PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 51Slide 531MSU CSE 803 Fall 2014Vectors [and more on masks]Vector space theory applies directly to several image processing/representation problems2MSU CSE 803 Fall 2014Image 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.3MSU CSE 803 Fall 2014Efaces100 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”4MSU CSE 803 Fall 2014The image as an expansion5MSU CSE 803 Fall 2014Different bases, different properties revealed6MSU CSE 803 Fall 2014Fundamental expansion7MSU CSE 803 Fall 2014Basis gives structural parts8MSU CSE 803 Fall 2014Vector space review, part 19MSU CSE 803 Fall 2014Vector space review, Part 2210MSU CSE 803 Fall 2014A 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 lost11MSU CSE 803 Fall 2014Orthonormal basis vectors help12MSU CSE 803 Fall 2014Represent S = [10, 15, 20]13MSU CSE 803 Fall 2014Projection of vector U onto V14MSU CSE 803 Fall 2014Normalized dot productCan now think about the angle between two signals, two faces, two text documents, …15MSU CSE 803 Fall 2014Every 2x2 neighborhood has some constant, some edge, and some line componentConfirm that basis vectors are orthonormal16MSU CSE 803 Fall 2014Roberts basis cont.If a neighborhood N has large dot product with a basis vector (image), then N is similar to that basis image.17MSU CSE 803 Fall 2014Standard 3x3 image basisStructureless and relatively useless!18MSU CSE 803 Fall 2014Frie-Chen basisConfirm that bases vectors are orthonormal19MSU CSE 803 Fall 2014Structure from Frie-Chen expansionExpand N using Frie-Chen basis20MSU CSE 803 Fall 2014Sinusoids provide a good basis21MSU CSE 803 Fall 2014Sinusoids also model well in images22MSU CSE 803 Fall 2014Operations using the Fourier basis23MSU CSE 803 Fall 2014A few properties of 1D sinusoidsThey are orthogonalAre they orthonormal?24MSU CSE 803 Fall 2014F(x,y) as a sum of sinusoids26MSU CSE 803 Fall 2014Continuous 2D Fourier TransformTo compute F(u,v) we do a dot product of our image f(x,y) with a specific sinusoid with frequencies u and v27MSU CSE 803 Fall 2014Power spectrum from FT28MSU CSE 803 Fall 2014Examples from imagesDone with HIPS in 199729MSU CSE 803 Fall 2014Descriptions of former spectra30MSU CSE 803 Fall 2014Discrete Fourier TransformDo 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.31MSU CSE 803 Fall 2014Bandpass filtering32MSU CSE 803 Fall 2014Convolution of two functions in the spatial domain is equivalent to pointwise multiplication in the frequency domain33MSU CSE 803 Fall 2014LOG or DOG filterLaplacian of GaussianApproxDifference of Gaussians34MSU CSE 803 Fall 2014LOG filter properties35MSU CSE 803 Fall 2014Mathematical model36MSU CSE 803 Fall 20141D model; rotate to create 2D model37MSU CSE 803 Fall 20141D Gaussian and 1st derivative38MSU CSE 803 Fall 20142nd derivative; then all 3 curves39MSU CSE 803 Fall 2014Laplacian of Gaussian as 3x340MSU CSE 803 Fall 2014G(x,y): Mexican hat filter41MSU CSE 803 Fall 2014Convolving LOG with region boundary creates a zero-crossingMask h(x,y)Input f(x,y) Output f(x,y) * h(x,y)42MSU CSE 803 Fall 201443MSU CSE 803 Fall 2014LOG relates to animal vision44MSU CSE 803 Fall 20141D EX.Artificial Neural Network (ANN) for computing g(x) = f(x) * h(x)level 1 cells feed 3 level 2 cellslevel 2 cells integrate 3 level 1 input cells using weights [-1,2,-1]45MSU CSE 803 Fall 2014Experience the Mach band effect46MSU CSE 803 Fall 2014Simple model of a neuron51MSU CSE 803 Fall 2014Canny edge detector uses LOG filter53MSU CSE 803 Fall 2014Summary 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