1. Image Coding/ Compression
David Hemmert Pradeep Suthram Tammo Heeren
All Mathcad files [MCD/PDF] can be found on:

2. Overview Review DCT (Discrete Cosine Transform)
JPEG compression/ decompression
Wavelet compression/ decompression

3. Review Linear Quantization
Quantization of gray levels in equidistance quantization steps

4. Review adaptive Quantization

5. DCT Discrete Cosine transform
Transformation of spatial image information into its spatial frequency components f f

6. DCT Math

7. DCT
Essentially taking the 2D fourier transform and only keeping the real part of the coefficients
Works with any orthogonal kernel (e.g. in wavelet compression/ decompression)
DCT used in JPEG coding/ decoding

8. DCT Results 0.005%/ 22000 / 8.8 dB 0.1%/ 864/ 10.3 dB

9. SNR and visual artifact
Procedure/ Transform
SNR of no visual artifacts
Compression
ratio
Linear quantization
35 dB
1.6
Adaptive quantization
31 dB 2
DCT
34 dB
JPEG
Wavelet
9.3

10. JPEG compression of Lenna
512 X 512 pixels
1 pixel = 8 bits
64 bytes = 8 x 8 submatrix = block
4096 submatrices
total/elements total

11. JPEG Algorithm Discrete Cosine Transform of every element
8x8 pixel block
DCT
Level-shift
Quantizer
Encoder
Data
Discrete Cosine Transform of every element
Gray scale image level-shifted by –128
for n = 8, 2^(n-1) = 128

12. using a typical normalization matrix
JPEG Algorithm
Quantization
using a typical normalization matrix [ ]
Normalization using a standard table

13. JPEG Algorithm Zig-zag pattern Removal of zeros Convert to binary
Compare the number of bits used

14. Discrete Wavelet Transform (DWT)
Orthogonal Basis
Area of basis equals zero
Low pass / High pass filtering scheme to generate basis coefficients
Compression by reducing the number of coefficient (zeroing least significant coefficients)

15. Common Orthogonal Wavelet Bases
Haar wavelet
(averaging)
Mexican Hat wavelet
(2nd derivative of Gaussian distribution)
Daub4 wavelet
(most common used)

16. (Daub4 Nth order matrix)
Filtering Scheme
Lowpass
N-1
times
Highpass
DWT
(Daub4 Nth order matrix)
Row 1
Pixels
(l to r)
Row 1
Coefficients
Row 1

17. Coefficients for First Row DWT Transformation

18. Generating Coefficients
DWT each row
Regroup coefficients into Low/Hi subvectors
DWT all columns of transformed matrix
Low-Low
Hi-Hi
90 % of the
coefficients
zeroed

19. Application to “Lenna”
original
50% coefficients
10% coefficients
2% coefficients
0.5% coefficients
0.1% coefficients

20. Signal to Noise Ratio 2%

21. References
Rafael C. Gonzalez, Richard E. Wood, “Digital Image Processing”, Addison Wesley, 1993
Geoffrey M. Davis, Aria Nosratinia, “Wavelet-based Image Coding: An Overview”,
Subhasis, Saha, “Image Compression - from DCT to Wavelets : A Review”,
Weidong Kou, “Digital Image Compression Algorithms and Standards,” Kluwer Academic Publishers, 1995.
“Selected Papers on Image Coding and Compression,” Majid Rabbani, Ed., Brian J. Thompson, Gen. Ed., SPIE Milestone Series, Vol MS-48, SPIE Optical Engineering Press, 1992.
“Fractal Image Compression Theory and Application,” Yuval Fisher, Ed., Springer-Verlag New York, 1995.
Bernd Jaehne, “Digital Image Processing”, Third Edition, Springer-Verlag, New York 1995