M. Spivak. Calculus on Manifolds. AddisonWesley, Redwood City, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Defined as surface integrals, the generalized irradiance tensors in equation (6) are computed by reducing them to boundary integrals, which yield closed form solutions in polyhedral environments. Let n denote the outward pointing normal of the boundary curve #A. Using generalized Stokes theorem [16], we have shown (see the Appendix) that T satisfies the recurrence relation (A, w) w j T (A) # mI (7) for n 0, where # ij is the Kronecker delta. The irradiance tensor T can be further expanded by [2] n 1)T Ij (A) # jI I n j ds, 8) with ....

Michael Spivak. Calculus on Manifolds. Benjamin/Cummings, Reading, Massachusetts, 1965.

....in u. The definition of the histogram in equation (1) can be extended trivially to 3D color spaces. The domain of an image # can be morphed. An interesting class of deformations are the differentiable vector fields. It is possible to express such fields in terms of flow equations whose solutions [12] give rise to families of transformations. A family or path of transformations # t is expressed as # t (#x) D ,where#x = x, y) D is a point in the image, and t R is the parameter of the transformation. Transformations that arise in this manner satisfy several properties. Clearly, they ....

....# 0 = Id, and are invertible. In this work we will only study transformations that satisfy these properties, for example rotations, scalings and other more exotic transformations we describe below. To study the effect of # t on the histogram we only need to study the effect of vector fields X [12]. 3. Transformations that preserve the histogram We would like to find the class of transformations that preserve the histogram of any image #. To simplify the analysis we break up the image into differential regions dxdy. If each differential region preserves its size under a transformation, ....

M. Spivak. Calculus on Manifolds. Benjamin/Cummings, 1965.

....k = k 1 ; Delta Delta Delta ; k n ) is another set of local coordinates at A then, dA = w(k)dk 1 Delta Delta Delta dk n ; w(k) w(g(k) j det( g i = k j )j; where the determinant is the Jacobian of the coordinate transformation. For a precise definition of integrals on manifolds see [S4] Two examples of left invariant integrals are well known. The group R 0 (example (3) in x3.1) with elements (4.4) A(x) 1 x 0 1 ; x 2 R is isomorphic to the real line. The continuous functions on R 0 are just the continuous functions f(x) on the real line. Here, dx is a ....

]M. Spivak, Calculus on Manifolds, Benjamin, New York, 1965.

....field then has the property that line integrals Z C F Delta dr depend only on the initial and terminal points of the curve C, see [26] and Delta refers to the fact that a line integral is involved. In this case the function H may be obtained from its partial derivatives, F = H x ; H y ) [25, 26]. Specifically for motion in an open connected region the potential function may be obtained, up to an additive constant, as H(x; y) Z (x;y) a;b) F Delta dr (2:2) where (a; b) is a point in the region. When a potential function exists, the path of the line integral taken from the starting ....

....(x; y) will not affect the final result. The function H may also be estimated, given H x ; H y via simulation experiments as described below. If F has components H x ; H y , then a necessary condition for the existence of a corresponding potential function is that y H x = x H y (2:3) [25, 26]. In the case that the region is simply connected, this condition is also sufficient. 2.2 Stochastic case: A pertinent probabilistic concept for dynamic situations is a stochastic differential equation (SDE) see [3, 16] Such equations lead to Markov processes and take the form dr(t) r(t) ....

Spivak, M. (1965). Calculus on Manifolds. Benjamin, New York.

....When using the above local parametric models, discontinuities occur at the borders of the local regions. To interpolate among the local models and maintain the continuity of the overall approximated function, we use locally defined influence functions, which form a partition of unity [17] over the whole Mach region. The partition of unity divides the whole operating region into nondisjoint regions, as defined in the following definition. Definition 2.1: Let a compact set exist. Then a collection of functions , defined in an open set containing , is called a partition of unity for ....

M. Spivak, Calculus on Manifold. New York: W. A. Benjamin, 1965.

.... recent work that have demonstrated analytic methods for computing swept volumes are (Ahn, et al. 1997, Elber 1997, Ling and Chase 1996, Sourin and Pasko 1996) Before proceeding with the analysis and to remain consistent with the terminology used in differential geometry and differential topology (Spivak 1968, Guillemin and Pollack 1974, and Lu, 1976) we define some terms that will be used throughout this work. Swept volume: The totality of points touched by a geometric entity while in motion. Variety: Subset of a Euclidean space defined by zeros of a finite number of differentiable functions. ....

Spivak, M. 1968, Calculus on Manifolds, Benjamin/Cummeings.

....a user defined parameter indicating the boundary of a local region. When using this type of influence function, a discontinuity occurs at the border of the local linear approximation regions. To maintain the continuity of the approximator, we use influence functions which form partitions of unity [21] over the approximation region D. The partition of unity divides the state space into non disjoint cells, as defined in the following. Definition 3.1 Let D ae R n . Then a collection Upsilon of functions i (z) defined in an open set containing D, is called a partition of unity for D, if it ....

M. Spivak, Calculus on Manifolds, W.A.Benjamin, New York, 1965.

.... coordinates such that ## ( ii ii uus# ( 6) where s ( i and u ( i may or may not have common terms since s ( i may contain variables that are defined as functions of other variables and are obtained by solving a set of analytic functions called varieties (Lu 1978 and Spivak 1968), and # ( i characterizes the equation of a singular surface. Furthermore, surface patches in closed form on the boundary of the envelope are a subset of Eq. 6) defined as # ( ii u ; im# 1, where m is the number of patches enveloping the workspace. Let [x y z] be the global ....

Spivak, M. 1968, Calculus on Manifolds, Benjamin/Cummeings.

....i then it is not clear what we mean by W = rP . This notion of the gradient of a spatial process is defined in the next section. We know from vector calculus that a vector field has zero curl if and only if it is the gradient of some potential field (see for example Weintraub [13] or Spivak [12]) In fluid dynamics, curl is known as vorticity. In essence, this means the fluid flow has no circulation. In our gradient based wind model, it is assumed that there is no circulation of the air because of the scale we are examining: we are not really interested in small scale turbulent ....

M. Spivak, Calculus on Manifolds, AddisonWesley, Reading, MA, 1965.

....in u. The definition of the histogram in equation (1) can be extended trivially to 3D color spaces. The domain of an image # can be morphed. An interesting class of deformations are the differentiable vector fields. It is possible to express such fields in terms of flow equations whose solutions [12] give rise to families of transformations. A family or path of transformations # t is expressed as # t (#x) D # R 2 , where #x = x, y) # D is a point in the image, and t # R is the parameter of the transformation. Transformations that arise in this manner satisfy several properties. ....

....# 0 = Id, and are invertible. In this work we will only study transformations that satisfy these properties, for example rotations, scalings and other more exotic transformations we describe below. To study the effect of # t on the histogram we only need to study the effect of vector fields X [12]. 3. Transformations that preserve the histogram We would like to find the class of transformations that preserve the histogram of any image #. To simplify the analysis we break up the image into differential regions dxdy. If each differential region preserves its size under a transformation, ....

M. Spivak. Calculus on Manifolds. Benjamin/Cummings, 1965.

.... A general construction of invariant measures in homogeneous spaces with applications to classical groups and related manifolds is given, for instance, in [11] A combinatorial approach is presented in [8] Necessary background material can be looked up in any textbook on geometry, for instance, [14], 16] We begin with the real case. Density for the submanifold of critical angles was calculated several times in statistics (see [7] Let a 1 ; a k and b 1 ; b n Gammak be orthonormal column vectors that span a plane p and its orthogonal complement. The invariant measure on G ....

M. Spivak, Calculus on manifolds, Benjamin Inc., New York, 1965.

.... set, A; for example: f F (x) m 1 (x)f(x) where m 1 (x) is an infinitely smooth function which is unity on A, vanishes outside a slightly larger set A 1 oe A, and is such that 0 m 1 (x) 1 on A 1 Gamma A (such classes of so called Schwartz functions can easily be shown to exist; see, e.g. [1, 36]) One thus has jf(x) Gamma f F (x)j = j1 Gamma m 1 (x)j jf(x)j = ff 1 (x) where ff 1 (x) 0 if x 2 A, and otherwise ff 1 (x) jf(x)j. While the transform of this f F exists, it may approach zero only asymptotically. In order to extend the class of reconstructors which can be used, allow the ....

Spivak, Michel, Calculus on Manifolds, Addison-Wesley, Reading, Mass., 1965.

....r.h.s. in case f is differentiable; rF then corresponds to rf ) Distributional differentiation is akin to a quite general recipe, extremely useful in image models, known in differential geometry as the carry along principle (e.g. pull back and push forward of forms and vectors, respectively [9, 10, 12, 24, 25]) In the context of a conservation principle, the natural setting for motion analysis [15] the proper differential tool is that of a Lie derivative. It can be defined within the framework of topological duality by the same token as any other derivative. Classically, if v is a vector field in ....

M. Spivak. Calculus on Manifolds. W. A. Benjamin, New York, 1965.

....Proposition 1 If g : A ae R n B ae R n is a diffeomorphism, then M A is a manifold if and only if g(M) is a manifold and N B is a manifold if and only if g Gamma1 (N) is a manifold. Another way to verify whether a subset of R n is a manifold is given by the following Theorem (e.g. Spivak, 1965). Theorem 1 Let A ae R m be open and let h : A R m Gamman be a smooth function such that h 0 (x) has rank m Gamma n whenever h(x) 0. Then h Gamma1 (0) is a n dimensional manifold in R m . Note that the rank of the Jacobian matrix h 0 in Theorem 1 is m Gamma n if h has the form ....

Spivak, M. (1965). Calculus on manifolds. Addison-Wesley, New York.

....of functions allows us to compute irradiance from non uniform luminaires. One direct application of our result is the exact computation of transfers from linear polygonal elements to di#erential areas [10] Common to all methods for computing surface integrals of this nature is Stokes theorem [13], by which surface integrals can be converted into boundary integrals. Such reductions are generally advantageous, as boundary integrals are better suited for computation and frequently lead to closed form solutions. Our approach is to use Stokes theorem to derive a recurrence relation for a ....

....A # kst r k dr s # dr t 2r 3 = Z A d# = #(A) 222 Equation (6) can be viewed as a generalization of Girard s formula for the area of spherical triangles to arbitrary regions on the sphere. More importantly, it tells us that although the 2 form d# is not exact over the whole sphere S 2 [13], it is locally exact over S 2 p, that is, the sphere minus one point, and thus can be represented as a boundary integral over any spherical region, except S 2 . 35 C Proof of Identity (42) Z #A #v, n# #w, u# 2 ds = #w, v# Z #A #w, n# #w, u# 2 ds, where w and v are ....

Michael Spivak. Calculus on Manifolds. Benjamin/Cummings, Reading, Massachusetts, 1965.

....the Lebesgue measure in R m ) It follows from n (D k ) p X i=1 n (U i D k ) ffl p X i=1 n Gamma1 (U i E) n Gamma1 p [ i=1 U i E that n (D k ) 0, so the proof is complete. Applying the Lebesgue s characterization of Riemann integrable functions (see [1] or [2]) we get the folowing result. Corollary. Let A be a nonempty Jordan measurable subset of R n and let f : A Gamma R m be a bounded monotone function. Then f is Riemann integrable. ....

Spivak, M., Calculus on Manifolds, Benjamin, New York, 1965.

....33, 34, 35, 38, 40, 41, 42, 43, 44, 46, 53, 54, 56, 57, 58, 84] Especially its interpretation in the context of topological duality will turn out to be crucial for our optic flow definition. 2. 2 Aperture Problem and Optic Flow Ambiguity It is taken for granted that optic flow is a vector field [73, 74]. This reflects the desire to link corresponding points whatever these may be separated by arbitrarily small temporal intervals. The motivation for this is of course that in the physical world such pointwise connections are actually meaningful; ideally they correspond to particle motion or ....

M. Spivak. Calculus on Manifolds. W. A. Benjamin, New York, 1965.

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M. Spivak. Calculus on Manifolds. AddisonWesley, Redwood City, 1965.

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M. Spivak. Calculus on Manifolds, Benjamin, New York, 1965.

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M. Spivak, Calculus on Manifolds, W. A. Benjamin, Menlo Park, CA, 1965.

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Spivak, M. 1968, Calculus on Manifolds, Benjamin/Cummeings.

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Spivak, M. 1968, Calculus on Manifolds, Benjamin/Cummeings.

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Spivak, M. 1968, Calculus on Manifolds, Benjamin/Cummeings.

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Spivak, M. 1968, Calculus on Manifolds, Benjamin/Cummeings.

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M. Spivak, Calculus on Manifolds, W. A. Benjamin, Inc, 1965.

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