8) block, it results in the matrix:

6. DCT Computation on an (8x8) Block

7. DCT Computation

8. DCT Computation

9. DCT Computation

10. Quantization

11. Quantization

12. Quantization Levels

13. Quantization Levels

14. Quantization

15. Quantization
Zeros represent less important higher frequencies that can be discarded.

16. Coding

17. Decompression

18. Inverse DCT and Decompression

19. Comparison

20. Finally ….

21. Further Levels