CS 4495 Computer Vision Principle Component Analysis
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1 CS 4495 Computer Vision Principle Component Analysis (and it s use in Computer Vision) Aaron Bobick School of Interactive Computing
2 Administrivia PS6 is out. Due *** Sunday, Nov 24th at 11:55pm *** PS7 is not optional. That was an aberration of last year s schedule. Tue Nov 26th, due technically Friday, December 5. Extension available until Sunday, Dec 7, 11:55pm. EXAM is Tuesday before Thanksgiving, Nov 26 th Covers concepts and basics Example questions will be posted on calendar by November 19 th In class Multiple choice/short answer
3 Today Eigen vectors and axes of inertia Principal components as dimensionality reduction in recognition Eigenfaces D >> N trick in tracking
4 Principal Components All principal components (PCs) start at the origin of the ordinate axes. First PC is direction of maximum variance from origin Subsequent PCs are orthogonal to 1st PC and describe maximum residual variance Wavelength 2 Wavelength PC Wavelength 1 PC Wavelength 1
5 2D example: fitting a line 2 E( a, b, d) = ( axi + byi d) i E = 0 2 ( ax + i by i d) = 0 d i d = ax + by Substitute (== subtract mean): = + = 2 E [ ax ( i x) by ( i y)] Bn i where B x1 x y1 y x x y y xn x yn y 2 2 = 2 y so minimize subject to n Bn d 2 =1 gives axis of least inertia n a = b x
6 Sound familiar???
7 Direct linear calibration - homogeneous A m 0 2n n This is a homogenous set of equations. When over constrained, defines a least squares problem minimize Am Since m is only defined up to scale, solve for unit vector m* Solution: m* = eigenvector of A T A with smallest eigenvalue Works with 6 or more points
8 Another interpretation Minimizing sum of squares of distances to the line is the same as maximizing the sum of squares of the projections on that line, thanks to Pythagoras. P Origin x (unit vector through origin) x T P
9 Algebraic Interpretation Trick: How is the sum of squares of projection lengths expressed in algebraic terms? L i n e P P P P t t t t m Point 1 Point 2 Point 3 : Point m L i n e x T B T = ( xp T ) i i B 2 x
10 Algebraic Interpretation Trick: How is the sum of squares of projection lengths expressed in algebraic terms? max( x T B T Bx), subject to x T x = 1 T T T maximize E = x Mx subject to x x = 1 ( M = B B) E x T E = x Mx T + λ(1 x x) E = 2Mx x + 2λx = 0 Mx = λx T ( xis an eigenvector of A A)
11 Another interpretation T BB n n 2 x i xy i i i= 1 i= 1 = n n 2 xy i i yi i= 1 i= 1 if about origin So the principal components are the orthogonal directions of the covariance matrix of a set points.
12 Algebraic Interpretation How many eigenvectors are there? For Real Symmetric Matrices of size NxN Except in degenerate cases when eigenvalues repeat, there are N distinct eigenvectors The eigenvectors are mutually orthogonal and therefore form a new basis Eigenvectors corresponding to the same eigenvalue have the property that any linear combination is also an eigenvector with the same eigenvalue; one can then find as many orthogonal eigenvectors as the number of repeats of the eigenvalue.
13 Eigenvalues 5 λ 1 λ
14 : General {a 11,a 12,...,a 1k } is 1st Eigenvector of correlation/covariance matrix, and coefficients of first principal component {a 21,a 22,...,a 2k } is 2nd Eigenvector of correlation/covariance matrix, and coefficients of 2nd principal component {a k1,a k2,...,a kk } is kth Eigenvector of correlation/covariance matrix, and coefficients of kth principal component
15 Dimensionality Reduction Dimensionality reduction We can represent the orange points with only their v 1 coordinates since v 2 coordinates are all essentially 0 This makes it much cheaper to store and compare points A bigger deal for higher dimensional problems
16 Higher Dimensions Suppose each data point is N-dimensional Same procedure applies: Outer product The eigenvectors of A define a new coordinate system eigenvector with largest eigenvalue captures the most variation among training vectors x eigenvector with smallest eigenvalue has least variation We can compress the data by only using the top few eigenvectors corresponds to choosing a linear subspace represent points on a line, plane, or hyper-plane these eigenvectors are known as the principal components We talk about the percentage of variance explained by the first k eigenvectors.
17 The space of all face images When viewed as vectors of pixel values, face images are extremely high-dimensional 100x100 image = 10,000 dimensions However, relatively few 10,000-dimensional vectors correspond to valid face images We want to effectively model the subspace of face images
18 The space of all face images We want to construct a low-dimensional linear subspace that best explains the variation in the set of face images
19 Principal Component Analysis Given: N data points x 1,,x N in R d where d is big We want to find a new set of features that are linear combinations of original ones: u(x i ) = u T (x i µ) (µ: mean of data points) What unit vector u in R d captures the most variance of the data?
20 Principal Component Analysis Direction that maximizes the variance of the projected data: N Projection of data point N Covariance matrix of data The direction that maximizes the variance is the eigenvector associated with the largest eigenvalue of Σ
21 Principal component analysis The direction that captures the maximum covariance of the data is the eigenvector corresponding to the largest eigenvalue of the data covariance matrix Furthermore, the top k orthogonal directions that capture the most variance of the data are the k eigenvectors corresponding to the k largest eigenvalues But first, we ll need the d>>>n trick
22 The dimensionality trick
23 Eigenfaces: Key idea Assume that most face images lie on a low-dimensional subspace determined by the first k (k<<<d) directions of maximum variance Use to determine the vectors or eigenfaces u 1, u k that span that subspace Represent all face images in the dataset as linear combinations of eigenfaces M. Turk and A. Pentland, Face Recognition using Eigenfaces, CVPR 1991
24 Eigenfaces example Training images x 1,,x N
25 Eigenfaces example Top eigenvectors: u 1, u k Mean: μ
26 Eigenfaces example Principal component (eigenvector) u k μ + 3σ k u k μ 3σ k u k
27 Eigenfaces example Face x in face space coordinates: =
28 Eigenfaces example Face x in face space coordinates: = Reconstruction: = + ^ x = µ + w 1 u 1 +w 2 u 2 +w 3 u 3 +w 4 u 4 +
29 Reconstruction example
30 Recognition with eigenfaces Process labeled training images: Find mean µ and covariance matrix Σ Find k principal components (eigenvectors of Σ) u 1, u k Project each training image x i onto subspace spanned by principal components: (w i1,,w ik ) = (u 1T (x i µ),, u kt (x i µ)) Given novel image x: Project onto subspace: (w 1,,w k ) = (u 1T (x µ),, u kt (x µ)) Optional: check reconstruction error x x to determine whether image is really a face ^ Classify as closest training face in k-dimensional subspace M. Turk and A. Pentland, Face Recognition using Eigenfaces, CVPR 1991
31 CS 4495 Computer Vision A. Bobick Recognition demo
32 Limitations Global appearance method: not robust to misalignment, background variation
33 Limitations The direction of maximum variance is not always good for classification
34 How to use this in tracking?
35 Visual Tracking Conventional approach Build a model before tracking starts Use contours, color, or appearance to represent an object Optical flow Incorporate invariance to cope with variation in pose, lighting, view angle View-based approach Solve complicated optimization problem Problem: Object appearance and environments are always changing Slide from Ming-Hsuan Yang
36 A few key inspirations: Eigentracking [Black and Jepson 96] View-Based learning method Learn to track a thing rather than some stuff Key insight: separate geometry from appearance use deformable But need to solve nonlinear optimization problem And fixed basis set Active contour [Isard and Blake 96] Use particle filter Propagate uncertainly over time But.. edge information is sensitive to lighting change Slide from Ming-Hsuan Yang
37 Incremental Visual Learning Aim to build a tracker that: Is not single image based Constantly updates the model Learns a representation while tracking Runs fast (close to real time) Operates on moving camera Challenge Pose variation Partial occlusion Adaptive to new environment Illumination change Drifts
38 Main Idea Adaptive visual tracker: Particle filter algorithm draw samples from distributions of deformation Subspace-based tracking learn to track the thing like the face in eigenfaces use it to determine the most likely sample With incremental update does not need to build the model prior to tracking handle variation in lighting, pose (and expression) Performs well with large variation in Pose Lighting (cast shadows) Rotation Expression change Slide from Ming-Hsuan Yang
39 For sampling-based method Nomenclature: Current location L t Current observation F t Predict the target location L t+1 in the next frame plt Ft, Lt ) pft Lt plt Lt ( ( ) ( ) 1 1 p(l t L t-1 ) dynamic model Use Brownian motion to model the dynamics p(f t L t ) observation model Use eigenbasis with approximation Slide from Ming-Hsuan Yang
40 Dynamic Model: p(l t L t-1 ) Representation of L t : Position (x t,y t ), rotation (r t ), and scaling (s t ) L t =(x t,y t,r t,s t ) Or affine transform with 6 parameters Simple dynamics model: Each parameter is independently Gaussian distributed L t L t+1 Slide from Ming-Hsuan Yang
41 Observation Model: p(f t L t ) Use probabilistic principal component analysis (P) to model our image observation process Given a location L t, assume the observed frame was generated from the eigenbasis The probability of observing a datum z given the eigenbasis B and mean µ, p(z B)=N (z ; µ, BB T +εi) where εi is additive Gaussian noise z Slide from Ming-Hsuan Yang µ B
42 Observation Model: p(f t L t ) The probability of observing a datum z given the eigenbasis B and mean µ, p(z B)=N (z ; µ, BB T +εi) where εi is additive Gaussian noise In the limit ε 0 (actually as the variance in the space is much greater than the variance out of the space) N (z ; µ, BB T +εi) is proportional to negative exponential of the square distance between z and the linear subspace B, i.e., Reconstruction p(z B) α exp(- (z- µ)- BB T (z- µ) ) z Slide from Ming-Hsuan Yang p(f t L t ) α exp(- (F t - µ)- BB T (F t - µ) ) µ B
43 Inference Fully Bayesian inference needs to compute p(l t F t, F t-1, F t-2,, L 0 ) Could do full particle filtering which is an approximation of full Bayesian inference. But still a lot of work (?). Instead approximate with p(l t F t, l * t-1) where l * t-1 is the best prediction at time t-1 This is different than what we did in original particle filters here we re keeping only the best from the last time frame. Slide from Ming-Hsuan Yang
44 Sampling Comes to the Rescue Remember particle filters? Instead we generate samples from a single location and then keep the best Maximum a posterior estimate. Basic tracking algorithm. Given a current location: 1. Consider a number of sample locations l s from our prior p(l t l * t-1). Their weight is from the prior. 2. For each sample l s, we compute the posterior p s = p(l s F t, l * t-1) by using Bayes rule: p s is the observation likelihood of l s under our P distribution, times the probability with which l s is sampled 3. Choose new best location: Maximum a posteriori estimate l * t = arg max p(l s F t, l * t-1) Slide from Ming-Hsuan Yang
45 The cool part Specifically do not assume the probability of observation remains fixed over time To allow for incremental update of our object model Given an initial eigenbasis B t-1, and a new observation w t-1, compute a new eigenbasis B t B t is then used in p(f t L t ) Slide from Ming-Hsuan Yang
46 Incremental Subspace Update To account for appearance change due to pose, illumination, shape variation Learn a representation while tracking Based on the R-SVD algorithm [Golub and Van Loan 96] and the sequential Karhunen-Loeve algorithm [Levy and Lindebaum 00] First assume zero (or fixed) mean Develop an update algorithm with respect to running mean Allow for decay Slide from Ming-Hsuan Yang
47 R-SVD Algorithm (in case you care) T Let X = UΣV and new data Y Decompose Y into projection of Y onto U and its T T complement, L = U Y, H = Y UL = ( I UU ) Y Let Y = UL+ JK where JK = QR(H ) SVD of [ X Y ] can be written as Compute SVD of Then SVD of [ X Y ] = [ U J ] " U = ' [ U J ] U Σ 0 Σ 0 L K L K Σ " V 0 ' ' = U ΣV " T [ X Y ] = U " Σ " V = Σ ' 0 I T ' T where V " V = 0 0 V I ' Slide from Ming-Hsuan Yang
48 Put All Together 1. (Optional) Construct an initial eigenbasis if necessary (e.g., for initial detection) 2. Choose initial location L 0 3. Generate possible locations: p(l t L t-1 ) 4. Evaluate possible locations: p(f t L t-1 ) 5. Select the most likely location by Bayes: p(l t F t, L t-1 ) 6. Update eigenbasis using R-SVD algorithm 7. Go to step 3 Slide from Ming-Hsuan Yang
49 Experiments 30 frame per second with pixel resolution on old machines Draw at most 500 samples 50 eigenvectors Update every 5 frames Runs XXX frames per second using Matlab Results Plush toy tracking Large lighting variation Slide from Ming-Hsuan Yang
50 Most Recent Work Subspace update with correct mean Sequential inference model Tracking without building a subspace a priori Learn a representation while tracking Better observation model Handling occlusion Slide from Ming-Hsuan Yang
51 Occlusion Handling An iterative method to compute a weight mask Estimate the probability a pixel is being occluded Given an observation I t, and initially assume there is no occlusion with W (0) where.* is a element-wise multiplication
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