1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems

1MSU CSE 803 Fall 2014

Vectors [and more on masks]

Vector space theory applies directly to several image

processing/representation problems


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.


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”


The image as an expansion


Different bases, different properties revealed


Fundamental expansion


Basis gives structural parts


Vector space review, part 1


Vector space review, Part 2

2


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


Orthonormal basis vectors help


Represent S = [10, 15, 20]


Projection of vector U onto V


Normalized dot product

Can now think about the angle between two signals, two faces, two text documents, …


Every 2x2 neighborhood has some constant, some edge, and some line component

Confirm that basis vectors are orthonormal


Roberts basis cont.

If a neighborhood N has large dot product with a basis vector (image), then N is similar to that basis image.


Standard 3x3 image basis

Structureless and relatively useless!


Frie-Chen basis

Confirm that bases vectors are orthonormal


Structure from Frie-Chen expansion

Expand N using Frie-Chen basis


Sinusoids provide a good basis


Sinusoids also model well in images


Operations using the Fourier basis


A few properties of 1D sinusoids

They are orthogonal

Are they orthonormal?


F(x,y) as a sum of sinusoids


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


Power spectrum from FT


Examples from images

Done with HIPS in 1997


Descriptions of former spectra


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.


Bandpass filtering


Convolution of two functions in the spatial domain is equivalent to pointwise multiplication in the frequency domain


LOG or DOG filter

Laplacian of GaussianApprox

Difference of Gaussians


LOG filter properties


Mathematical model


1D model; rotate to create 2D model


1D Gaussian and 1st derivative


2nd derivative; then all 3 curves


Laplacian of Gaussian as 3x3


G(x,y): Mexican hat filter


Convolving LOG with region boundary creates a zero-crossing

Mask h(x,y)

Input f(x,y) Output f(x,y) * h(x,y)



LOG relates to animal vision


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]


Experience the Mach band effect


Simple model of a neuron


Canny edge detector uses LOG filter


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

Documents

1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems