1. Digital Image Processing Lecture 21: Lossy Compression
Prof. Charlene Tsai

2. Reminder …
The Thursday class (5/29) is moved to Monday (5/28) for next week.

3. JPEG Algorithm
Lossy compression trades some acceptable data loss for greater rate of compression.
There are many available, but the most popular is the one developed by JPEG – transform coding.
Coding not done on pixel values, but on a transform.

4. Transform Discrete Cosine Transform (DCT)
Applying to 8x8 blocks, the forward and inverse DCT are
DCT are real-valued, high information-packing capability, and separable.

5. Comparison with FFT Given the following sequence
If using FFT, the inverse of the first 4 FFT coefficients gives
If using DCT, the result is

6. JPEG Compression For each 8x8 image block, perform the compression
Subtracting 128 from each value
Apply DCT
Normalization by dividing by matrix Q (the lossy part)
Turning most of the elements zero
Change the matrix into a vector by reading the nonzero element in a zigzag fashion. O …

7. (con’d)
The first element (DC coefficient) of the vector is encoded as the difference between itself and the DC of previous block.
Keeps all values (except the very first one) small
Compressed using RLE
Other values (AC coefficients) are compressed using Huffman coding
Rate of compression is controlled by scaling Q => more scaling, more compression

8. Decompression Inverse the operations: Decode Huffman encoding and RLE
Put the vector back to 8x8 matrix
Multiply by Q
Inverse DCT
Shift back by 128

9. The Lossless Parts
Huffman Coding and RLE for AC and DC terms, respectively.
We’ll focus the discussion on the AC terms
Each nonzero value x is assigned a category k
The number of preceding 0s (run).
Values:
Category:
Run:

10. Huffman Code Table Part of JPEG baseline standard
See the handout for the code table and application.

11. Wavelet Transform
For both FT and DCT, we assume so kind of periodicity in the image.
Wavelet Transform keeps the wave concept, but drop the eriodicity.
Wavelet is a little part of a wave
wave
wavelet

12. What to do with a wavelet?
Given a wavelet function , we can
Dilate it by applying a scaling factor to x.
e.g would squash and would expand.
Translate by adding/subtracting a value from x. e.g shift the wavelet 2 to the right, and shift 3 to the left.
Change it height by multiplying the function by a constant.
All together, we get

13. Applications Noise reduction Edge detection Compression
Adopted by JPEG2000

14. A Simple Wavelet Transform
All wavelet transform work by taking weighted averages of input values and providing extra information for inversion.
Here the example is: averaging of two values and differencing.
If a and b are two numbers, we get average s and difference d by:

15. (cont)
Given a simple vector v of 8 elements, we create two new vectors v1 and v2 of 4 elements each:
v1 containing the averages
v2 containing the differences
DWT at 1 scale is:
We can keep going for another 2 levels.
Let’s go through the example in pg425 of the handout. (available online)

16. Haar Wavelet Simplest wavelet, defined as
Haar wavelet can be written in terms of a simpler pulse function

17. Mother wavelet
Father wavelet (scaling function)

18. Discrete Wavelet transform
The forward transformation
The inverse is

19. are called the filter coefficients (or taps)
A wavelet is completely specified by its taps.

20. Back to Haar Wavelet where values are what we have in a DWT matrix
Low pass
where
values are what we have in a DWT matrix
Let’s look at the example in pg 429.
High pass

21. Two-Dimensional Wavelet (STD)

22. Two-Dimensional Wavelet (non-STD)

23. Example: one scale

24. Image Compression
For a given value d, set all values x in the DWT for which to 0.

25. High-Pass Filtering
Except from the top left image, the rest is high-frequency information.
If setting the top-left corner to 0, the result after inversion would be a high-pass filtered image.

26. Denoising Very similar to compression by using thresholding.
An example is given in pg446.