Introduction to the Mathematics of Image and Data Analysis Math 5467, Spring 2008 Instructor: Gilad Lerman lerman@umn.edu

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

Digital Images and Problems

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

Problem 2: Denoising From X.Li http://www.ee.princeton.edu/~lixin/denoising.htm

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).

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

Beyond Functions… Decompositions of Data…

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)

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

What’s a Digital Image?

Mechanism for digitizing

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

Digital Image Acquisition

Sampling and Quantization

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

Effect of Quantization

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 http://en.wikipedia.org/wiki/Pixels_per_inch and http://en.wikipedia.org/wiki/Dots_per_inch (the textbook does not make this distinction)

Subsampling

Resampling

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

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

Clearer Example

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);