Introduction to the Mathematics of Image and Data Analysis Math 5467, Spring 2008 Instructor: Gilad Lerman [email protected].

Introduction to the Mathematics of Image and

Data Analysis

Math 5467, Spring 2008

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

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

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)

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);