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EigenFaces and EigenPatches Useful model of variation in a region –Region must be fixed shape (eg rectangle) Developed for face recognition Generalised for –face location –object location/recognition

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Overview Model of variation in a region

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Overview of Construction Mark face region on training set Sample region Normalise Statistical Analysis

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Sampling a region Must sample at equivalent points across region Place grid on image and rotate/scale as necessary Use interpolation to sample image at each grid node

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Interpolation Pixel values are known at integer positions –What is a suitable value at non-integer positions? Values known at integer values Estimate value here

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Interpolation in 1D Estimate continuous function, f(x), that passes through a set of points (i,g(i)) f(x) x

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1D Interpolation techniques f(x) x Nearest Neighbour f(x) x Linear f(x) x Cubic

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2D Interpolation Extension of 1D case Nearest Neighbour Bilinear y interp at x=0 y interp at x=1

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Representing Regions Represent each region as a vector –Raster scan values n x m region: nm vector g

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Normalisation Allow for global lighting variations Common linear approach –Shift and scale so that Mean of elements is zero Variance of elements is 1 Alternative non-linear approach –Histogram equalization Transforms so similar numbers of each grey-scale value

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Review of Construction Mark face region on training set Sample region Normalise Statistical Analysis The Fun Step

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Multivariate Statistical Analysis Need to model the distribution of normalised vectors –Generate plausible new examples –Test if new region similar to training set –Classify region

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Fitting a gaussian Mean and covariance matrix of data define a gaussian model

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Principal Component Analysis Compute eigenvectors of covariance, S Eigenvectors : main directions Eigenvalue : variance along eigenvector

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Eigenvector Decomposition If A is a square matrix then an eigenvector of A is a vector, p, such that Usually p is scaled to have unit length,|p|=1

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Eigenvector Decomposition If K is an n x n covariance matrix, there exist n linearly independent eigenvectors, and all the corresponding eigenvalues are non-negative. We can decompose K as

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Eigenvector Decomposition Recall that a normal pdf has The inverse of the covariance matrix is

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Fun with Eigenvectors The normal distribution has form

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Fun with Eigenvectors Consider the transformation

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Fun with Eigenvectors The exponent of the distribution becomes

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Normal distribution Thus by applying the transformation The normal distribution is simplified to

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Dimensionality Reduction Co-ords often correllated Nearby points move together

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Dimensionality Reduction Data lies in subspace of reduced dim. However, for some t,

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Approximation Each element of the data can be written

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Normal PDF

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Useful Trick If x of high dimension, S huge If No. samples, N<dim(x) use

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Building Eigen-Models Given examples Compute mean and eigenvectors of covar. Model is then P – First t eigenvectors of covar. matrix b – Shape model parameters

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Eigen-Face models Model of variation in a region

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Applications: Locating objects Scan window over target region At each position: –Sample, normalise, evaluate p(g) Select position with largest p(g)

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Multi-Resolution Search Train models at each level of pyramid –Gaussian pyramid with step size 2 –Use same points but different local models Start search at coarse resolution –Refine at finer resolution

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Application: Object Detection Scan image to find points with largest p(g) If p(g)>p min then object is present Strictly should use a background model: This only works if the PDFs are good approximations – often not the case

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Application: Face Recognition Eigenfaces developed for face recognition –More about this later

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