Introduction to the Mathematics of Image and Data Analysis Math 5467, Spring 2008 Instructor: Gilad...
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Transcript of Introduction to the Mathematics of Image and Data Analysis Math 5467, Spring 2008 Instructor: Gilad...
Introduction to the Mathematics of Image and
Data Analysis
Math 5467, Spring 2008
Instructor: Gilad Lerman
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
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)
1( ) ( ), e.g. ( ) exp( ).n n nnf x a e x e x inx
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…
Common in Digital CameraCharged-Couple Device (CCD)
photodiode
Digital Image Acquisition
Sampling and Quantization
Basic Notation and DefinitionBasic 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)
SubsamplingSubsampling
ResamplingResampling
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
y
x
020
4060
80
0
50
1000
50
100
150
200
250
columnsrows
inte
nsity
y x
I(x,y)
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);