Introduction to the Mathematics of Image and Data Analysis

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# Introduction to the Mathematics of Image and Data Analysis - PowerPoint PPT Presentation

Introduction to the Mathematics of Image and Data Analysis. Math 5467, Spring 2013 Instructor: Gilad Lerman [email protected] What’s the course is about?. Mathematical techniques (Fourier, wavelets, SVD, etc.) Problems from data analysis (mainly image analysis). Digital Images and Problems.

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

Math 5467, Spring 2013

[email protected]

• 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

(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).

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)
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)
• Sparse approximation and compressed sensing
• 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)

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 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

y

x

I(x,y)

y

x

Image as a function

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);