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 [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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what s the course is about
What’s the course is about?
  • Mathematical techniques (Fourier, wavelets, SVD, etc.)
  • Problems from data analysis (mainly image analysis)
problem 1 compression
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
Problem 2: Denoising

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

problem 3 error concealment

(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
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
Beyond Functions…
  • Decompositions

of Data…

class plan
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
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
Examples of Sensors

Well known from physics classes…

photodiode

Common in Digital Camera

Charged-Couple Device (CCD)

slide16

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
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
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

I(x,y)

y

x

Image as a function

Based on slides by W. Trappe

few matlab commands
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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