On the minimization over SO(3) Manifolds


 Percival Gibson
 2 years ago
 Views:
Transcription
1 On the minimization over SO(3) Manifolds Frank O. Kuehnel RIACS NASA Ames Research Center, MS 69, Moffett Field, CA 9435, USA Abstract. In almost all imagemodel and modelmodel registration problems the question arises as to what optimal rigid body transformation applies to bring a physical 3dimensional model in alignment with the observed one. Data may also be corrupted by noise. Here I will present the exponential and quaternion representations for the SO(3) group. I will present the technique of compounding derivatives and demonstrate that it is most suited for dealing with numerical optimization problems that involve rotation groups. 1 Introduction A common task in computer vision is matching images or features and estimating essential transformation parameters [1]. In the weak perspective regime the dimensional affine image transformation with 6 parameters is applicable; in general a perspective transformation applies. Amongst Fig. 1. Flyby satellite view on a rendered region of duckwater. The main difference between both images is a rotation about the viewing axis. those parameters are 3 Eulerangles that describe the orientation of the
2 viewer with respect to a worldcoordinate system, see Fig. 1 as an example. Then neglecting other free parameters, we could formulate the image matching problem as finding a minimum to the log likelihood log P p (I p Îp(φ,θ,ψ)), (1) where we sum over all pixels in the observed image data I p and the expected image Îp. It is well known that numerical optimiztion algorithms with Eulerangle representation have numerical problems. Several alternatives to Eulerangle representations exist. However, a prior it is not clear how well various representations and minimization methods will perfom. The subject of my investigation is the types of suitable rotation group representations and their application in numerical minimization algorithms. I will compare the convergence rate of various methods in the special case of matching pointclouds. In the next section, I will present suitable rotation group representations, followed by their application within numerical optimization algorithms. SO(3) reprsentations I briefly introduce the euler, the exponential and the quaternion representation. An extensive introduction to rotation groups and parametrization can be found in []..1 Eulerangle representation The rotation matrix R represents an orthonormal transformation, RR T = I, det(r) = 1, as such it can be decomposed into simpler rotation matrices, R = R z (φ) R y (θ)r z (ψ), () with cos φ sinφ cos θ sin θ R z (φ) = sinφ cos φ, R y (θ) = 1. (3) 1 sinθ cos θ The decomposition () is not unique. We adhere to the NASA Standard Aerospace convention [3] with ψ the precession, θ the nutation and φ as the spin. The derivatives with respect to the Eulerangles are easily obtained and will not be given here.
3 . Exponential representation, (Axisangle) The axisangle representation is frequently used in kinematics [4] and commonly referred to as the exponential representation. The rotation matrix R is obtained by exponentiation of a matrix H. This generating matrix H must be antisymmetric. and in 3dimensions constitutes the cross product operator, r z r y H = [r] = r z r x, r = ω/θ,θ = ω, (4) r y r x where ω is called the Rodrigues vector. We notice that in the limit θ the Rodrigues vector is illdefined and we need proper limit considerations [5]. The rotation matrix R and its generating matrix H are related by, R = exp(θh) = I + sin θh + (1 cosθ)h. (5) In minimization problems that utilize gradients we require the knowledge of derivatives with respect to the components of the Rodrigues vector ω xyz. In the following I introduce the index notation: Greek indices run over {1,, 3} which is synonymous for {x, y, z}, repeated indices are implicitly summed over 1. Now the derivatives are easily obtained dr = sin θ dh dr β + (1 cos θ) dh dr β + cos θhr α + sin θh r α, (6) dω α dr β dω α dr β dω α where Hence, using and dr β dω α = (δ αβ r α r β )/θ, dθ dω α = r α. (7) dh dr x r x + dh dr y r y + dh dr z r z = H (8) dh dr x r x + dh dr y r y + dh dr z r z = H (9) we obtain the rotation matrix derivative with respect to the Rodrigues vector component dr = dh sin θ dω α dr α θ ( + dh (1 cos θ) + dr α θ H(cos θ sin θ/θ) + H (sin θ 1 cos θ θ 1 summation over recurrent indicies is call Einstein summation ) ) r α. (1)
4 . It is emphasized that the natural endowed algebra sturcture of the vector space ω IR 3 is not isomorphic to the multiplication in the SO(3) group, R(ω 1 ) R(ω ) R(ω 1 + ω )! (11).3 Quaternion representation Quaternions can compactly represent rotation matrices. We will see that the compounding (noncommutative multiplication) operation is isomorphic to the matrix multiplication in SO(3). Let s introduce general quaternions by ˆQ = (q,q), q = q 1 e 1 + q e + q 3 e 3, (1) where q is a regular 3dimensional vector with basis {e α } in IR 3. Addition of quaternions is componentwise, a product (compounding) of two quaternions as follows, ˆP ˆQ = (p q p T q,p q + q p + p q). (13) The notation in (13) utilizes the dot and cross product defined in IR 3. It is remarked that the product (13) is noncommutative but associative! In addition we have a conjugate operation, hence we can write the squared norm as, ˆQ = (q,q) ˆQ = (q, q), (14) ˆQ ˆQ = (q + q T q,) q + q T q. (15) We identify a scalar value c IR with a quaternion that has zero vector components, (c,) c and a vector q IR 3 with a quaternion that has zero scalar component (, q) q. Further, we will be sloppy with the quaternion product notation but implicitly assume that ˆQ ˆP means ˆQ ˆP. The claim is that a rotation can be represented by ˆR = (cos θ,sin θ r), ˆR ˆR 1 (16) with θ the magnitude and r the direction of the Rodirigues vector given in (4) and below, such that Rv = ˆRˆV ˆR, ˆV = (,v). (17)
5 Proof of (17) is easy and proceeds as follows: Writing equation (5) in the form with halfangles, Rv = cos θ Iv + cos θ sin θ r v + sin θ r (r v) + sin θ rrt v = (cos θ,sin θ r) (,v) (cos θ, sin θ r), (18) which completes the proof! Any quaternion ˆQ with ˆQ ˆQ = 1 constitutes a rotation in SO(3), the components of the rotation matrix R are easily obtained from (18), ˆQ = (q,q 1,q,q 3 ), q + q 1 + q + q 3 = 1 R( ˆQ) 1 q q 3 (q 1q q q 3 ) (q 1 q 3 + q q ) = (q q 1 + q q 3 ) 1 q1 q 3 (q q 3 q q 1 ) (19) (q 3 q 1 q q ) (q 3 q + q q 1 ) 1 q1 q Back to our initial statement, that we can easily idenitify an isomorphic structure between quaternions and rotation matrices, R( ˆQ 1 ) R( ˆQ ) = R( ˆQ 1 ˆQ ). ().4 Quaternion differential algebra As in the case for the Rodrigues vector, we are interested in the differential structure of the rotation matrix with respect to the quaternion components. Since we have 4 quaternion components and only 3 rodrigues vector ones, we also have to address the problem of overparametrization. The quaternion ring is endowed with a commutative operation (addition) and a noncommutative multiplicative operation (composition), it is natural to consider the two differential structures. Additive differential rotation We can cast the rotation matrix R in a particular well suited quaternion form, essentially it s restating (18) R = Iq + qqt + q [q] + [q] [q], (1) I is the unit matrix. Using tensor notation in (1), I µν = δ µν, ( qq T ) µν = q µq ν, ([q] ) µν = ε µρν q ρ, ()
6 where δ µν is the Kroneckerdelta and ε µνρ the total antisymmetric tensor. Now we consider a differential rotation R( ˆQ + dˆq) R( ˆQ) and obtain R µν q = q δ µν + ε µρν q ρ, We used the following tensor contraction in (3) R µν q η = (δ ηµ q ν + q µ δ ην + q ε µην δ µν q η ). (3) ε αβγ ε αµν = δ βµ δ γν δ βν δ γµ. (4) Compounded differential rotation Alternatively, we can compose the differential rotation as the compound of the deviation from the unitquaternion Î (this represents the identity operation, nullrotation), by δ ˆQ = (Î + δˆq) ˆQ ˆQ = δˆq ˆQ, (Î + δˆq)(î + δˆq) = 1. (5) Then, we can write a differential rotation R((Î + δˆq) ˆQ) R( ˆQ) with the change of δˆq (5), [R( ˆQ + δ ˆQ) R( ˆQ)]v = δ ˆQ (,v) ˆQ + ˆQ (,v) δ ˆQ = δˆq (,v ) + (,v ) δˆq = δq v + δq v = δq R( ˆQ)v + δq (R( ˆQ)v), (6) hence the derivatives have a particularly simple form compared to (3), R µν δq = R µν, R µν δq α = ε αρµ R ρν. (7) Overparametrization, connection with the Rodrigues vector The rotation group SO(3) is a 3parameter group, that is reflected in the number of Eulerangles and the vector components of the Rodrigues vector. Quaternions have 4 real components, seemingly we have increased the number of parameters. However, for a quaternion ˆQ to represent a rotation it must lie on the normalizedsphere S 3, ˆQ ˆQ = 1. (8) This represents an additional constraint, that is not built into the quaternioncomponent derivatives (3,7).
7 We can choose to parameterize the quaternion with the Rodrigues vector (16), then the derivatives of a general rotation quaternion ˆq with respect to the independent rodrigues vector ω components are, ˆq ω α = ( r α sin θ, r α cos θ r + 1 θ sin θ (δ αβ r α r β )e β ). (9) Here we assume that < θ or equivalently < q. Equivalently we can express (9) in quaternion component form, q = 1 ω α q q η α, = 1 ω α (q sincx) q αq η q + sincx q δ αη, x = arctan q. (3) We notice, that (9,3) show that the quaternionconstraint manifests in nonlinear functions. On the other hand, if we were to parametrize the infinitesimal deviation from a unitquaternion, ˆq = Î + δˆq = (q,q) = (cos δθ,sin δθ r), r = δω/δθ, δθ = ω. (31) then the quaternionconstraint is reflected by a rather simple linear dependence, δq δq η δω =, α q= δω = 1 α q= δ αη. (3) Notice, that (3) is the limiting case of (3) for q. 3 Nonlinear function minimization over S 3 Given a bounded scalar function with SO(3) group parameters as arguments M f( ˆQ) = f(r( ˆQ)), ˆQ S 3, M IR (33) we are to find the (not necessarily unique) optimal quaternion ˆQ (f) which minimizes the scalar function f( ˆQ) in (33). In most cases analytical closed form solutions to (33) are not available. Numerical minimization algorithms utilize gradient information and at most deal with quadratic expansions of (33). Therefore it is natural to lay focus on functions of the form f = a + b µν R µν + c ηρ µν R µνr ηρ. (34) The arguments of f are implicitly subsumed in the rotation matrix components R µν. Equation (34) is frequently the result of a quasinewton approximation to (1). Despite of the seemingly simple structure of Eqn. (34)
8 with rotation matrix components occuring at most in order, hence we could call them R µν harmonic functions, it generally doesn t possess a closed form solution. Only for the particular case presented below the solution is known. 3.1 Pseudoquadratic functions The special case of finding a unit quaternion ˆQ min which minimizes f s (R( ˆQ)) N = a + Rc i d i (35) i=1 can be solved analytically [6, 7]. (35) represents the case where a cloud of N points c i in IR 3 are mapped by a rotation such that the result most closely resembles the cloud d i. We can write (35) in tensor notation, f s = a + d µ d µ d µ c ν R µν + c ν c ρ δ µη R µν R ηρ, (36) where we have omitted the sum over point index i, thus in terms of tensor coefficients (34) we have b µν = c µ d ν, Now, we use the orthogonality relation, c ηρ µν = c νc ρ δ µη. c ηρ µν R µνr ηρ = c ν c ρ δ νρ (37) and obtain an expression that is linear in the matrix elements R µν, hence (35) constitutes only a pseudoquadratic function, n f s = a + b µν R µν, b µν = d i c T i, a = a + c T i c i + d T i d i. (38) i=1 According to [7] we can find a solution to the R µν linear problem (38) by simply decomposing the general rank tensor b µν and restating the problem as in (35). Then we solve for ˆQ min by noticing that f s is quadratic in the quaternion components and therefore can be written as n f s = a + (q,q) Bi T B i(q,q) T, (39) i=1 with 4 4 matricies B i, [ ] (ci d B i = i ) T. d i c i [d i + c i ] A solution to (35) is given by the eigenvector ˆq of B T i B i with minimal eigenvalue.
9 4 Gradient search methods on the S 3 manifold In the absence of analytical solutions we resort to numerical minimization methods. To measure the minimization search performance of various methods we focus on the pseudoquadratic function f s and argue that the results are representative even in the general scenario of (34). 4.1 Iterative quadratic expansion in the Rodrigues vector ω or Eulerangles We start with a Rodrigues vector ω and expand the function f s to quadratic order in ω about ω. We keep the rotation matrix entries R µν to second order, (this basically means that we don t use the orthogonality relationship), f s ( ω) ã + (b µν + R µρ c ρ c ν ) R µν ω α } {{ } b T R µν R µρ ω α + ω α c ν c ρ ω β, ω α ω β } {{ } C (4) where ω = ω ω. Omitted are all contributions from R µν / ω α ω β. We need to calculate the exponential derivatives R µν ω α = R µν q,η q,η ω α, (41) which is exactly expression (1). Having obtained the minimal ω we then update the rodrigues vector ω ω 1 = ω + ω, and iteratively proceed with a local expansion (4) until convergence is reached. expansion with Eulerangles The same outline as depicted above applies to a local quadratic expansion with Eulerangles. Instead of the Rodrigues vector derivatives (41) we need to calculate the Eulerangle derivatives, R µν φ, R µν θ Then the update procedure follows the rule, and R µν ψ. (4) {φ,θ,ψ } {φ 1,θ 1,ψ 1 } = {φ + φ,θ + θ,ψ + ψ}.
10 4. Quaternionpath gradient search We construct a product sequence (path) from a suitable starting point ˆQ (), ˆQ (k) = (Î + ˆq(k) ) (Î + ˆq(k 1) )... ˆQ (), (43) with the constraint ˆQ (k) S 3 k, (44) meaning that ˆQ (k) at any step represents a rotation. NewtonRaphson, Lagrange multiplier technique One way to constrain quaternions on a unit sphere is by a Lagrange multiplier technique. As a test example we aim to obtain a quaternion sequence to find the minimum of the pseudoquadratic functional (35). As usual with the Lagrange multiplier technique, given f s and the normalization contraint for δˆq, we can deal with the function g g = f s + λ((1 + δq ) + δq 1). (45) Now, if we write derivatives of g with respect to the multiplicative differential quaternions, we obtain dg dδˆq = { bµν R µν + λ(1 + δq ) = ε αµρ b ρν R µν + λδq α =, α {1,,3} (46) Assuming R being constant for the gradient search, we find an easy solution to (46), i.e. starting with R( ˆQ () ), in the first step, then we can solve for the update ˆq, ˆq = Î + δˆq, λ = b µνr µν q, q α q = ε αµρ R µν b ρν R ηξ b ηξ. (47) Having found ˆq and properly normalized, we successively obtain a new starting point ˆQ (1) = ˆq ˆQ () with rotation matrix R( ˆQ (1) ) for the next search step and solve for another quaternion until convergence is reached. Experimental results follow. QuasiNewton method through infinitesimal exponential parametrization Instead of imposing the quaternion constraint by a Lagrange multiplier technique we could parametrize the deviation quaternion (Î + ˆq) in
11 (43) by its Rodriguesvector and assume that we are in the linear regime (3), f s ã + (b µν + R µρ c ρ c ν ) R µν R µν R µρ ω α + ω α c ν c ρ ω β, (48) δω } {{ α δω } α δω β } {{ } b T C where ã = a c T i c i. Using (7) and (3), R µν δω α = R µν δq,η δq,η δω α = (R µν δq δω α ε ηµρ R ρν δq η δω α ) = ε αµρ R ρν, (49) and we obtain the expression for b b α = ε αµρ (R ρν b µν R µξ c ξ R ρν c ν ). (5) To emphasize the structure of the symmetric matrix C we write, D αµ = R µν δω α c ν = ε αµρ R ρν c ν, C αβ = D αµ D T µβ. (51) We find ω in equation (48) is quickly solved using SVD, f s ω =, ω = 1 C 1 b, (5) and calculate the finite step update quaternion (Î + ˆq(k) ) using (31). With a new quaternion ˆQ (k+1) as obtained in (43), we calculate rotation matrices and derivatives thereof. This process is repeated until suffient convergence is reached. 5 Experimental results Here I present numerical experiments on the convergence of the various gradient minimization methods. We compare the standard methods as described in section 4.1 with the ones utilizing the quaternionpath methods, see section 4.. The iterative Eulerangle expansion and the iterative Rodrigues vector expansion are referred to as Standard 1 and Standard. The two methods utilizing quaternionpath gradients are referred to as Lagrange and Compounding. We initialize all experiments by a random selection of N points {c i } and choosing a random transformation R( ˆQ f ) to obtain a point cloud {d i } according to (35). Then we randomly initialize our first guess R( ˆQ () )
12 and employ the minimization methods. The graphs show the minimization trajectory in the Rodrigues vector space. The green dota represents the initial guess, the endpoints of the two red lines the two possible exact solutions. We track the convergence by counting the number of steps it takes to obtain the accuracy f s. This corresponds to the residual value of mismatching the euclidian distance (35). We clearly notice that in all scenarios the compounding quaternion derivative method is vastly superior and has superlinear convergence. Also it never failed in all tested cases. Most rarely does the Eulerangle method succeed. 6 Conclusions I have concisely introduced important rotation group SO(3) representations and presented their differential structure. I ve developed the new idea of a product (compounding) pathquaternions and demonstrated its advantages application towards numerical minimization methods. In the special pointmatching case this method is vastly superior than all its alternatives. It is expected that the superlinear convergence of the compounding pathquaternion approach is retained even in the case of more general function types that depend on SO(3) parameters.
13 6.1 N = data points NewtonRaphson 4 Lagrange w x w y 4  wx Compounding 43 derivatives wy w z wz   wy .5 Exponential 5 derivatives.5  wy Euler derivatives 5 wz  4 wz wx wx Accuracy Standard 1 Standard Lagrange Compounding
14 6. N = 5 data points Ω z Newton Raphson Lagrange Ω y 4 Ω z Ω x Compounding Ω 3derivatives y ΩxΩ Ω z Ω x Exponential Ω 3 derivatives y 4 Ω z Ω x Ω Euler derivatives y Accuracy Standard 1 Standard Lagrange Compounding
15 6.3 N = 1 data points w NewtonRaphson Lagrange x w y w Compounding derivatives x w y w z w z w Exponential derivatives x w y w y w Euler derivatives x w z w z   Accuracy Standard Standard Lagrange Compounding References 1. K. Kanatani, Analysis of 3D Rotation Fitting, Transactions of pattern analysis and machine intelligence, IEEE, Vol. 16, no. 5, S. L. Altman, Rotations, Quaternions and Double Groups. Clarendon Press, Oxford, 1986.
16 3. G. J. Minkler, Aerospace Coordinate Systems and Transformations. Magellan Book Company, Balitmore, MD, L. D. Landau and E. M. Lifshitz, Course of Theoretical Physiscs, Mechanics, ed. L.D. Landau, ButterworthHeinemann, 3rd edtion, X. Pennec and J. P. Thirion, A Framework for Uncertainty and Validation of 3D Registration Methods based on Points and Frames, International Journal of Computer Vision, 5(3), pp 39, B. K. P. Horn, Closed Form Solutions of Absolute Orientations Using Unit Quaternions, Journal of Optical Society of America, A4(4), pp 6964, J. Weng, T. S. Huang and N. Ahuja, Motion and Structure from Two Perspective Views: Algorithms, Error Analysis, and Error Estimation IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. II, No 5, pp , 1989.
Metrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More information3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala)
3D Tranformations CS 4620 Lecture 6 1 Translation 2 Translation 2 Translation 2 Translation 2 Scaling 3 Scaling 3 Scaling 3 Scaling 3 Rotation about z axis 4 Rotation about z axis 4 Rotation about x axis
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More information1 Scalars, Vectors and Tensors
DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical
More informationSO(3) Camillo J. Taylor and David J. Kriegman. Yale University. Technical Report No. 9405
Minimization on the Lie Group SO(3) and Related Manifolds amillo J. Taylor and David J. Kriegman Yale University Technical Report No. 945 April, 994 Minimization on the Lie Group SO(3) and Related Manifolds
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical
More informationContinuous Groups, Lie Groups, and Lie Algebras
Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups
More informationRegression Using Support Vector Machines: Basic Foundations
Regression Using Support Vector Machines: Basic Foundations Technical Report December 2004 Aly Farag and Refaat M Mohamed Computer Vision and Image Processing Laboratory Electrical and Computer Engineering
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and expressions. Permutations and Combinations.
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationVector algebra Christian Miller CS Fall 2011
Vector algebra Christian Miller CS 354  Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More information(Quasi)Newton methods
(Quasi)Newton methods 1 Introduction 1.1 Newton method Newton method is a method to find the zeros of a differentiable nonlinear function g, x such that g(x) = 0, where g : R n R n. Given a starting
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationSubspace Analysis and Optimization for AAM Based Face Alignment
Subspace Analysis and Optimization for AAM Based Face Alignment Ming Zhao Chun Chen College of Computer Science Zhejiang University Hangzhou, 310027, P.R.China zhaoming1999@zju.edu.cn Stan Z. Li Microsoft
More informationPrincipal Rotation Representations of Proper NxN Orthogonal Matrices
Principal Rotation Representations of Proper NxN Orthogonal Matrices Hanspeter Schaub Panagiotis siotras John L. Junkins Abstract hree and four parameter representations of x orthogonal matrices are extended
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationOrthogonal Projections and Orthonormal Bases
CS 3, HANDOUT A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).
More information3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 PreAlgebra 4 Hours
MAT 051 PreAlgebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
More informationDynamics. Basilio Bona. DAUINPolitecnico di Torino. Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30
Dynamics Basilio Bona DAUINPolitecnico di Torino 2009 Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30 Dynamics  Introduction In order to determine the dynamics of a manipulator, it is
More informationRigid body dynamics using Euler s equations, RungeKutta and quaternions.
Rigid body dynamics using Euler s equations, RungeKutta and quaternions. Indrek Mandre http://www.mare.ee/indrek/ February 26, 2008 1 Motivation I became interested in the angular dynamics
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationMachine Learning and Pattern Recognition Logistic Regression
Machine Learning and Pattern Recognition Logistic Regression Course Lecturer:Amos J Storkey Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh Crichton Street,
More informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEYINTERSCIENCE A John Wiley & Sons, Inc.,
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationG.A. Pavliotis. Department of Mathematics. Imperial College London
EE1 MATHEMATICS NUMERICAL METHODS G.A. Pavliotis Department of Mathematics Imperial College London 1. Numerical solution of nonlinear equations (iterative processes). 2. Numerical evaluation of integrals.
More informationMeanShift Tracking with Random Sampling
1 MeanShift Tracking with Random Sampling Alex Po Leung, Shaogang Gong Department of Computer Science Queen Mary, University of London, London, E1 4NS Abstract In this work, boosting the efficiency of
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement Primary
Shape, Space, and Measurement Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two and threedimensional shapes by demonstrating an understanding of:
More informationgeometric transforms
geometric transforms 1 linear algebra review 2 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3 matrix operations addition
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 03 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More information2D Geometric Transformations. COMP 770 Fall 2011
2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrixvector multiplication Matrixmatrix multiplication
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationThinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks
Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationConstrained Least Squares
Constrained Least Squares Authors: G.H. Golub and C.F. Van Loan Chapter 12 in Matrix Computations, 3rd Edition, 1996, pp.580587 CICN may05/1 Background The least squares problem: min Ax b 2 x Sometimes,
More informationCS 5614: (Big) Data Management Systems. B. Aditya Prakash Lecture #18: Dimensionality Reduc7on
CS 5614: (Big) Data Management Systems B. Aditya Prakash Lecture #18: Dimensionality Reduc7on Dimensionality Reduc=on Assump=on: Data lies on or near a low d dimensional subspace Axes of this subspace
More informationOptimal linearquadratic control
Optimal linearquadratic control Martin Ellison 1 Motivation The lectures so far have described a general method  value function iterations  for solving dynamic programming problems. However, one problem
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationSolving Polynomial Equations
Solving Polynomial Equations Xun Sam Zhou Department of Computer Science & Engineering University of Minnesota 12/16/2010 1 Motivation Polynomials widely appear in Robotics GPS Manipulators Computer Vision
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationDynamic Analysis. Mass Matrices and External Forces
4 Dynamic Analysis. Mass Matrices and External Forces The formulation of the inertia and external forces appearing at any of the elements of a multibody system, in terms of the dependent coordinates that
More information2.2 Creaseness operator
2.2. Creaseness operator 31 2.2 Creaseness operator Antonio López, a member of our group, has studied for his PhD dissertation the differential operators described in this section [72]. He has compared
More informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More informationMATH MathematicsNursing. MATH Remedial Mathematics IBusiness & Economics. MATH Remedial Mathematics IIBusiness and Economics
MATH 090  MathematicsNursing MATH 091  Remedial Mathematics IBusiness & Economics MATH 094  Remedial Mathematics IIBusiness and Economics MATH 095  Remedial Mathematics IScience (3 CH) MATH 096
More informationChapter 7. Lyapunov Exponents. 7.1 Maps
Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories a key component of chaotic dynamics. For one dimensional maps the exponent is simply the average
More informationNOV  30211/II. 1. Let f(z) = sin z, z C. Then f(z) : 3. Let the sequence {a n } be given. (A) is bounded in the complex plane
Mathematical Sciences Paper II Time Allowed : 75 Minutes] [Maximum Marks : 100 Note : This Paper contains Fifty (50) multiple choice questions. Each question carries Two () marks. Attempt All questions.
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationDepth from a single camera
Depth from a single camera Fundamental Matrix Essential Matrix Active Sensing Methods School of Computer Science & Statistics Trinity College Dublin Dublin 2 Ireland www.scss.tcd.ie 1 1 Geometry of two
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationComponent Ordering in Independent Component Analysis Based on Data Power
Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals
More informationApplications to Data Smoothing and Image Processing I
Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationEpipolar Geometry. Readings: See Sections 10.1 and 15.6 of Forsyth and Ponce. Right Image. Left Image. e(p ) Epipolar Lines. e(q ) q R.
Epipolar Geometry We consider two perspective images of a scene as taken from a stereo pair of cameras (or equivalently, assume the scene is rigid and imaged with a single camera from two different locations).
More informationThe cover SU(2) SO(3) and related topics
The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationLinear Codes. Chapter 3. 3.1 Basics
Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length
More informationUNIVERSITY GRADUATE STUDIES PROGRAM SILLIMAN UNIVERSITY DUMAGUETE CITY. Master of Science in Mathematics
1 UNIVERSITY GRADUATE STUDIES PROGRAM SILLIMAN UNIVERSITY DUMAGUETE CITY Master of Science in Mathematics Introduction The Master of Science in Mathematics (MS Math) program is intended for students who
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationAccurate and robust image superresolution by neural processing of local image representations
Accurate and robust image superresolution by neural processing of local image representations Carlos Miravet 1,2 and Francisco B. Rodríguez 1 1 Grupo de Neurocomputación Biológica (GNB), Escuela Politécnica
More informationKinematics of Robots. Alba Perez Gracia
Kinematics of Robots Alba Perez Gracia c Draft date August 31, 2007 Contents Contents i 1 Motion: An Introduction 3 1.1 Overview.......................................... 3 1.2 Introduction.........................................
More informationFactor analysis. Angela Montanari
Factor analysis Angela Montanari 1 Introduction Factor analysis is a statistical model that allows to explain the correlations between a large number of observed correlated variables through a small number
More informationLecture 11: Graphical Models for Inference
Lecture 11: Graphical Models for Inference So far we have seen two graphical models that are used for inference  the Bayesian network and the Join tree. These two both represent the same joint probability
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationCHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationThe Trimmed Iterative Closest Point Algorithm
Image and Pattern Analysis (IPAN) Group Computer and Automation Research Institute, HAS Budapest, Hungary The Trimmed Iterative Closest Point Algorithm Dmitry Chetverikov and Dmitry Stepanov http://visual.ipan.sztaki.hu
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationMehtap Ergüven Abstract of Ph.D. Dissertation for the degree of PhD of Engineering in Informatics
INTERNATIONAL BLACK SEA UNIVERSITY COMPUTER TECHNOLOGIES AND ENGINEERING FACULTY ELABORATION OF AN ALGORITHM OF DETECTING TESTS DIMENSIONALITY Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree
More informationComputer Animation. Lecture 2. Basics of Character Animation
Computer Animation Lecture 2. Basics of Character Animation Taku Komura Overview Character Animation Posture representation Hierarchical structure of the body Joint types Translational, hinge, universal,
More informationAn Iterative Image Registration Technique with an Application to Stereo Vision
An Iterative Image Registration Technique with an Application to Stereo Vision Bruce D. Lucas Takeo Kanade Computer Science Department CarnegieMellon University Pittsburgh, Pennsylvania 15213 Abstract
More informationPrincipal Components Analysis (PCA)
Principal Components Analysis (PCA) Janette Walde janette.walde@uibk.ac.at Department of Statistics University of Innsbruck Outline I Introduction Idea of PCA Principle of the Method Decomposing an Association
More informationNumerical Analysis An Introduction
Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationBy choosing to view this document, you agree to all provisions of the copyright laws protecting it.
This material is posted here with permission of the IEEE Such permission of the IEEE does not in any way imply IEEE endorsement of any of Helsinki University of Technology's products or services Internal
More informationAn Application of Linear Algebra to Image Compression
An Application of Linear Algebra to Image Compression Paul Dostert July 2, 2009 1 / 16 Image Compression There are hundreds of ways to compress images. Some basic ways use singular value decomposition
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationEnhancing the SNR of the Fiber Optic Rotation Sensor using the LMS Algorithm
1 Enhancing the SNR of the Fiber Optic Rotation Sensor using the LMS Algorithm Hani Mehrpouyan, Student Member, IEEE, Department of Electrical and Computer Engineering Queen s University, Kingston, Ontario,
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationEM Clustering Approach for MultiDimensional Analysis of Big Data Set
EM Clustering Approach for MultiDimensional Analysis of Big Data Set Amhmed A. Bhih School of Electrical and Electronic Engineering Princy Johnson School of Electrical and Electronic Engineering Martin
More informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
More informationComputer programming course in the Department of Physics, University of Calcutta
Computer programming course in the Department of Physics, University of Calcutta Parongama Sen with inputs from Prof. S. Dasgupta and Dr. J. Saha and feedback from students Computer programming course
More information