CS 205 Mathematical Methods for Robotics and Vision - Chapter 5 docx

... collection of mathematical tools for both understandingand solving problems in robotics and computer vision. Several classes at Stanford cover the topics presented in this class, and do so in ... want to understandrobotics or vision, youshould take classes in these subjects, since this course is not on robotics or vision. On the other hand, if you do plan to study robotics,...

Ngày tải lên: 14/03/2014, 14:20

... own right. 1. 1 Who Should Take This Class The main goal of this class is to present a collection of mathematical tools for both understandingand solving problems in robotics and computer vision. ... and D. Hilbert, Methods of Mathematical Physics, Volume I and II, John Wiley and Sons, 19 89. D. A. Danielson, Vectors and Tensors in Engineering and Physics, Addison-Wes...

Ngày tải lên: 10/08/2014, 02:20

... the equalityof left- and right-hand side. When this process is finished, is in echelon form. In particular, if the matrix is square and if all columns have a pivot, then is upper-triangular. “Stop” ... procedure Orthonormal means orthogonal and with unit norm. 18 CHAPTER 2. ALGEBRAIC LINEAR SYSTEMS 2. 6 .2 Backsubstitution A system x c (2. 10) inechelon formiseasilysolvedfor x. Tosee...

Ngày tải lên: 10/08/2014, 02:20

... the discrepancies between left- and right-hand sides of the equations. 3 .4. LEAST-SQUARES SOLUTION OF A HOMOGENEOUS LINEAR SYSTEMS 33 which is the projection of the right-hand side vector b onto the ... following components: for for When written as a function of the vector c, this is y c Notice that there is no other choice for y, which is therefore unique: minimum residual forces the...

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... x where and are, respectively, the largest and smallest singular value of . Proof. From the definitions y x x and y y y (4.10) we immediately obtain the expression for steepest descent in terms of and ... most problems in robotics, vision, and arguably every other science or endeavor take on the form of optimization problems. One is that the desired goal may not be achievable, a...

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... can be reduced to a first-order system of the form (6.1) by introducing the -dimensional vector x . . . . . . With this definition, we have for 69 68 CHAPTER 5. EIGENVALUES AND EIGENVECTORS ... is the -th column of , and e is the -th column of , which is the conjugate of the -th row of . Since conjugationdoes not change the norm of a vector, the equality (5.13) implies that the -th col...

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... Re Im Re where Re, Im denote the real and imaginary part and where the two-argument function is defined as follows for if if if and if and and is undefined for . This function returns the arctangent ... STATE ESTIMATION 89 k | k-1 ^ y H k x k | k-1 ^ x ^ k | k k | k P x ^ P k+1 | k k+1 | k propagatepropagate x ^ P k-1 | k-1 k-1 | k-1 y k y k-1 y k+1 k update k | k-1 P k-1 k+1 time...

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... solve a particular type of estimation problem, namely, the estimation problem for the observationequation(7.12)with a linear function and n Gaussian zero-mean noise with the indentity matrix for ... matrices, and the Kalman filter makes optimal use of this informationfor a proper weightingof each of the scalar equations in (7.23). Better information ought to yield more accurate results...

Ngày tải lên: 10/08/2014, 02:20

... a differential linear problem. 3 CS 205 Mathematical Methods for Robotics and Vision Carlo Tomasi Stanford University Fall 2000 4 CHAPTER 1. INTRODUCTION 1. 2 Syllabus Here is the ideal syllabus, ... want to understandrobotics or vision, youshould take classes in these subjects, since this course is not on robotics or vision. On the other hand, if you do plan to stu...

Ngày tải lên: 14/08/2014, 11:20

... that a 0 (2. 3) For later reference, it is useful to rewrite the last two equalities in a different form. Equation (2. 2) is the same as x b (2. 4) and equation (2. 3) is the same as x 0 (2. 5) where a ... sides b , c , and b c yields b c b c b c and from equation (2. 8) we obtain b b c c b c b b c c b c Canceling equal terms and dividing by -2 yields the desired result. Corollary...

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... is by letting (3. 13) From y x we obtain x y v v , so that equation (3. 13) is equivalent to equation (3. 11) with , and the unit-norm constraint on y yields equation (3. 12). Section 3. 5 shows a sample ... minimizes x b 3. 4. LEAST-SQUARES SOLUTION OF A HOMOGENEOUS LINEAR SYSTEMS 33 which is the projection of the right-hand side vector b onto the complement of the range of . 3. 4...

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... 50 CHAPTER 4. FUNCTION OPTIMIZATION so that orthogonalityfor q becomes p p or p p (4. 19) This condition is called -conjugacy,or -orthogonality: if equation (4. 19) holds, then p and p are ... the procedure, due to Hestenes and Stiefel (Methods of conjugate gradients for solving linear systems,J. Res. Bureau National Standards, section B, Vol 49 , pp. 40 9 -4 36, 1952), which als...

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... transformation. The columns of are called eigenvectors, and the diagonal entries of are called eigenvalues. −2 −1 0 1 2 −2 −1 .5 −1 −0 .5 0 0 .5 1 1 .5 2 Figure 5. 1: Effect of the transformation (5. 5) ... generally look for a decomposition of of the form (5. 2) Chapter 5 Eigenvalues and Eigenvectors Given a linear transformation b x the singular value decomposition of trans...

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... have and Exercise: verifythatthis solution satisfies boththe differentialequation (6. 22) and the initialvalueequation y y . Thus, the solutions to system (6. 22) for and for have different forms. ... other hand, depend on the constants as follows: Im Re Im Re where Re, Im denote the real and imaginary part and where the two-argument function is defined as follows for if if if...

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... n x n x 7. 3. STATE ESTIMATION 89 k | k-1 ^ y H k x k | k-1 ^ x ^ k | k k | k P x ^ P k+1 | k k+1 | k propagatepropagate x ^ P k-1 | k-1 k-1 | k-1 y k y k-1 y k+1 k update k | k-1 P k-1 k+1 time Figure ... Kalman filter computes up-to-date information about the current state. Chapter 7 Stochastic State Estimation Perhaps the most important part of studying a problem in robotics or vi...

Ngày tải lên: 14/08/2014, 11:20