1. Filtering and enhancement of color images in the block DCT domain
Jayanta Mukhopadhyay
Dept. of Computer Science and Engg.

2. Processing with compressed image: Compresed domain approach
J. Mukhopadhyay, “Image and video processing in the compressed domain”, CRC Press, 2011.

3. Motivations Computation with reduced storage.
Avoid overhead of forward and reverse transform.
Exploit spectral factorization for improving the quality of result and speed of computation.
DCT domain processing under consideration.
Image Resizing

4. 2D DCT
Type-II Even:
Type-II DCT of x(m,n):

5. Useful properties of DCT blocks

6. 2D DCT: Sub-band relation
Sub-band approximation:
2D DCT of xLL(m,n)
Low-pass truncated approximation:
S.-H. Jung, S.K. Mitra, and D. Mukherjee, Subband DCT: Definition, analysis and applications. IEEE Trans. on Circuits and systems for VideoTechnology, 6(3):273–286, June 1996.

7. Image downsampling Sub-band approximation 8x8 8x8 8x8 8x8 4x4 4x4 4x4
J. Mukherjee and S.K. Mitra. Image resizing in the compressed domain using subband DCT. IEEE Transactions on Circuits and systems for Video Technology, 12(7):620–627, July 2002.

8. Image upsampling
Sub-band approximation
4x4
4x4
8x8

9. 2D DCT: Block composition and decomposition
J. Jiang and G. Feng. The spatial relationships of DCT coefficients between a block and its sub-blocks. IEEE Trans. on Signal Processing, 50(5):1160–1169, May 2002.

10. Block composition and decomposition
4x4
8x8

11. Image Resizing

12. Image Halving Use of linear and distributive properties. X00 X01 X10Xd

13. Not so sparse matrix multiplication!
DCT(p0): Not so sparse.
No gain!
DCT(p1)

14. Typical result:
Original
Bi-linear
Linear and distributive method

15. 2D DCT: Sub-band relation
Low-pass truncated approximation:

16. Block composition and decomposition
Block composition and decomposition
To convert M adjacent N-point DCT blocks to a single MxN-point DCT block.
NxN zero matrix

17. 2D DCT: Block composition and decomposition

18. Useful conversion for halving or doubling 8-point DCT blocks.
Composition
Decomposition

19. Image Halving: Approximation followed by Composition (IHAC)

20. Image Halving: Composition followed by Approximation (IHAC)

21. Image Doubling: Decomposition followed by Approximation (IDDA)x2

22. Image Doubling: Approximation followed by Decomposition (IDAD)x2

24. IDDA

25. IDAD

26. Resizing with integral factors
To convert NxN block to LNxMN block.
LN x MN block
NxN DCT block
LxM D/S (LMDS)
1. Merge LxM adjacent DCT blocks.
2. Sub-band approximation to a NxN DCT block.

27. LMDS

28. LxM U/S (LMUS) 1. Convert NxN to LNxMN block
Efficiently compute exploiting large blocks of zeroes.
2. Decompose into LxM NxN blocks.

29. LMUS

30. An example: 3x2 D/S and U/S

31. Arbitrary Resizing (P/Q x R/S)
U/S-D/S Resizing Algorithm (UDRA)
U/S by PxR
D/S by QxS
D/S-U/S Resizing Algorithm (DURA)
U/S PxR

32. HDTV (1080x920) to NTSC (480x640)
DURA
UDRA

33. Hybrid Resizing (HRA) More general sub-band relation
Truncated DCT block of X or padded with zeroes, if required.
X: DCT block of QNxSN
Y: DCT block of PNxRN

34. HRAS

35. HRAC

36. Original image (Watch)

37. HRAC: A few examples

38. UDRA
HRAS
HRAC

39. Color Image Resizing

40. Color encoding in JPEG
Y-Cb-Cr color space: Cb Y Cr

41. Baseline JPEG Compression: Usually  the chromatic components Cb and Cr are at lower resolution than the Y component.
 Cascaded stages of down-sampling and up-sampling(the DURA algorithm) faces a problem of dimensionality mismatch.

42. DURA

43. HRAS
HRAC

44. Thank you!