1. Department of Electrical and Computer Engineering
EEL 6562 Image Processing and Computer Vision Lecture 4 Image Quantization
Dr. Dapeng Oliver Wu
University of Florida
Department of Electrical and Computer Engineering
Fall 2016

2. Outline Introduction Uniform quantization
Minimum Mean Squared Error (MMSE) quantization
Scalar quantization
Vector quantization
Machine learning, data mining, bioinformatics, source coding, computer vision, pattern recognition, artificial intelligence
Implementation
Visual quantization
Half toning

3. Image Sampling
Sampling by
i = 0 Δx 1 …
N - 1 Δy y’ x’
j = 0 1 2
M-1

4. Examples i
N-1
M-1 j

5. Matrix Representation
fmin ≤ f(i, j) ≤ fmax , value in a continuum
Popular image size
N = 64, M = 64 (26), # of pixels = 4,096.
N = 512, M = 512 (29), # of pixels = 262,144.
N = 1280, M = 1024, # of pixels = 1,310,720.

6. Purposes of Quantization
Analog to digital conversion (ADC)
Use a finite number of (discrete) values to represent infinite number of (continuous) values
Compression:
Use less number of (discrete) values to represent a continuous/discrete signal
It is like election in human society
Question: how to choose representative values and the regions being represented?
There are infinite number of ways. Which is optimal?

7. Quantization
fmax
fmin

8. A Quantizer Q(f) Quantizer f Q(f) r8 r7
Decision Levels {tk, k = 1, …, L+1} r6
Reconstruction Levels {rk, k = 1, …, L} r5 If r4 f
Then Q(f) = rk t1 t2 t3 t4 t5 t6 t7 t8
L levels need bits r3
returns the smallest integer
that is bigger than or equal to x r2
Quantizer
error r1 r0 r0 r1 r2 r3 r4 r5 r6 r7 r8 f t1 t2 t3 t4 t5 t6 t7 t8

9. Matrix Representation
For grayscale image, 256 levels or 8 bits/pixel is sufficient for most applications
For color image, each component (R, G, B) needs 256 levels or 8 bits/pixel (bpp)
Storage for typical images
512 x 512, 8 bits grayscale image: 262,144 Bytes
1024 x 768, 24 bits true color image: 2,359,296 Bytes

10. Uniform Quantization
Equal distances between adjacent decision levels and between adjacent reconstruction levels
tl – tl-1 = rl – rl-1 = q
Parameters of Uniform Quantization
L: levels (L = 2R)
B: dynamic range B = fmax – fmin
q: quantization interval (stepsize), which is uniform
q = B/L = B2-R
Quantization function
Note:
returns the biggest
integer that is smaller
than or equal to x

11. A Uniform Quantizer Q(f) r7=fmax-q/2 r6 r5 r4 stepsize q=(fmax-fmin)/8
r0=fmin+q/2 f t0 t1 t2 t3 t4 t5 t6 t7 t8
fmin
fmax

12. Truncated Uniform Quantizer
Signal with infinite dynamic range
Truncate the lower and higher values to fmin and fmax
Parameters
fmin, fmax
L (R)
Quantization function

13. A Truncated Uniform Quantizer
overload
region
Q(f)
r7=fmax-q/2 r6 r5 r4 r3 r2 r1
r0=fmin+q/2 f
t0 =-∞
fmin t1 t2 t3 t4 t5 t6 t7
fmax
t8 =∞
overload
region

14. Example 1 of Uniform Quantizer
The output of a CCD camera is in the range of 0.0 to 5.0 volt
L = 256
q = 5 / 256 (volt)
The output value in the interval (l * q, (l + 1) * q] is represented by l, l = 0, …, 255.
The reconstruction level rl = l * q + q/2, l = 0,…, 255.

15. Example 2 of Uniform Quantizer
Digital Image of 256 gray levels
Quantize it into 4 levels
fmin= 0,
fmax = 256,
q = 256 / 4 = 64,
q/2 = 32,
Compress the image from 8 bits/pixel to 2 bits/pixel. The compression ratio is 4.

16. Examples of Uniform Quantizer
q=8, L=32
Original, L=256
q=16, L=16
q=64, L=4

17. Performance Measures for a Quantizer
Mean squared error (MSE)
Peak Signal to Noise Ratio (PSNR)

18. Implementation of Uniform Quantization
Setup the quantization function Q(f) for all possible input level first.
Matlab code
x = imread('lena_gray.bmp');
[height, width] = size(x);
B = 256;
x = double(x);
% This is the quantization table
Q = zeros(256, 1);
% Quantized to 16 levels
L = 16;
q = B / L;
for i = 0:255,
Q(i+1, 1) = floor(i / q) * q + q /2;
end
y = zeros(size(x));
for i = 1:height,
for j = 1:width,
y(i, j) = Q(x(i,j) + 1);
MSE = mean(mean((x-y).^2))
PSNR =10*log10(255^2/MSE) % in dB
y=uint8(y);
imshow(y)
L = 16, MSE = 21.6, PSNR=34.8 dB

19. Quantization Effect – False Contour
1-D Signal
2-D Image
f(t) t t
f(i,j)=max(I,j)

20. Minimum Mean Square Error Quantizer
Mean squared error (MSE) of a quantizer
Where p(f) is the probability density function of f
Uniform distribution
Gaussian distribution
Laplacian distribution
MSE for a specific image
Minimize E in terms of {tk} and {rk}

21. Minimum Mean Square Error Quantizer
MMSE Quantizer
Nearest neighbor
condition
Centriod
condition

22. MMSE Quantizer for A Uniform Source

23. MMSE Quantizer Example for a Uniform Source
Original Image
L=64, MSE=1.507
L=16, MSE=21.624
L=2, MSE=
L=8, MSE=83.725
L=4, MSE=368.50

24. MMSE Quantizer for a Non-uniform Source
Following conditions may not have a closed form solution
Numerical procedures are necessary
Tables 4.1 and 4.2 in Jain’s book give the optimal tl and rl for Gaussian and Laplacian sources with unit variance and zero mean.
Anil K. Jain, ``Fundamentals of digital image processing,'' Englewood Cliffs, NJ : Prentice Hall, 1989. 

25. MMSE Quantizer Example for a Gaussian Source
Original Image
PDF: Gaussian
μf=128, σf=36
Uniform Quantization
L=16, MSE=22.73
MMSE Quantization
L=16, MSE=14.35

26. Types of Quantization Scalar quantization: Vector quantization
Quantize a scalar random variable into representative scalars
The scalar random variable can be continuous or discrete
Vector quantization
Quantize a random vector into representative vectors
The random vector can be continuous or discrete

27. Llyod Algorithm When the distribution of the signal is unknown
The quantizer is designed based on a training set containing representative samples
tl and rl are determined iteratively, based on the nearest neighbor and centroid conditions, respectively.

28. Lloyd Algorithm (for scalar quantization)
Iterative algorithms for determining MMSE quantizer parameters
Can be based on a pdf or training data
Iterate between centroid condition (election) and nearest neighbor condition (partitioning)

29. Generalized Lloyd Algorithm (for vector quantization)
Start with initial codewords
Iterate between centroid condition (senator election) and nearest neighbor condition (constituency partitioning)

41. Visual Quantization Reduce the artifact of false contour
Better visual perception (Human eyes are very sensitive to edges)
Might produce worse MSE

42. Pseudo Random Noise Quantizerf
Uniform Quantizer
Q()
Step size: q + g + + + + - h
Random Noise
Generator
Uniform in (-q/2, q/2)

43. Example 8 Level Pseudo Random 8 Level Uniform Quantizer
MSE=85.84
8 Level Pseudo Random
Noise Quantizer
MSE=86.09

44. Halftoning Halftoning Use 2 levels to render a multi-level image
Create the illusion of continuous-tone pictures on bi-level printers or display devices.
E.g, representing gray scale image using 2 levels (black and white)
CMYB color printer
(using cyan, magenta, yellow, black toners)
Black/white printer
(using black toner)
monochrome monitor

45. Dithering Halftoning is achieved by dithering Two Level Quantizer f +fw +
(Multiple levels) - fb
(Two levels; applied to one dot)
Dither Signal d
For a B/W printer, 600 dpi (dots per inch); one square inch has 360,000 dots. # of dots in black determines gray level.

46. Types of Dithering
Ordered dithering
Error-diffusion-based dithering

47. d Ordered Dither Signal
Dither signal d(m,n) is obtained by repeating a kxk dither matrix Dk(m,n)
d(m, n) = Dk([m]k, [n]k),
where [m]k means m modulo k.
Criteria for the dither matrix
Contains sufficiently
distinct values
Similar values should be separated
as far as possible Dk d Dk Dk Dk Dk Dk … … …

48. Recursive Design of the Dither Matrix
Define D2
Recursion
where Uk is a kxk matrix of 1’s.
Usage
Dk needs to be scaled and shifted, q=256/2k

49. Example K = 4, q = 16 fb=64 fw=192 Halftoning with k=4, q=16
2 Level Uniform Quantization
Halftoning with k=4, q=16

50. Halftoning Using Error Diffusion+ -
e[n-1]
Delay

51. Quantization Summary
Typically L = 28 = 256 and we have log2(L) = log2(28) = 8 bit quantization.
From now on omit references to fmin; fmax and unless otherwise stated assume that the original digital images are quantized to 8 bits or 256 levels.
To denote this refer to as taking integer values k where 0≤k≤255, i.e., let us say that {0, …, 255}

52. Homework 2
Using Matlab or C or any other computer language, to implement uniform quantization of the grayscale image in “YourLastNameGray.jpg” obtained in HW1. Your program should do the following: i) read a grayscale image into an array; ii) quantize and save the quantized image in a different array; iii) compute the MSE between the original and quantized images; and iv) display and print the quantized image. Your program should allow you to vary the reconstruction level (L). Record the MSE obtained with L=64, 32, 16, 8, and save the quantized images with corresponding L values, MSE values, and PSNR values.
Your submission should include 1) your computer program file and 2) an MS WORD file that contains L values, MSE values, PSNR values, and the images. Please put the images in the WORD file, instead of in JPEG files. JPEG files are not allowed to upload as separate files.

53. Homework 2
Put your solution in a single WORD file or a single pdf file.
Submit your homework in the format of WORD file, Matlab file, or pdf through E-Learning web site under the directory of Homework 2.
Please submit your homework in separate files (e.g., *.m or *.doc or *.pdf). Please do not submit zip files or JPEG files. JPEG files should be embedded in a WORD file or pdf file.
Due: 4pm, 9/14

54. Reading
“Digital Image Processing”, Chapter 2 (Section – 2.4.3)