Vectors [and more on masks]

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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 persons 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.
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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 defs., part 1
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Vector space defs. Part 2
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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 apply
• 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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Spatial direction and frequency in 2D
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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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(No Transcript)
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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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Output conditioning threshold versus smoother
output signal
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3D situation in the eye
Neuron c has input to neuron A but - input to
neuron B. Neuron d has input to neuron B but
input to neuron A. Neuron b gives no input to
neuron B it is not in the receptive field of B.
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Receptive fields
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Experiments with cats/monkeys

• Stabilize/drug animal to stare
• Place delicate probe in visual network
• Move step edge across FOV
• Probe shows response function when the edge
  images to receptive field
• Slightly moving the probe produces similar signal
  when edge is nearby

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Canny edge detector uses LOG filter
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Summary of LOG filter

• Convenient filter shape
• Boundaries detected as 0-crossings
• Psychophysical evidence that animal visual
  systems might work this way (your testimony)
• Physiological evidence that real NNs work as the
  ANNs