Introduction to the Mathematics of Image and Data Analysis

1. Introduction to the Mathematics of Image and Data Analysis Math 5467, Spring 2013 Instructor: Gilad Lerman lerman@umn.edu

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 • 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 http://www.ee.princeton.edu/~lixin/denoising.htm

6. (a) original lenna image (b) corrupted lenna image (c) concealed lenna image 25% blocks in a checkerboard pattern are corrupted corrupted blocks are concealed via edge-directed interpolation Problem 3: Error Concealment 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)

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) • Sparse approximation and compressed sensing

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: http://www.math.umn.edu/~lerman/math5467

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)

19. Subsampling

20. Resampling

21. Back to Compression • Color image of 600x800 pixels • 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. y x I(x,y) y x Image as a function 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);