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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 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.

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## PowerPoint Slideshow about 'Introduction to the Mathematics of Image and Data Analysis' - Jims

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

Math 5467, Spring 2013

lerman@umn.edu

• Mathematical techniques (Fourier, wavelets, SVD, etc.)

• Problems from data analysis (mainly image analysis)

• 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

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 Concealment

Slide by W. Trappe (using the source codes provided by W.Zeng).

Starting point:

Questions:

• Effectiveness of reconstruction in different spaces

• “Reconstruction” of f from partial data

• Adaptive Reconstruction (not using one fixed basis)

• Decompositions

of Data…

• 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

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

dpi = dots per inch

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

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

• 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

x

I(x,y)

y

x

Image as a function

Based on slides by W. Trappe

• 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);