msu cse 803 fall 2014
DESCRIPTION
Vectors [and more on masks]. Vector space theory applies directly to several image processing/representation problems. MSU CSE 803 Fall 2014. Image as a sum of “ basic images ”. - PowerPoint PPT PresentationTRANSCRIPT
1 MSU CSE 803 Fall 2015
Vectors [and more on masks]
Vector space theory applies directly to several image processing/representation problems
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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.
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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”
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The image as an expansion
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Different bases, different properties revealed
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Fundamental expansion
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Basis gives structural parts
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Vector space review, part 1
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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
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Orthonormal basis vectors help
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Represent S = [10, 15, 20]
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Projection of vector U onto V
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Normalized dot product
Can now think about the angle between two signals, two faces, two text documents, …
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Every 2x2 neighborhood has some constant, some edge, and some line component
Confirm that basis vectors are orthonormal
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Roberts basis cont.
If a neighborhood N has large dot product with a basis vector (image), then N is similar to that basis image.
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Standard 3x3 image basis
Structureless and relatively useless!
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Frie-Chen basis
Confirm that bases vectors are orthonormal
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Structure from Frie-Chen expansion
Expand N using Frie-Chen basis
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Sinusoids provide a good basis
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Sinusoids also model well in images
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Operations using the Fourier basis
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A few properties of 1D sinusoids
They are orthogonal
Are they orthonormal?
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F(x,y) as a sum of sinusoids
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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
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Power spectrum from FT
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Examples from images
Done with HIPS in 1997
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Descriptions of former spectra
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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.
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Bandpass filtering
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Convolution of two functions in the spatial domain is equivalent to pointwise multiplication in the frequency domain
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LOG or DOG filter
Laplacian of Gaussian Approx
Difference of Gaussians
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LOG filter properties
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Mathematical model
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1D model; rotate to create 2D model
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1D Gaussian and 1st derivative
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2nd derivative; then all 3 curves
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Laplacian of Gaussian as 3x3
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G(x,y): Mexican hat filter
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Convolving LOG with region boundary creates a zero-crossing
Mask h(x,y)
Input f(x,y) Output f(x,y) * h(x,y)
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LOG relates to animal vision
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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]
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Experience the Mach band effect
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Simple model of a neuron
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Canny edge detector uses LOG filter
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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