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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 ”.
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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” 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 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” MSU CSE 803 Fall 2014
The image as an expansion MSU CSE 803 Fall 2014
Different bases, different properties revealed MSU CSE 803 Fall 2014
Fundamental expansion MSU CSE 803 Fall 2014
Basis gives structural parts MSU CSE 803 Fall 2014
Vector space review, part 1 MSU CSE 803 Fall 2014
Vector space review, Part 2 2 MSU 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 apply • The 2D structure of the image is NOT lost MSU CSE 803 Fall 2014
Orthonormal basis vectors help MSU CSE 803 Fall 2014
Represent S = [10, 15, 20] MSU CSE 803 Fall 2014
Projection of vector U onto V MSU CSE 803 Fall 2014
Normalized dot product Can now think about the angle between two signals, two faces, two text documents, … MSU CSE 803 Fall 2014
Every 2x2 neighborhood has some constant, some edge, and some line component Confirm that basis vectors are orthonormal MSU 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. MSU CSE 803 Fall 2014
Standard 3x3 image basis Structureless and relatively useless! MSU CSE 803 Fall 2014
Frie-Chen basis Confirm that bases vectors are orthonormal MSU CSE 803 Fall 2014
Structure from Frie-Chen expansion Expand N using Frie-Chen basis MSU CSE 803 Fall 2014
Sinusoids provide a good basis MSU CSE 803 Fall 2014
Sinusoids also model well in images MSU CSE 803 Fall 2014
Operations using the Fourier basis MSU CSE 803 Fall 2014
A few properties of 1D sinusoids They are orthogonal Are they orthonormal? MSU CSE 803 Fall 2014
F(x,y) as a sum of sinusoids MSU 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 MSU CSE 803 Fall 2014
Power spectrum from FT MSU CSE 803 Fall 2014
Examples from images Done with HIPS in 1997 MSU CSE 803 Fall 2014
Descriptions of former spectra MSU 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. MSU CSE 803 Fall 2014
Bandpass filtering MSU CSE 803 Fall 2014
Convolution of two functions in the spatial domain is equivalent to pointwise multiplication in the frequency domain MSU CSE 803 Fall 2014
LOG or DOG filter Laplacian of Gaussian Approx Difference of Gaussians MSU CSE 803 Fall 2014
LOG filter properties MSU CSE 803 Fall 2014
Mathematical model MSU CSE 803 Fall 2014
1D model; rotate to create 2D model MSU CSE 803 Fall 2014
1D Gaussian and 1st derivative MSU CSE 803 Fall 2014
2nd derivative; then all 3 curves MSU CSE 803 Fall 2014
Laplacian of Gaussian as 3x3 MSU CSE 803 Fall 2014
G(x,y): Mexican hat filter MSU 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) MSU CSE 803 Fall 2014
LOG relates to animal vision MSU 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] MSU CSE 803 Fall 2014
Experience the Mach band effect MSU CSE 803 Fall 2014
Simple model of a neuron MSU CSE 803 Fall 2014
Canny edge detector uses LOG filter MSU CSE 803 Fall 2014
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 MSU CSE 803 Fall 2014
