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Introduction to the Mathematics of Image and Data Analysis

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### Introduction to the Mathematics of Image and Data Analysis

Math 5467, Spring 2013

Instructor: Gilad Lerman

What’s the course is about?

- Mathematical techniques (Fourier, wavelets, SVD, etc.)
- Problems from data analysis (mainly image analysis)

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

Problem 2: Denoising

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

(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 ConcealmentSlide 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)

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

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

Examples of Sensors

Well known from physics classes…

photodiode

Common in Digital Camera

Charged-Couple Device (CCD)

- 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 Sampling

dpi = dots per inch

(top left image is 3692*2812 pixels & 1250dpi)

bottom right image is 213*162 pixels & 72dpi)

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

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);

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