1. Introduction to the Mathematics of Image and Data Analysis
Math 5467, Spring 2008
Instructor: Gilad Lerman

2. What’s the course is about?
Mathematical techniques (Fourier, wavelets, SVD, etc.)
Problems from data analysis (mainly image analysis)

3. Digital Images and Problems

4. Problem 1: Compression Color image of 600x800 pixels Movie
Without compression
1.44M bytes
After JPEG compression (popularly used on web)
only 89K bytes
compression ratio ~ 16:1
Movie
Raw video ~ 243M bits/sec
DVD ~ about 5M bits/sec
Compression ratio ~ 48:1
“Library of Congress” by M.Wu (600x800)
Based on slides by W. Trappe

5. Problem 2: Denoising
From X.Li

6. Problem 3: Error Concealment
25% blocks in a checkerboard pattern are corrupted
corrupted blocks are concealed via edge-directed interpolation
(a) original lenna image
(c) concealed lenna image
(b) corrupted lenna image
Slide by W. Trappe (using the source codes provided by W.Zeng).

7. Problems from mathematics
Starting point:
Questions:
Effectiveness of reconstruction in different spaces
“Reconstruction” of f from partial data
Adaptive Reconstruction (not using one fixed basis)
By “reconstruction” of f from partial data, I mean what can we tell about f from any partial information about the coefficients a_n

8. Beyond Functions…
Decompositions
of Data…

9. Class plan Quick introduction to images
Singular value decomposition (adaptive representation)
Hilbert spaces and normed spaces
Basic Fourier analysis and image analysis in the frequency domain
Convolution and low/high pass spatial filters
Image restoration
Wavelet analysis
Image compression (if time allows)

10. Grade 10% Homework 10% Project 10% Class Participation
20% Exam 1 (date may change)
20% Exam 2 (date may change)
30% Final Exam
More Class Info:

11. What’s a Digital Image?

12. Mechanism for digitizing

13. Examples of Sensors Well known from physics classes… photodiode
Common in Digital Camera
Charged-Couple Device (CCD)

14. Digital Image Acquisition

15. Sampling and Quantization

16. Basic Notation and Definition
Image is a function f(xi,yj), i=1,…,N, j=1,…,M
Image = matrix ai,j = f(xi,yj)
In gray level image: range of values 0,1,….,L-1, where L=2k.
(these are k-bits images, most commonly k=8)
Number of bits to store an M*N image with L=2k levels:
Number of bits to store an M*N color image with L=2k levels:
M*N*k
3*M*N*k

17. Effect of Quantization

18. Effect of Sampling dpi = dots per inch
(top left image is 3692*2812 pixels & 1250dpi)
bottom right image is 213*162 pixels & 72dpi)
Some people make the distinction between dpi for printer and ppi (pixels per inch) for computer display
See and
(the textbook does not make this distinction)

19. Subsampling

20. Resampling

21. Back to Compression Color image of 600x800 pixels Movie
Without compression
(600*800 pixels) * (24 bits/pixel) = 11.52M bits = 1.44M bytes
After JPEG compression (popularly used on web)
only 89K bytes
compression ratio ~ 16:1
Movie
720x480 per frame,
30 frames/sec,
24 bits/pixel
Raw video ~ 243M bits/sec
DVD ~ about 5M bits/sec
Compression ratio ~ 48:1
“Library of Congress” by M.Wu (600x800)
Based on slides by W. Trappe

22. Image as a function y x
I(x,y) y x
Based on slides by W. Trappe

23. Clearer Example

24. Few Matlab Commands imread (from file to array)
imshow(‘filename’), image/sc(matrix)
colormap(‘gray’)
imwrite (from array to a file)
Subsampling B = A(1:2:end,1:2:end);