PCA to Eigenfaces. CS 510 Lecture #16 March 23 th A 9 dimensional PCA example

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PCA to Eigenfaces CS 510 Lecture #16 March 23 th 2015 A 9 dimensional PCA example is dark around the edges and bright in the middle. is light with dark vertical bars. is light with dark horizontal bars. All classes initially use 2 for low value, 7 for high value. Each instance is corrupted by sigma=1 Guassian Noise. 2 1

2 Eigenspace Example 1 Consider 3 examples from the 3 classes. 3 The Image Matrices Here they are as matrices. 4,,, 1.65 3.11 2.25 3.22 5.79 3.09 1.10 2.47 2.96 1.55 3.29 1.62 2.91 3.88.71 2.35 3.60 2.46.80 2.43 2.04 1.59 8.17.79.69 1.96 4.34,,, 6.36 2.39 9.36 6.05.55 6.60 5.97 3.49 7.33 6.43 1.43 7.01 7.66 3.20 6.66 6.96 1.82 7.52 6.52.89 7.74 4.80 1.97 7.58 5.75 1.06 7.24,, 8.11 8.94 5.85 2.63 2.60 5.16 7.20 6.09 6.12 6.94 6.68 5.99 3.63 3.15 1.37 8.50 6.89 6.49 7.02 7.73 7.08 2.75 2.10 1.91 5.92 6.85 7.16

3 Normalized Image Vectors Each as a 9x1 vector, an unrolled image. Each has zero mean and unit length. 5 = X,,,,,,,, -.133.0500 -.102.0750.324.0910 -.188.00100 -.0630 -.117.127 -.141.0930.186 -.152 -.0140.185 -.0740 -.231 -.0450 -.143 -.114.505 -.163 -.239 -.0710.0450.0480 -.149.184.0710 -.266.131.0300 -.0670.0320.0530 -.202.0520.162 -.116.135.0860 -.161.0440.0840 -.229.124.0200 -.178.217.0410 -.200.0570.126.197 -.0290 -.129 -.157.0370.0800.0630 -.0520.0810.0930 -.00600 -.0660 -.120 -.163.172.124 -.0150.0900.157.0600 -.113 -.177 -.132.0310.126.0270 Singular Value Decomposition 3/23/15 -.29 -.080 -.31 -.51.40.30 -.17 -.15 -.51 -.14.46.10 -.48 -.27 -.17 -.46 -.30.35 -.23 -.060.34.050 -.49 -.29 -.18.16 -.66 -.23 -.52.64 -.060.12.16.050 -.44.17.84.040.16 -.10 -.090.21 -.12 -.32 -.30.080 -.47 -.42 -.36 -.45 -.29.35 -.22.080 -.14 -.10 -.38.60 -.080 -.020 -.40 -.54 -.050 -.14.50.080.080.13 -.21.62 -.47 -.23 -.20.16 -.050.050 -.53.77.21 0.040 1.2 0 0 0 0 0 0 0 0 0.55 0 0 0 0 0 0 0 0 0.19 0 0 0 0 0 0 0 0 0.12 0 0 0 0 0 0 0 0 0.030 0 0 0 0 0 0 0 0 0.010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -.29 -.14 -.23 -.23.84.080 -.14 -.14 -.20 -.080.46 -.060 -.52.040 -.47 -.10.50.16 -.31.10.34.64.16 -.42 -.38.080 -.050 -.51 -.48.050 -.060 -.10 -.36.60.080.050.40 -.27 -.49.12 -.090 -.45 -.080.13 -.53.30 -.17 -.29.16.21 -.29 -.020 -.21.77 -.17 -.46 -.18.050 -.12.35 -.40.62.21 -.15 -.30.16 -.44 -.32 -.22 -.54 -.47 0 -.51.35 -.66.17 -.30.080 -.050 -.23.040 U D U T Ø Actual values for this example. Ø The Eigenvalues are 1.2, 0.55, 0.19 etc. Ø The Eigenvectors are columns of U matrix. 6 CS 510, Image Computa4on, Ross Beveridge & Bruce Draper

Subspace Projection Pictures Legend 7 Eigenspace Example 2 Consider 12 4x4 images. Ø Low value is 2, high is 7, noise sigma 1.0 8 4

Example 2 Subspace 3D Legend 9 Eigen Space Example 3 10 5

Example 3 Subspace Legend 11 Example 3 Subspace: 99 Samples Legend 12 6

Example 3: Dimensions 3, 4 & 5 Legend 13 Example 3 Observations The first two Principle Components carry no information with respect to image class. However, Principle Components 3 and 4 carry all the information necessary for a nearest neighbors classifier [ 4.775, 3.846,.4245,.3970,.07430 ] 1 2 3 4 5 The Eigenvalues, which record variance along each axis, show higher PC s have more variance: 14 7

On to Faces - Preprocessing Integer to float conversion Converts 256 gray levels to singlefloats Geometric Normalization Aligns human chosen eye coordinates Masking Crop with elliptical mask leaving only face visible. Histogram Equalization Histogram equalizes unmasked pixels: 256 levels. Pixel normalization Shift and scale pixel values so mean pixel value is zero and standard deviation over all pixels is one. 15 Standard Eigenfaces Training Training images Eigenspace Tes+ng Distance Matrix PCA space projec4on 16 8

Linear Discriminant Analysis 17 The PCA-LDA Variant Training Training images Eigenspace Combined space (PCA+LDA) Tes4ng PCA+LDA space projec4on Distance Matrix 18 9

More Recently Self Quo4ent Preprocessing Local Region PCA Algorithm 13 Local Features + Hole Face Self Quo4ent - Ligh4ng Removal PCA based whitening Local Features 250 basis vectors per feature. 3500 total basis vectors. Fisher Criterion Weigh4ng All features combined Similarity based upon Correla4on 19 LRPCA generates 3500 Features............ 20... 10

Summary PCA can be applied to any set of registered images It extracts the dimensions of maximum covariance In some sense, the structure of the domain Dimensions of co-variance may or may not be related to classification No longer competitive by itself, but still a common part of classification strategies in high-dimensional data. 21 11

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