1. Discrete Wavelet Transform (DWT)
Presented by
Sharon Shen
UMBC

2. Overview Introduction to Video/Image Compression DWT Concepts
Compression algorithms using DWT
DWT vs. DCT
DWT Drawbacks
Future image compression standard
References

3. Need for Compression
Transmission and storage of uncompressed video would be extremely costly and impractical.
Frame with 352x288 contains 202,752 bytes of information
Recoding of uncompressed version of this video at 15 frames per second would require 3 MB. One minute180 MB storage. One 24-hour day262 GB
Using compression, 15 frames/second for 24 hour1.4 GB, 187 days of video could be stored using the same disk space that uncompressed video would use in one day

4. Principles of Compression
Spatial Correlation
Redundancy among neighboring pixels
Spectral Correlation
Redundancy among different color planes
Temporal Correlation
Redundancy between adjacent frames in a sequence of image

5. Classification of Compression
Lossless vs. Lossy Compression
Lossless
Digitally identical to the original image
Only achieve a modest amount of compression
Lossy
Discards components of the signal that are known to be redundant
Signal is therefore changed from input
Achieving much higher compression under normal viewing conditions no visible loss is perceived (visually lossless)
Predictive vs. Transform coding

6. Classification of Compression
Predictive coding
Information already received (in transmission) is used to predict future values
Difference between predicted and actual is stored
Easily implemented in spatial (image) domain
Example: Differential Pulse Code Modulation(DPCM)

7. Classification of Compression
Transform Coding
Transform signal from spatial domain to other space using a well-known transform
Encode signal in new domain (by string coefficients)
Higher compression, in general than predictive, but requires more computation (apply quantization)
Subband Coding
Split the frequency band of a signal in various subbands

8. Classification of Compression
Subband Coding (cont.)
The filters used in subband coding are known as quadrature mirror filter(QMF)
Use octave tree decomposition of an image data into various frequency subbands.
The output of each decimated subbands quantized and encoded separately

9. Discrete Wavelet Transform
The wavelet transform (WT) has gained widespread acceptance in signal processing and image compression.
Because of their inherent multi-resolution nature, wavelet-coding schemes are especially suitable for applications where scalability and tolerable degradation are important
Recently the JPEG committee has released its new image coding standard, JPEG-2000, which has been based upon DWT.

10. Discrete Wavelet Transform
Wavelet transform decomposes a signal into a set of basis functions.
These basis functions are called wavelets
Wavelets are obtained from a single prototype wavelet y(t) called mother wavelet by dilations and shifting:
(1)
where a is the scaling parameter and b is the shifting parameter

11. Discrete Wavelet Transform
Theory of WT
The wavelet transform is computed separately for different segments of the time-domain signal at different frequencies.
Multi-resolution analysis: analyzes the signal at different frequencies giving different resolutions
MRA is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies
Good for signal having high frequency components for short durations and low frequency components for long duration.e.g. images and video frames

12. Discrete Wavelet Transform
Theory of WT (cont.)
Wavelet transform decomposes a signal into a set of basis functions.
These basis functions are called wavelets
Wavelets are obtained from a single prototype wavelet y(t) called mother wavelet by dilations and shifting:
(1)
where a is the scaling parameter and b is the shifting parameter

13. Discrete Wavelet Transform
The 1-D wavelet transform is given by :

14. Discrete Wavelet Transform
The inverse 1-D wavelet transform is given by:

15. Discrete Wavelet Transform
Discrete wavelet transform (DWT), which transforms a discrete time signal to a discrete wavelet representation.
it converts an input series x0, x1, ..xm, into one high-pass wavelet coefficient series and one low-pass wavelet coefficient series (of length n/2 each) given by:

16. Discrete Wavelet Transform
where sm(Z) and tm(Z) are called wavelet filters, K is the length of the filter, and i=0, ..., [n/2]-1.
In practice, such transformation will be applied recursively on the low-pass series until the desired number of iterations is reached.

17. Discrete Wavelet Transform
Lifting schema of DWT has been recognized as a faster approach
The basic principle is to factorize the polyphase matrix of a wavelet filter into a sequence of alternating upper and lower triangular matrices and a diagonal matrix .
This leads to the wavelet implementation by means of banded-matrix multiplications

18. Discrete Wavelet Transform
Two Lifting schema:

19. Discrete Wavelet Transform
where si(z) (primary lifting steps) and ti(z) (dual lifting steps) are filters and K is a constant.
As this factorization is not unique, several {si(z)}, {ti(z)} and K are admissible.

20. Discrete Wavelet Transform
2-D DWT for Image

21. Discrete Wavelet Transform

22. Discrete Wavelet Transform
2-D DWT for Image

23. Discrete Wavelet Transform
Integer DWT
A more efficient approach to lossless compression
Whose coefficients are exactly represented by finite precision numbers
Allows for truly lossless encoding
IWT can be computed starting from any real valued wavelet filter by means of a straightforward modification of the lifting schema
Be able to reduce the number of bits for the sample storage (memories, registers and etc.) and to use simpler filtering units.

24. Discrete Wavelet Transform
Integer DWT (cont.)

25. Discrete Wavelet Transform
Compression algorithms using DWT
Embedded zero-tree (EZW)
Use DWT for the decomposition of an image at each level
Scans wavelet coefficients subband by subband in a zigzag manner
Set partitioning in hierarchical trees (SPHIT)
Highly refined version of EZW
Perform better at higher compression ratio for a wide variety of images than EZW

26. Discrete Wavelet Transform
Compression algorithms using DWT (cont.)
Zero-tree entropy (ZTE)
Quantized wavelet coefficients into wavelet trees to reduce the number of bits required to represent those trees
Quantization is explicit instead of implicit, make it possible to adjust the quantization according to where the transform coefficient lies and what it represents in the frame
Coefficient scanning, tree growing, and coding are done in one pass
Coefficient scanning is a depth first traversal of each tree

27. Discrete Wavelet Transform
DWT vs. DCT

28. Discrete Wavelet Transform
Disadvantages of DCT
Only spatial correlation of the pixels inside the single 2-D block is considered and the correlation from the pixels of the neighboring blocks is neglected
Impossible to completely decorrelate the blocks at their boundaries using DCT
Undesirable blocking artifacts affect the reconstructed images or video frames. (high compression ratios or very low bit rates)

29. Discrete Wavelet Transform
Disadvantages of DCT(cont.)
Scaling as add-onadditional effort
DCT function is fixedcan not be adapted to source data
Does not perform efficiently for binary images (fax or pictures of fingerprints) characterized by large periods of constant amplitude (low spatial frequencies), followed by brief periods of sharp transitions

30. Discrete Wavelet Transform
Advantages of DWT over DCT
No need to divide the input coding into non-overlapping 2-D blocks, it has higher compression ratios avoid blocking artifacts.
Allows good localization both in time and spatial frequency domain.
Transformation of the whole image introduces inherent scaling
Better identification of which data is relevant to human perception higher compression ratio

31. Discrete Wavelet Transform
Advantages of DWT over DCT (cont.)
Higher flexibility: Wavelet function can be freely chosen
No need to divide the input coding into non-overlapping 2-D blocks, it has higher compression ratios avoid blocking artifacts.
Transformation of the whole image introduces inherent scaling
Better identification of which data is relevant to human perception higher compression ratio (64:1 vs. 500:1)

32. Discrete Wavelet Transform
Performance
Peak Signal to Noise ratio used to be a measure of image quality
The PSNR between two images having 8 bits per pixel or sample in terms of decibels (dBs) is given by:
PSNR = 10 log10
mean square error (MSE)
Generally when PSNR is 40 dB or greater, then the original and the reconstructed images are virtually indistinguishable by human observers

33. Discrete Wavelet Transform
Improvement in PSNR using DWT-JEPG over DCT-JEPG at S = 4

34. Discrete Wavelet Transform

35. Discrete Wavelet Transform
images.
Discrete Wavelet Transform
Compression ratios used for 8-bit 512x Lena image. 8 16 32 64
128
PSNR (dBs) performance of baseline JPEG using on Lena image.
38.00
35.50
31.70
22.00
2.00
PSNR (dBs) performance of Zero-tree coding using arithmetic coding on Lena image.
39.80
37.00
34.50
33.00
29.90
PSNR (dBs) performance of bi-orthogonal filter bank using VLC on Lena image.
35.00
34.00
32.50
28.20
26.90
PSNR (dBs) performance of bi-orthogonal filter bank using FLC on Lena image.
33.90
32.80
27.70
26.20
PSNR (dBs) performance of W6 filter bank using VLC on Lena image.
33.60
32.00
30.90
27.00
25.90
PSNR (dBs) performance of W6 filter bank using FLC on Lena image.
29.60
29.00
27.50
25.00
23.90
Comparison of image compression results using DCT and DWT

36. Discrete Wavelet Transform
Visual Comparison
(a)
(b)
(c)
(a) Original Image256x256Pixels, 24-BitRGB (b) JPEG (DCT) Compressed with compression ratio 43:1(c) JPEG2000 (DWT) Compressed with compression ratio 43:1

37. Discrete Wavelet Transform
Implementation Complexity
The complexity of calculating wavelet transform depends on the length of the wavelet filters, which is at least one multiplication per coefficient.
EZW, SPHIT use floating-point demands longer data length which increase the cost of computation
Lifting schemea new method compute DWT using integer arithmetic
DWT has been implemented in hardware such as ASIC and FPGA

38. Discrete Wavelet Transform
Resources of the ASIC used and data processing rates for DCT and DWT encoders
Type of encoders using ASIC
No. of Logic gates of the ASIC used
Amount of on chip RAM used by the encoders
Data processing rates of the encoders using ASIC
DCT
34000
128 byte
210 MSa/sec
DWT
55000
55 kbyte
150 MSa/sec

39. Discrete Wavelet Transform
Number of logic gates

40. Discrete Wavelet Transform
Processing Rate

41. Discrete Wavelet Transform
Disadvantages of DWT
The cost of computing DWT as compared to DCT may be higher.
The use of larger DWT basis functions or wavelet filters produces blurring and ringing noise near edge regions in images or video frames
Longer compression time
Lower quality than JPEG at low compression rates

42. Discrete Wavelet Transform
Future video/image compression
Improved low bit-rate compression performance
Improved lossless and lossy compression
Improved continuous-tone and bi-level compression
Be able to compress large images
Use single decompression architecture
Transmission in noisy environments
Robustness to bit-errors
Progressive transmission by pixel accuracy and resolution
Protective image security

43. Discrete Wavelet Transform
References
M. Martina, G. Masera , A novel VLSI architecture for integer wavelet transform via lifting scheme, Internal report, VLSI Lab., Politecnico diTor i no, Jan. 2000, unpublished.

44. Discrete Wavelet Transform
THANK YOU !
Q & A