msu cse 803 fall 2014
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1 MSU CSE 803 Fall 2015
Vectors [and more on masks]
Vector space theory applies directly to several image processing/representation problems
2 MSU CSE 803 Fall 2015
Image as a sum of “basic images”
What if every person’s portrait photo could be expressed as a sum of 20 special images? è We would only need 20 numbers to model any photo è sparse rep on our Smart card.
3 MSU CSE 803 Fall 2015
Efaces
100 x 100 images of faces are approximated by a subspace of only 4 100 x 100 “images”, the mean image plus a linear combination of the 3 most important “eigenimages”
4 MSU CSE 803 Fall 2015
The image as an expansion
5 MSU CSE 803 Fall 2015
Different bases, different properties revealed
6 MSU CSE 803 Fall 2015
Fundamental expansion
7 MSU CSE 803 Fall 2015
Basis gives structural parts
8 MSU CSE 803 Fall 2015
Vector space review, part 1
9 MSU CSE 803 Fall 2015
Vector space review, Part 2
2
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A space of images in a vector space
n M x N image of real intensity values has dimension D = M x N
n Can concatenate all M rows to interpret an image as a D dimensional 1D vector
n The vector space properties apply
n The 2D structure of the image is NOT lost
11 MSU CSE 803 Fall 2015
Orthonormal basis vectors help
12 MSU CSE 803 Fall 2015
Represent S = [10, 15, 20]
13 MSU CSE 803 Fall 2015
Projection of vector U onto V
14 MSU CSE 803 Fall 2015
Normalized dot product
Can now think about the angle between two signals, two faces, two text documents, …
15 MSU CSE 803 Fall 2015
Every 2x2 neighborhood has some constant, some edge, and some line component
Confirm that basis vectors are orthonormal
16 MSU CSE 803 Fall 2015
Roberts basis cont.
If a neighborhood N has large dot product with a basis vector (image), then N is similar to that basis image.
17 MSU CSE 803 Fall 2015
Standard 3x3 image basis
Structureless and relatively useless!
18 MSU CSE 803 Fall 2015
Frie-Chen basis
Confirm that bases vectors are orthonormal
19 MSU CSE 803 Fall 2015
Structure from Frie-Chen expansion
Expand N using Frie-Chen basis
20 MSU CSE 803 Fall 2015
Sinusoids provide a good basis
21 MSU CSE 803 Fall 2015
Sinusoids also model well in images
22 MSU CSE 803 Fall 2015
Operations using the Fourier basis
23 MSU CSE 803 Fall 2015
A few properties of 1D sinusoids
They are orthogonal
Are they orthonormal?
24 MSU CSE 803 Fall 2015
F(x,y) as a sum of sinusoids
26 MSU CSE 803 Fall 2015
Continuous 2D Fourier Transform
To compute F(u,v) we do a dot product of our image f(x,y) with a specific sinusoid with frequencies u and v
27 MSU CSE 803 Fall 2015
Power spectrum from FT
28 MSU CSE 803 Fall 2015
Examples from images
Done with HIPS in 1997
29 MSU CSE 803 Fall 2015
Descriptions of former spectra
30 MSU CSE 803 Fall 2015
Discrete Fourier Transform
Do N x N dot products and determine where the energy is.
High energy in parameters u and v means original image has similarity to those sinusoids.
31 MSU CSE 803 Fall 2015
Bandpass filtering
32 MSU CSE 803 Fall 2015
Convolution of two functions in the spatial domain is equivalent to pointwise multiplication in the frequency domain
33 MSU CSE 803 Fall 2015
LOG or DOG filter
Laplacian of Gaussian Approx
Difference of Gaussians
34 MSU CSE 803 Fall 2015
LOG filter properties
35 MSU CSE 803 Fall 2015
Mathematical model
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1D model; rotate to create 2D model
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1D Gaussian and 1st derivative
38 MSU CSE 803 Fall 2015
2nd derivative; then all 3 curves
39 MSU CSE 803 Fall 2015
Laplacian of Gaussian as 3x3
40 MSU CSE 803 Fall 2015
G(x,y): Mexican hat filter
41 MSU CSE 803 Fall 2015
Convolving LOG with region boundary creates a zero-crossing
Mask h(x,y)
Input f(x,y) Output f(x,y) * h(x,y)
42 MSU CSE 803 Fall 2015
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LOG relates to animal vision
44 MSU CSE 803 Fall 2015
1D EX.
Artificial Neural Network (ANN) for computing
g(x) = f(x) * h(x)
level 1 cells feed 3 level 2 cells
level 2 cells integrate 3 level 1 input cells using weights [-1,2,-1]
45 MSU CSE 803 Fall 2015
Experience the Mach band effect
46 MSU CSE 803 Fall 2015
Simple model of a neuron
51 MSU CSE 803 Fall 2015
Canny edge detector uses LOG filter
53 MSU CSE 803 Fall 2015
Summary of LOG filter
n Convenient filter shape n Boundaries detected as 0-crossings n Psychophysical evidence that animal
visual systems might work this way (your testimony)
n Physiological evidence that real NNs work as the ANNs
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