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MSU CSE 803 Stockman Fall 2009
Vectors [and more on masks]
Vector space theory applies directly to several image processing/representation
problems
MSU CSE 803 Stockman Fall 2009
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.
MSU CSE 803 Stockman Fall 2009
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”
MSU CSE 803 Stockman Fall 2009
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
MSU CSE 803 Stockman Fall 2009
Normalized dot product
Can now think about the angle between two signals, two faces, two text documents, …
MSU CSE 803 Stockman Fall 2009
Every 2x2 neighborhood has some constant, some edge, and some line component
Confirm that basis vectors are orthonormal
MSU CSE 803 Stockman Fall 2009
Roberts basis cont.
If a neighborhood N has large dot product with a basis vector (image), then N is similar to that basis image.
MSU CSE 803 Stockman Fall 2009
A few properties of 1D sinusoids
They are orthogonal
Are they orthonormal?
MSU CSE 803 Stockman Fall 2009
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
MSU CSE 803 Stockman Fall 2009
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.
MSU CSE 803 Stockman Fall 2009
Convolution of two functions in the spatial domain is equivalent to pointwise multiplication in the frequency domain
MSU CSE 803 Stockman Fall 2009
LOG or DOG filter
Laplacian of GaussianApprox
Difference of Gaussians
MSU CSE 803 Stockman Fall 2009
Convolving LOG with region boundary creates a zero-crossing
Mask h(x,y)
Input f(x,y) Output f(x,y) * h(x,y)
MSU CSE 803 Stockman Fall 2009
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]
MSU CSE 803 Stockman Fall 2009
3D situation in the eyeNeuron 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.
MSU CSE 803 Stockman Fall 2009
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