1. Lempel–Ziv–Welch (LZW)
Universal lossless data compression algorithm
Created by Abraham Lempel, Jacob Ziv, and Terry Welch.
It was published by Welch in 1984 as an improved implementation of
the LZ78 algorithm published by Lempel and Ziv in 1978.
The algorithm is designed to be fast to implement but is not usually
optimal because it performs only limited analysis of the data.

2. Fixed Length: LZW Coding
Error Free Compression Technique
Remove Inter-pixel redundancy
Requires no priori knowledge of probability distribution of pixels
Assigns fixed length code words to variable length sequences
Patented Algorithm US 4,558,302
Included in GIF and TIFF and PDF file formats

3. The scenario described in Welch's 1984 paper[1] encodes sequences of 8-
bit data as fixed-length 12-bit codes.
The codes from 0 to 255 represent 1-character sequences consisting of
the corresponding 8-bit character, and the codes 256 through 4095 are
created in a dictionary for sequences encountered in the data as it is
encoded.
At each stage in compression, input bytes are gathered into a sequence
until the next character would make a sequence for which there is no
code yet in the dictionary.
The code for the sequence (without that character) is emitted, and a new
code (for the sequence with that character) is added to the dictionary.

4. LZW Coding Coding Technique
A codebook or a dictionary has to be constructed
For an 8-bit monochrome image, the first 256 entries are assigned to the gray levels 0,1,2,..,255.
As the encoder examines image pixels, gray level sequences that are not in the dictionary are assigned to a new entry.
For instance sequence can be assigned to entry 256, the address following the locations reserved for gray levels 0 to 255.

5. Dictionary Location Entry
LZW Coding
Example
Consider the following 4 x 4 8 bit image
Dictionary Location Entry
0 0
1 1
. .
Initial Dictionary

6. Dictionary Location Entry
LZW Coding
Is 39 in the dictionary……..Yes
What about 39-39………….No
Then add in entry 256
And output the last recognized symbol…39
Dictionary Location Entry
0 0
1 1
. .
39-39

7. Workshop Code the following image using LZW codes 39 39 126 126
* How can we decode the compressed sequence to obtain the original image ?
Software Research

8. LZW Coding

9. Vector Quantization: Definitions
X={x1,x2,…,xN } is set of input vectors in d-D space
c={c1,c2,…,cM} is set of code vectors in the space
P is a partition of the space into M code cells (cluster)
C={C1,C2,…,CM}           

10. Structure of vector quantizer
Code cells (clusters) are non-overlapping regions so
that the input space is completely covered.
A codevector cj is assigned to each code cell (cluster) Cj.
Vector quantization maps each input vector xi in code cell
(cluster) Cj to the code vector: xi cj 
Set of code vectors is codebook         

11. Structure of vector quantizer
Code cells (clusters) are non-overlapping regions so
that the input space is completely covered.
A codevector cj is assigned to each code cell (cluster) Cj.
Vector quantization maps each input vector xi in code cell
(cluster) Cj to the code vector: xi cj 
Set of code vectors is codebook         

12. Encoding/Decoding with codebook xi  cj
Distortion measure:
Image
Codebook
Image
Codebook j xi cj
xiCj

13. Distortion with measure L2
Distortion (quantization error) D for a cluster Cj:
Distortion D for data X: 
Optimal partition:

14. The optimal partition into clusters
1. Nearest neighbor condition:
Point is assigned to the nearest codevector
2. Centroid condition:
Code vector cj is centroid of the cluster Cj:
Voronoi cells

15. Code book generation Random code book
GLA, LBG, k-means, c-means algorithm
Merge approach: Pairwise Nearest Neighbor method
Split approach
Split-and-Merge approach
Stochastic Relaxation, simulated annealing
Genetic algorithm
Stuctured quantizers
Lattice quantizers

16. GLA: Generalized Lloyd algorithm
Other names: LBG, k-means, or c-mean algorithm
GLA (X,P,C): returns (P,C)
REPEAT
FOR i:=1 TO N DO // For each point xi
j  Find_Nearest_Centroid(xi,C)
FOR j:=1 TO M DO // For each cluster Cj
cj  Calculate_Centroid(X,Cj)
UNTIL no improvement.      C      P   C  P                          

17. Complexity of GLA 1) For each point xi find the nearest centroid Cj
The number of operation: O(NM)
2) For each cluster Cj calculate centroid cj
The number of operation: O(N)
Totally: O(kMN), where k is the number of iterations
Data format: use (d+1)th coordinate xi,d+1 to store the
cluster number ni which the point xi belongs to: xiCn
x1: x1,1, x1,2, …, x1,d, x1,d+1=n1
x2: x2,1, x2,2, …, x2,d, x2,d+1 =n1 …
xN: xN,1, xN,2, …, xN,d, xN,d+1=nN

18. Wavelet transform: wavelet mother function
How to obtain a set of wavelet functions?
Translation () and dilation (scaling, s)

19. Scaling (stretching or compressing)

20. Translation (shift)

21. Examples of mother wavelets

22. Discrete Wavelet Transform
Subband Coding
The spectrum of the input data is decomposed into a set of bandlimitted components, which is called subbands
Ideally, the subbands can be assembled back to reconstruct the original spectrum without any error
The input signal will be filtered into lowpass and highpass components through analysis filters
The human perception system has different sensitivity to different frequency band
The human eyes are less sensitive to high frequency-band color components
The human ears is less sensitive to the low-frequency band less than 0.01 Hz and high-frequency band larger than 20 KHz

23. Subband Transform
Separate the high freq. and the low freq. by subband decomposition

24. Subband Transform
Filter each row and downsample the filter output to obtain two N x M/2 images.
Filter each column and downsample the filter output to obtain four N/2 x M/2 images

25. Haar wavelet transform
Average : resolution
Difference : detail
Example for one dimension

26. Haar wavelet transform
Example: data=( )
-average:(5+7)/2, (6+5)/2, (3+4)/2, (6+9)/2
-detail coefficients:
(5-7)/2, (6-5)/2, (3-4)/2, (6-9)/2
n’= ( | )
n’’= (23/4 22/4 | )
n’’’= (45/8 | 1/ )

27. Haar wavelet transform

28. Subband Transform The standard image wavelet transform
The Pyramid image wavelet transform

29. Subband Transform

30. Wavelet Transform and Filter Banks
h0(n) is scaling function, low pass filter (LPF)
h1(n) is wavelet function, high pass filter (HPF)
is subsampling (decimation)

31. 2-D Wavelet transform
Horizontal filtering
Vertical filtering

32. Subband Transform

33. 2-D wavelet transform
LL3
HH4
LH2
HH3
LH1
HL2
HH2
HL1
HH1

34. Wavelet Coding
Software Research

35. Wavelet Transform 1 2 3 4
Put a pixel in each quadrant- No size change

36. Wavelet Transform Now let a = (x1+x2+x3+x4)/4 b =(x1+x2-x3-x4)/4
c =(x1+x3-x2-x4)/4
d =(x1+x4-x2-x3)/4 c d
Software Research

37. Wavelet Transform

38. Wavelet Transform

39. Wavelet Transform

40. 2-D Wavelet Transform via Separable Filters
From Matlab Wavelet Toolbox Documentation

41. 2-D Example

42. Wavelet Transform

43. JPEG 2000 Image Compression Standard

44. JPEG 2000 JPEG 2000 is a new still image compression standard
”One-for-all” image codec:
* Different image types: binary, grey-scale, color,
multi-component
* Different applications: natural images, scientific,
medical remote sensing text, rendered graphics
* Different imaging models: client/server, consumer
electronics, image library archival, limited buffer
and resources.

45. History Call for Contributions in 1996
The 1st Committee Draft (CD) Dec. 1999
Final Committee Draft (FCD) in March 2000
Accepted as Draft International Standard in Aug. 2000
Published as ISO Standard in Jan. 2002

46. Key components Transform Wavelet Wavelet packet Wavelet in tiles
Quantization
Scalar
Entropy coding
(EBCOT) code once, truncate anywhere
Rate-distortion optimization
Context modeling
Optimized coding order

47. Key components Visual Weighting Masking Region of interest (ROI)
Lossless color transform
Error resilience
The codestream obtained after compression of an image with JPEG 2000 is scalable in nature, meaning that it can be decoded in a number of ways; for instance, by truncating the codestream at any point, one may obtain a representation of the image at a lower resolution, or signal-to-noise ratio

48. JPEG 2000: A Wavelet-Based New Standard
4/11/2017
JPEG 2000: A Wavelet-Based New Standard
Targets and features
Excellent low bit rate performance without sacrifice performance at higher bit rate
Progressive decoding to allow from lossy to lossless
For details
David Taubman: “High Performance Scalable Image Compression with EBCOT”, IEEE Trans. On Image Proc, vol.9(7), 7/2000.
JPEG2000 Tutorial by IEEE Sig. Proc Magazine 9/2001
Taubman’s book on JPEG 2000 (on library reserve)
Links and
Ref: see also JPEG 2000 Tutorial by Christopolous et IEEE Trans. on consumer electronics 11/00

49. Superior compression performance: at high bit rates, where artifacts become nearly imperceptible, JPEG 2000 has a small machine-measured fidelity advantage over JPEG.
At lower bit rates (e.g., less than 0.25 bits/pixel for grayscale images), JPEG 2000 has a much more significant advantage over certain modes of JPEG: artifacts are less visible and there is almost no blocking
Multiple resolution representation: JPEG 2000 decomposes the image into a multiple resolution representation in the course of its compression process.
This representation can be put to use for other image presentation purposes beyond compression as such.

50. Progressive transmission by pixel and resolution accuracy, commonly referred to as progressive decoding and signal-to-noise ratio (SNR) scalability: JPEG 2000 provides efficient code-stream organizations which are progressive by pixel accuracy and by image resolution (or by image size).
This way, after a smaller part of the whole file has been received, the viewer can see a lower quality version of the final picture.
The quality then improves progressively through downloading more data bits from the source.

53. 2-D wavelet transform Original Transform Coeff.
128, 129, 125, 64, 65, …
Transform Coeff.
4123, -12.4, -96.7, 4.5, …

54. Quantization of wavelet coefficients
Transform Coeff.
4123, -12.4, -96.7, 4.5, …
Quantized Coeff.(Q=64)
64, 0, -1, 0, …

55. Entropy coding 0 1 1 0 1 1 0 1 0 1 . . . Coded Bitstream
Quantized Coeff.(Q=64)
64, 0, -1, 0, …

56. EBCOT
Key features of EBCOT: Embedded Block Coding with Optimized Truncation
Low memory requirement in coding and decoding
Easy rate control
High compression performance
Region of interest (ROI) access
Error resilience
Modest complexity

57. Block structure in EBCOT
Encode each block separately & record the bitstream of each block.
Block size is 64x64.

58. Progressive encoding

59. ROI: Region of interest
Scale-down the coefficients outside the ROI so those are
in lowerer bit-planes.
Decoded or refined ROI bits before the rest of the image.

60. ROI: Region of interest
Sequence based code
ROI coefficients are coded as independent sequences
Allows random access to ROI without fully decoding
Can specify exact quality/bitrate for ROI and the BG
Scaling based mode:
Scale ROI mask coefficients up (decoder scales down)
During encoding the ROI mask coefficients are found significant at early stages of the coding
ROI always coded with better quality than BG
Can't specify rate for BG and ROI

61. Tiling
Image  Component  Tile  Subband  Code-Block  Bit-Planes

62. JPEG 2000 vs JPEG
DCT WT

63. JPEG 2000 vs JPEG: Quantization

64. JPEG 2000 vs JPEG: Bitrate=0.3 bpp
MSE= MSE=73
PSNR=26.2 db PSNR=29.5 db

65. JPEG 2000 vs JPEG: Bitrate=0.2 bpp
MSE= MSE=113
PSNR=23.1 db PSNR=27.6 db

66. Examples JPEG2K vs. JPEG

69. - 17 Mbps data rate through 1994, 20
- 17 Mbps data rate through 1994, 20.4 Mbps from 1995 to 2004, 16 bit from 2005 onward
- 10 bit data encoding through 1994, 12 bit from 1995
- Silicon (Si) detectors for the visible range, indium gallium arsenide (InGaAr) for the NIR, and indium-antimonide (InSb) detectors for the SWIR
- "Whisk broom" scanning
- 12 Hz scanning rate
- Liquid Nitrogen (LN2) cooled detectors
- 10 nm nominal channel bandwidth, calibrated to within 1 nm
- 34 degrees total field of view (full 677 samples)
- 1 milliradian Instantaneous Field Of View (IFOV, one sample), calibrated to within 0.1 mrad
- 76GB hard disk recording medium

76. The End