1 msu cse 803 fall 2014 vectors [and more on masks] vector space theory applies directly to several...
TRANSCRIPT
![Page 1: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/1.jpg)
1MSU CSE 803 Fall 2014
Vectors [and more on masks]
Vector space theory applies directly to several image
processing/representation problems
![Page 2: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/2.jpg)
2MSU CSE 803 Fall 2014
Image as a sum of “basic images”
What if every person’s portrait photo could be expressed as a sum of 20 special images? We would only need 20 numbers to model any photo sparse rep on our Smart card.
![Page 3: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/3.jpg)
3MSU CSE 803 Fall 2014
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”
![Page 4: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/4.jpg)
4MSU CSE 803 Fall 2014
The image as an expansion
![Page 5: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/5.jpg)
5MSU CSE 803 Fall 2014
Different bases, different properties revealed
![Page 6: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/6.jpg)
6MSU CSE 803 Fall 2014
Fundamental expansion
![Page 7: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/7.jpg)
7MSU CSE 803 Fall 2014
Basis gives structural parts
![Page 8: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/8.jpg)
8MSU CSE 803 Fall 2014
Vector space review, part 1
![Page 9: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/9.jpg)
9MSU CSE 803 Fall 2014
Vector space review, Part 2
2
![Page 10: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/10.jpg)
10MSU CSE 803 Fall 2014
A space of images in a vector space
M x N image of real intensity values has dimension D = M x N
Can concatenate all M rows to interpret an image as a D dimensional 1D vector
The vector space properties applyThe 2D structure of the image is
NOT lost
![Page 11: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/11.jpg)
11MSU CSE 803 Fall 2014
Orthonormal basis vectors help
![Page 12: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/12.jpg)
12MSU CSE 803 Fall 2014
Represent S = [10, 15, 20]
![Page 13: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/13.jpg)
13MSU CSE 803 Fall 2014
Projection of vector U onto V
![Page 14: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/14.jpg)
14MSU CSE 803 Fall 2014
Normalized dot product
Can now think about the angle between two signals, two faces, two text documents, …
![Page 15: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/15.jpg)
15MSU CSE 803 Fall 2014
Every 2x2 neighborhood has some constant, some edge, and some line component
Confirm that basis vectors are orthonormal
![Page 16: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/16.jpg)
16MSU CSE 803 Fall 2014
Roberts basis cont.
If a neighborhood N has large dot product with a basis vector (image), then N is similar to that basis image.
![Page 17: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/17.jpg)
17MSU CSE 803 Fall 2014
Standard 3x3 image basis
Structureless and relatively useless!
![Page 18: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/18.jpg)
18MSU CSE 803 Fall 2014
Frie-Chen basis
Confirm that bases vectors are orthonormal
![Page 19: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/19.jpg)
19MSU CSE 803 Fall 2014
Structure from Frie-Chen expansion
Expand N using Frie-Chen basis
![Page 20: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/20.jpg)
20MSU CSE 803 Fall 2014
Sinusoids provide a good basis
![Page 21: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/21.jpg)
21MSU CSE 803 Fall 2014
Sinusoids also model well in images
![Page 22: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/22.jpg)
22MSU CSE 803 Fall 2014
Operations using the Fourier basis
![Page 23: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/23.jpg)
23MSU CSE 803 Fall 2014
A few properties of 1D sinusoids
They are orthogonal
Are they orthonormal?
![Page 24: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/24.jpg)
24MSU CSE 803 Fall 2014
F(x,y) as a sum of sinusoids
![Page 25: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/25.jpg)
26MSU CSE 803 Fall 2014
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
![Page 26: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/26.jpg)
27MSU CSE 803 Fall 2014
Power spectrum from FT
![Page 27: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/27.jpg)
28MSU CSE 803 Fall 2014
Examples from images
Done with HIPS in 1997
![Page 28: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/28.jpg)
29MSU CSE 803 Fall 2014
Descriptions of former spectra
![Page 29: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/29.jpg)
30MSU CSE 803 Fall 2014
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.
![Page 30: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/30.jpg)
31MSU CSE 803 Fall 2014
Bandpass filtering
![Page 31: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/31.jpg)
32MSU CSE 803 Fall 2014
Convolution of two functions in the spatial domain is equivalent to pointwise multiplication in the frequency domain
![Page 32: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/32.jpg)
33MSU CSE 803 Fall 2014
LOG or DOG filter
Laplacian of GaussianApprox
Difference of Gaussians
![Page 33: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/33.jpg)
34MSU CSE 803 Fall 2014
LOG filter properties
![Page 34: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/34.jpg)
35MSU CSE 803 Fall 2014
Mathematical model
![Page 35: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/35.jpg)
36MSU CSE 803 Fall 2014
1D model; rotate to create 2D model
![Page 36: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/36.jpg)
37MSU CSE 803 Fall 2014
1D Gaussian and 1st derivative
![Page 37: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/37.jpg)
38MSU CSE 803 Fall 2014
2nd derivative; then all 3 curves
![Page 38: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/38.jpg)
39MSU CSE 803 Fall 2014
Laplacian of Gaussian as 3x3
![Page 39: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/39.jpg)
40MSU CSE 803 Fall 2014
G(x,y): Mexican hat filter
![Page 40: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/40.jpg)
41MSU CSE 803 Fall 2014
Convolving LOG with region boundary creates a zero-crossing
Mask h(x,y)
Input f(x,y) Output f(x,y) * h(x,y)
![Page 41: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/41.jpg)
42MSU CSE 803 Fall 2014
![Page 42: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/42.jpg)
43MSU CSE 803 Fall 2014
LOG relates to animal vision
![Page 43: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/43.jpg)
44MSU CSE 803 Fall 2014
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]
![Page 44: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/44.jpg)
45MSU CSE 803 Fall 2014
Experience the Mach band effect
![Page 45: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/45.jpg)
46MSU CSE 803 Fall 2014
Simple model of a neuron
![Page 46: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/46.jpg)
51MSU CSE 803 Fall 2014
Canny edge detector uses LOG filter
![Page 47: 1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems](https://reader035.vdocuments.mx/reader035/viewer/2022062715/56649d745503460f94a54db1/html5/thumbnails/47.jpg)
53MSU CSE 803 Fall 2014
Summary of LOG filter
Convenient filter shapeBoundaries detected as 0-
crossingsPsychophysical evidence that
animal visual systems might work this way (your testimony)
Physiological evidence that real NNs work as the ANNs