Image Analysis & Retrieval. CS/EE 5590 Special Topics (Class Ids: 44873, 44874) Fall 2016, M/W Lec 13

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1 Image Analysis & Retrieval CS/EE 5590 Special Topics (Class Ids: 44873, 44874) Fall 2016, M/W 0012 Lec 13 Dimension Reduction: SVD and PCA Zhu Li Dept of CSEE, UMKC Office: FH560E, Ph: x Z. Li, Image Analysis & Retrv p.1

2 Outline Recap: About HW-2 Quiz-1 Linear Algebra Refresher SVD Principal Component Analysis (PCA) Z. Li, Image Analysis & Retrv p.2

3 Homework-2: Aggregation Data Set: 6 class, 6x20=120 images Can train global Kmeans and GMM from other image source Aggregation PCA: will cover today. GMM and Kmeans model: prob using smaller number of components, nc=24, 48 Other issues? Z. Li, Image Analysis & Retrv p.3

4 HW-2 extension CDVS data set available on box.com for download What we will do with this? Course Project Options : o New aggregation schemes Improved AKULA Super Vector aggregation Graph Cut As an Aggregation Point Cloud Geometry Aggregation o IEEE ICME: due 12/02 (prob 12/09) Image Analysis & Understanding, 2015 p.4

5 Quiz-1 Retrieval System Performance Metrics books, 50 are related to the query, retrieved 20, 15 are correct p n p 15 5 n Precision = tp/(tp+fp) = 0.75 Recall=tp/(tp+fn) = 0.3 Z. Li, Image Analysis & Retrv p.5

6 Cheating sheet award Quiz-1 Z. Li, Image Analysis & Retrv p.6

7 Outline Recap: About HW-2 Quiz-1 Linear Algebra Refresher SVD Principal Component Analysis (PCA) Z. Li, Image Analysis & Retrv p.7

8 Vector and Matrix Notations Vector Matrix Z. Li, Image Analysis & Retrv p.8

9 Vector Products Inner Product Outer Product Z. Li, Image Analysis & Retrv p.9

10 y=ax Matrix-Vector Product So, y is a linear combination of basis {a k } with weights from x Z. Li, Image Analysis & Retrv p.10

11 C=AB Matrix Product A: nxp B: pxm A: nxm = Associative: ABC = (AB)C = A(BC) Distributive: A(B+C) = AB + AC Z. Li, Image Analysis & Retrv p.11

12 Vector outer product: Outer Product/Kron Example Z. Li, Image Analysis & Retrv p.12

13 Matrix Transpose Transpose Z. Li, Image Analysis & Retrv p.13

14 Matrix Trace and Determinant Trace:Tr(A): only for nxn square matrix Determinant: Det(A): The size of volumes spanned by A, All possible linear combinations of a 1 and a 2 Det(A) = 2-9 = 7; Z. Li, Image Analysis & Retrv p.14

15 Eigen Values and Eigen Vectors Definition: for nxn matrix A: In Matlab: [P, V]=eig(A); Z. Li, Image Analysis & Retrv p.15

16 Eigen Vectors of Symmetric Matrix If square matrix A: nxn is symmetric A=A T Then its Eigen Values are real, and Eigen Vectors are othonormal: A = USU T where S is a diagonal matrix with eigen values of A. Application: solution to the Quadratic form maximization: will be the largest eigen value, and x* will be the corresponding eigen vector of A. Z. Li, Image Analysis & Retrv p.16

17 SVD for non square matrix: A mxn : A = UΣV T V T Σ Z. Li, Image Analysis & Retrv p.17

18 SVD as Signal Expansion The Singular Value Decomposition (SVD) of an nxm matrix A, is, A = USV T = i σ i u i v i t Where the diagonal of S are the eigen values of AA T, [σ 1, σ 2,, σ n ], called singular values U are eigenvectors of AA T, and V are eigen vectors of A T A, the outer product of u i v it, are basis of A in reconstruction: A (mxn) = U (mxm) S (mxn) V (nxn) The 1 st order SVD approx. of A is: σ 1 U :, 1 V :, 1 T Z. Li, Image Analysis & Retrv p.18

19 SVD approximation of an image Very easy function [x]=svd_approx(x0, k) dbg=0; if dbg x0= fix(100*randn(4,6)); k=2; end [u, s, v]=svd(x0); [m, n]=size(s); x = zeros(m, n); sgm = diag(s); for j=1:k x = x + sgm(j)*u(:,j)*v(:,j)'; end Z. Li, Image Analysis & Retrv p.19

20 Norm Vector Norm: Length of the vector Euclidean Norm (L2 Norm): norm(x, 2) L p norm: Matrix Norm: Forbenius Norm Z. Li, Image Analysis & Retrv p.20

21 Quadratic Form Quadratic form f(x)=x T Ax in R: Positive Definite (PD): For non-zero x, x T Ax > 0 Positive Semi-Definite (PSD): For non-zero x, x T Ax >= 0 Indefinite: Exists x 1, x 2 non zero, but x 1T Ax 1 >0, while x 2T Ax 2 < 0; Z. Li, Image Analysis & Retrv p.21

22 Matrix Calculus Gradient of f(a): Matrix Gradient Properties Z. Li, Image Analysis & Retrv p.22

23 Hessian of f(x) For function:f: R n R Gradient & Hessian of Quadratic Form: f(x)= x T Ax Z. Li, Image Analysis & Retrv p.23

24 Outline Recap: About HW-2 Quiz-1 Linear Algebra Refresher SVD Principal Component Analysis (PCA) Z. Li, Image Analysis & Retrv p.24

25 PCA -Dimension Reduction in Retrieval A typical image retrieval pipeline R d -> R p Image Formation Feature Computing Feature Aggregation Classification e.g, dense SIFT: x 128 e.g, Fisher Vector: k=64, d=128 Knowledge/ Data Base Image Analysis & Understanding, 2015 p.25

26 Principal Component Analysis The formulation: for data points {x 1, x 2,, } in R n, find a lower dimensional representation in R m, via a projection P,: mxn, s.t., the energy of the data is preserved Image Analysis & Understanding, 2015 p.26

27 PCA solution Take the Lagrangian of the problem L w, λ = w T Sw λ(w T w I) Take the derivative w.r.t. to w, and KKT condition gives us, Sw = λw This is an Eigen problem, finding projection s.t. it is just a scaling along the scatter matrix eigen vectors. Image Analysis & Understanding, 2015 p.27

28 PCA how to compute PCA via SVD on the Covariance matrix S: covariance, nxn Image Analysis & Understanding, 2015 p.28

29 2d Data

30 Principal Components 5 Gives best axis to project Minimum RMS error Principal vectors are orthogonal st principal vector nd principal vector

31 PCA on HoGs Matlab Implementation of PCA: [A, s, eig_values]=princomp(hogs); HoG basis function Image Analysis & Understanding, 2015 p.31

32 PCA Application in Aggregation SIFT aggregation Usually a PCA is done on SIFT features, to reduce the dimension from 128 to say 24, 32. Then a GMM is trained in R 32 space, for FV encoding Homework-2 Aggregation Fisher Vector Aggregation of SIFT load../../dataset/cdvs_sift_aggregation_test _data.mat; [n_sift, kd_sift]=size(gd_sift_cdvs); offs = randperm(n_sift); offs = offs(1:200*2^10); % PCA [A1, s1, lat1]=princomp(double(gd_sift_cdvs(offs, :))); figure(41); hold on; grid on; stem(lat1, '.'); title('sift pca eigen values'); Image Analysis & Understanding, 2015 p.32

33 Eigen values SIFT PCA Z. Li, Image Analysis & Retrv p.33

34 SIFT PCA Basis Functions Capturing max variation directions Z. Li, Image Analysis & Retrv p.34

35 Visualizing SIFT in lower dimensional space Project SIFTs from 2 images to 2D space Z. Li, Image Analysis & Retrv p.35

36 Summary HW-2 and Project SIFT dimension reduction via PCA first, before aggregation with VLAD and FV CDVS data set already pre-processed, easy to use. Ext into Project: super vector, geometry coding. SVD and PCA SVD non-square matrix decomposition, left transform and right transform, with scaling in between SVD as an image decomposition, linear combination of outer-product basis PCA eigen values indicate amount of info/energy in each dimension, PCA basis are eigen vectors to the covariance matrix Many applications Z. Li, Image Analysis & Retrv p.36