Introduction to the
Mathematics of Image and
Data Analysis
Math 5467, Spring 2015
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
– 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
View more recent results
Problem 3: Inpainting
20th century slide by W. Trappe (using the source codes provided by W.Zeng):
(a) original lenna image
(b) corrupted lenna image
25% blocks in a checkerboard
pattern are corrupted
See 21th century inpainting
(c) concealed lenna image
corrupted blocks are concealed
via edge-directed interpolation
Problems from mathematics
Starting point:
f ( x)  n1 an  en ( x), e.g. en ( x)  exp(inx).

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
(if time allows)
Grade
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•
•
•
•
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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: M*N*k
• Number of bits to store an M*N color image with L=2k levels: 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)
Subsampling
Resampling
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
Image as a function
250
intensity
200
150
y
100
50
I(x,y)
0
100
80
y
60
50
rows
40
20
0
0
columns
x
x
Based on slides by W. Trappe
Clearer Example
Few Matlab Commands
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•
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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);