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Procrustes Analysis Procrustes Analysis General Procrustes Analysis Robert Collins CSE586 CSE 586, Spring 2011 Advanced Computer Vision Procrustes Shape Analysis Robert Collins CSE586 Credits lots of slides are due to Lecture material from Tim Cootes University of Manchester. For more info, see http://www.isbe.man.ac.uk/~bim/ (includes code for exploring active shape/appearance models). Robert Collins CSE586 Statistical Shape Modelling • Represent the shape using a set of points • Align a set of training examples, and compute a “mean shape” • Model distribution of the variation of shapes in the training set wrt the mean shape • This allows us to analyze the major “modes” of shape variation and to generate realistic new shapes similar to ones in training set Robert Collins CSE586 Procrustes Analysis Procrustes Analysis General Procrustes Analysis R1, t1, s1 R, t, s R2, t2, s2 Rm, tm, sm Align one shape with another (not symmetric) Align a set of shapes with respect to some unknown “mean” shape (independent of ordering of shapes) Robert Collins CSE586 Why “Procrustes”? http://www.procrustes.nl/gif/illustr.gif Robert Collins CSE586 Aligning Two Shapes • Procrustes analysis: – Find transformation which minimises 2 | x1 −T (x2 ) | – T is a particular type of transformation that you want “shape” to be invariant to. – Typically a similarity transformation – Involves solving for best rotation, translation and scale between the two sets of points [we talked about how to do this in earlier lectures] Robert Collins CSE586 Steps in Similarity Alignment Given a set of K points: Configuration Translation normalization: Centered Configuration (center of mass at origin) Scale normalization: Pre-shape (divide by Sqrt of SSQ centered coordinates) Rotation normalization: Shape (rotate to align with a second normalized shape) Robert Collins CSE586 Generalized Procrustes Analysis • Generalised Procrustes Analysis – Find the transformations Ti which minimise | m T (x ) |2 ∑ − i i 1 – Where m = T (x ) n ∑ i i – Under the constraint that | m |=1 Robert Collins CSE586 Aligning Shapes : Algorithm • Normalise all so center of mass is at origin, and size=1 • Let m = x1 • Align each shape with m (via a rotation) 1 • Re-calculate m = Ti (xi ) n ∑ • Normalise m to default location, size, and orientation • Repeat until convergence Robert Collins CSE586 Hand shape model • 72 points placed around boundary of hand – 18 hand outlines obtained by thresholding images of hand on a white background • Primary landmarks chosen at tips of fingers and joint between fingers – Other points placed equally between 6 5 4 3 2 1 Robert Collins CSE586 Labeling Examples From Stegmann and Gomez Robert Collins CSE586 Labeling Examples From Stegmann and Gomez Robert Collins CSE586 Point Alignment Training points before and after alignment From Stegmann and Gomez Robert Collins CSE586 Hand Shape Model Samples from training set, after alignment Robert Collins CSE586 Statistical Shape Analysis Independent PCA on each data point cluster Problem: each point DOES NOT move independently from all the other points. In fact, the movement of points when the shape undergoes a deformation is often highly correlated. From Stegmann and Gomez Robert Collins CSE586 Statistical Shape Analysis Model how the entire ensemble of points varies. We will concatenate all points into a single vector in a high dimensional space. The full hand shape model is then built by computing PCA on all model points collectively, which captures correlations of motion between points. Will describe how in a moment, but first let’s look at the results. From Stegmann and Gomez Robert Collins CSE586 Hand Shape Model First 3 modes of variation Varying b1 Varying b2 Varying b3 Robert Collins CSE586 Face Shape Example Sample face training image 1st mode of variation Robert Collins CSE586 Shape Space Consider K landmark points (xi,yi) in 2D Consider each shape as a concatenated vector x, viewed as a point in 2*K dimensional space. (note: the “shape” really lives in 2K-4 dimensional space, but that is outside the scope of this lecture) We would like a parameterized model that describes the class of shapes we have observed. Robert Collins CSE586 Shape Space • Each observed shape is now a point (vector) x in 2*K dimensional space. • The “mean shape” is the center of mass of all these points x shape space Robert Collins CSE586 Shape Models • For shape synthesis – Parameterised model preferable – We will use a linear model x = x + Pb – That is, x = x + b1 p1 + b2 p2 + ... + bm pm € Robert Collins CSE586 Principal Component Analysis • Matrix X contains each shape as one column • Compute eigenvectors of scatter matrix XT X • Eigenvectors : main directions • Eigenvalue : spread or variation along that direction p1 p2 ∝ λ2 ∝ λ1 Robert Collins CSE586 Shape Space • The eigenvectors p1, p2 form a new (rotated) basis set of coordinate axes • Each shape x can now be assigned coordinates (b1,b2) according to the eigenvector axes p2 p1 x x = x + b1 p1 + b2 p2 shape space Robert Collins CSE586 Dimensionality Reduction • Coordinates often highly correlated • Some dimensions account for most of the observed spread (variation) of the data • We don’t lose much by approximating the shape using only those dimensions b1 p1 x x x ≈ x + p1b1 Robert Collins CSE586 Dimensionality Reduction • Data lies in subspace of reduced dim. x = x + p1b1 +L... + pnbn • However, for some t, bj ≈ 0 if j > t (Variance of bj is λ j ) λi i t Robert Collins CSE586 Building Shape Models • Given aligned shapes, {x i} • Apply PCA x ≈ x + Pb • P – First t eigenvectors of covar. matrix • b – Shape model parameters = coordinates of shape along eigenvector axes Robert Collins CSE586 Statistical Shape Models • Represent likelihood of observing a given shape with a probability distribution • Learn p(b) from training set • If x multivariate gaussian, then – b gaussian with diagonal covariance Sb = diag(λ1 L ... λt ) • Or, can use mixture model for p(b) Robert Collins CSE586 Background Information on PCA • principal components analysis (PCA) is a technique that can be used to simplify a dataset • It is a linear transformation that chooses a new coordinate system for the data set such that greatest variance by any projection of the data set comes to lie on the first axis (then called the first principal component), the second greatest variance on the second axis, and so on. • PCA can be used for reducing dimensionality by eliminating principal components that have small variance. Robert Collins CSE586 PCA Toy Example Robert Collins CSE586 Geometrical Interpretation: Robert Collins CSE586 Geometrical Interpretation: Robert Collins CSE586 Principal Component Analysis (PCA) • Given a set of points, how do we know if they can be compressed like in the previous example? – The answer is to look into the correlation between the points – The tool for doing this is called PCA Robert Collins CSE586 Principal component in 2d Robert Collins CSE586 PCA Theorem Robert Collins CSE586 PCA Theorem Robert Collins CSE586 PCA Theorem Robert Collins CSE586 PCA Theorem Robert Collins CSE586 Using PCA to Compress Data • Expressing x in terms of e1 … en has not changed the size of the data • However, if the points are highly correlated many of the coordinates of x will be zero or closed to zero. Robert Collins CSE586 Using PCA to Compress Data • Sort the eigenvectors ei according to their eigenvalue: Robert Collins CSE586 Example of Approximation Robert Collins CSE586 Implementing PCA • Need to find “first” k eigenvectors of Q: Robert Collins CSE586 Singular Value Decomposition (SVD) Robert Collins CSE586 SVD Properties Robert Collins CSE586 More about Shape Space Consider K landmark points (xi,yi) in 2D Dimension of the space of observations is 2*K But what is dimension of the space of the shapes? Recall, two point configurations are equivalent (have the same shape) if one can be translated, rotated and scaled into the other. Thus, each unique “shape” is an equivalence class. Robert Collins CSE586 Shape Space The shape space is then the quotient space induced by this equivalence class on the observation space k rotation translation scaling So shape space has dimension 2*K - 4 This is often called Kendall’s Shape Space. Robert Collins CSE586 Shape Space For triangles (K=3), Kendall’s shape space can be visualized as a sphere. For more points, the space is a more complicated manifold (is it a hypersphere?) Rather than measure variation in shape on the sphere (or higher dimensional nonlinear manifold), it is common to project points into the tangent plane taken at the mean shape. This is a linear space, so can use PCA, Gaussians, mixture of Gaussians, etc. Robert Collins CSE586 Shape Space http://www.york.ac.uk/res/fme/resources/morphologika/helpfiles/shapespace.html Robert Collins CSE586 Shape Space Tangent projection can be carried out by subtracting the vector of normalized shape coordinates of a figure from the mean vector, and taking just the component perpendicular to the mean. tangent plane nonlinear shape space mean shape shape sample projected into tangent plane shape sample Robert Collins CSE586 Shape Space If shape variation is relatively small, we can just take the shape difference vector as an approximation to the vector in the tangent plane. tangent plane nonlinear shape space mean shape shape sample .
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