Careful numerical implementation of parallel transport algorithm is also needed. Schilds ladder for the parallel transport of deformations. The idea is rather similar to those drafting instruments one sees to draw parallel lines, its based on the principle that if you draw a parallelogram where the opposite sides have equal length, the resulting lines are parallel. Request pdf schilds ladder for the parallel transport of deformations in time series of images followup imaging studies require the evaluation of the anatomical changes over time for. For validation, we apply our approach to rs scans of fifty one subjects before and after a memory, and cotask mpare it against directly using pearsons correlation and its regularized variants as features. I am learning about parallel transport on a riemannian manifold. Or, vice versa, parallel transport is the local realization of a connection. The general theory of relativity arpan saha 1st year engineering physics dd iit bombay monday, november 9, 2009 room 202, physics dept. Schild s ladder describes the parallel transport with respect to the symmetric part of the connection of the space 4.
User mie5199 uploaded this creative ladder schilds ladder parallel transport differential geometry general relativity curve png image on october 10. The longitudinal changes for a specific subject can be evaluated. I must admit, id never heard of schilds ladder before, but it is apparently a constructive explanation of the notion of parallel transport of vectors on a manifold. However, as each rung of schild s ladder requires two imperfect, computationallyintensive image registrations, the schild s. The schild s ladder computes the parallel transport of a along c. It provides a straighforward method to compute a rst order approximation of the parallel transport of a vector along. Greg egan wrote a sf novel by that title, although he was mostly using it as a metaphor for the question of retaining personal identity even as one undergoes changes due to ones path in life.
One way to characterise the geometry of space is to describe how vectors are carried along any path, a process known as parallel transport. The concept of tangent vectors in a manifold also involves a tangent bundle which is a sort of tangent space at each point on that manifold. Parallel transport on the cone manifold of spd matrices. Schilds ladder parallel transport differential geometry general. Schilds ladder is a way to approximate the parallel transport pxy. E cient parallel transport in the group of di eomorphisms via. The book derives its name from schilds ladder, a construction in differential geometry, devised by. Inria efficient parallel transport of deformations in.
Motivation for parallel transport why does one need parallel transport. As an alternative, as i mentioned previously, we can use the geometric construction known as schilds ladder to do the parallel transport. The title, schilds ladder, is taken from a method in differential geometry which is used to approximate the parallel transport of a vector along a curve. Twenty thousand years into the future, an experiment in quantum physics has had a catastrophic result, creating an enormous, rapidly expanding vacuum that devours everything it comes in contact with. The angle by which it twists, is proportional to the area inside the loop. The pole ladder is a way to approximate the parallel transport of a tangent vector to since the original function might be hard to compute. However, as each rung of schilds ladder requires two imperfect, computationallyintensive image registrations, the schilds. What do a parallel transport of vectors on a manifold got to do with the novel. Pole ladder is a modification of schild s ladder for the parallel transport along geodesic curves which is based on the observation that this geodesic can be taken as one of the diagonals of the geodesic parallelogram.
This scheme was called schilds ladder since it was in the spirit of the work of. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Schild s ladder is a 2002 science fiction novel by australian author greg egan. Schild s ladder parallel transport can be discretely approximated by schild s ladder, which takes finite steps along a curve, and approximates levicivita. Schilds ladder parallel transport procedure for an. Efficient parallel transport of deformations in time series.
I tend to think of parallel transport as being defined by schilds ladder. The modern notion of tangent vectors in mathematics involves a tangent bundle, which is to say a separate tangent space at each point on the manifold. We demonstrate that the procedure, while it can be performed forany connection, in fact is only capable of. Parallel transport by schild s ladder it is not possible to directly compare svfs from different subjects, since they have different coordinate systems. In physical terms, it denotes the transport of the deformation vof the shape f0to a shape ft. Parallel transport of a vector on a sphere physics forums. Frontiers detection of conversion from mild cognitive. Parallel transport of deformations in shape space of. I must admit, id never heard of schild s ladder before, but it is apparently a constructive explanation of the notion of parallel transport of vectors on a manifold. Schilds ladder for the parallel transport of deformations in. It is therefore necessary to normalize the svfs to a common template space, for which we employ the parallel transport method do carmo, 1992. The book derives its name from schild s ladder, a construction in differential geometry, devised by the mathematician and physicist alfred schild. Schild s ladder for the parallel transport of di eomorphic deformations parame terized by tangent velocity elds, based on the construction of a geodesic parallel ogram on a manifold.
The schild s ladder is based on the construction of a geodesic parallelogram. Parallel transport of surface deformations from pole. I dont understand how the schild s ladder for parallel transporting a vector through a geodesic can be applied for example in a sphere where the tangent vector seems to be out of the manifold the sphere itself. In this last class, pole ladder is a simplification of schilds ladder for the parallel transport along geodesics that was shown to be particularly. However, the parallel transport can be locally approximated by schilds ladder 29,40 or by the pole ladder 46. A copy of the license is included in the section entitled gnu free documentation license. The schild ladder parallel transports a vector a along the curve c by. Numerical methods based on jacobi elds or geodesics parallelograms are currently used in geomet ric data processing. For example, the schilds ladder see schilds ladder parallel transport. We illustrate the computational advantages and demonstrate the numerical accuracy of this very simple method by comparing with standard methods of transport on simulated images with progressing brain atrophy. The schilds ladder provides a straightforward method to compute a first order approximation of the parallel transport of a vector along a curve using geodesics. We previously showed that the schild s ladder is an e cient and simple method for the parallel transport of diffeomorphic deformations parameterized by tangent velocity elds. Reference request for parallel transport mathoverflow. In this last class, pole ladder is a simplification of schild s ladder for the parallel transport along geodesics that was shown to be particularly simple and numerically.
Parallel transport is an important step in many discrete algorithms for statistical computing on manifolds. The effect of each holonomy on a particle with total spin j is determined by a unitary matrix, u j, which is found by using the appropriate representation of su2 a homomorphism from the group su2 to the group of unitary linear operators on the hilbert space that contains the particles spin state. E cient parallel transport in the group of di eomorphisms. Spin networks are states of quantum geometry in a theory of quantum gravity, discovered by lee smolin and carlo rovelli, which is the conceptual ancestor of the imaginary physics of schilds ladder. Parallel transport on the cone manifold of spd matrices for domain adaptation or yair, student member, ieee, mirela benchen, and ronen talmon, member, ieee. Numerical methods based on jacobi fields or geodesics parallelograms are currently used in geometric data processing. Cartan connections are ehresmann connections with additional structure which allows the parallel transport to be though of as a map rolling a certain model space along a curve in the manifold. Pdf efficient parallel transport of deformations in time. We illustrate the computational advantages and demonstrate the numerical accuracy of this very simple method by comparing with standard methods of. We analyze the schild s ladder parallel transport procedure for an arbitraryconnection. Efficient parallel transport of deformations in time series of images. As another alternative, you can also try reading a lessvague description of how to do parallel transport geometrically, namely schild s ladder to be able to utilize this construction, all youll be able to need to do is to draw great circles on a sphere, and find their midpoints. Iit bombay slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising.
Followup imaging studies require the evaluation of the anatomical changes over time for specific clinical groups. From an identity between parallel transport and jacobi 18. These methods use schild s ladder to approximate parallel transport along a curve by taking small steps in the associated tangent space at each time 6. Archive ouverte hal parallel transport with pole ladder. We demonstrate that the procedure, while it can be performed forany.
The comparison of longitudinal trajectories is therefore the transport of longitudinal deformations in a common reference frame. The book derives its name from schild s ladder, a construction in differential geometry. In this work, we propose the schild s ladder framework as an effective method to transport longitudinal deformations in time series of images in a common space using diffeomorphic registration. In the 70s of the past century 30 proposed a scheme for performing the parallel transport with a very simple geometrical constructions. We previously proposed an effective computational scheme based on the schild s ladder for the parallel transport of diffeomorphic deformations parameterized by tangent velocity fields, based on the construction of a geodesic parallel ogram on a manifold. These methods are rst order approximation schemes which need to be.
The fact that this angle stays constant during parallel transport can be used to recover the connection from its geodesics after all geodesics in riemannian geometry can be defined without the connection. What it is, is an exemplary novel that showcases the brilliance of greg egan as a writer. By lorenzi marco, nicholas ayache and xavier pennec. Riemannian connection on a surface parallel transport. Similar to schild s ladder, used for parallel transport in curved spaces. If the manifold is equipped with an affine connection a covariant derivative or connection on the tangent bundle, then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection.
Schilds ladder may be however ine cient for transporting longitudinal deformations from image time series of multiple time points, in which. From schilds to pole ladder article pdf available in journal of mathematical imaging and vision 5012 october 20. Here is a partial list of applications in statistical shape analysis that require transport. Alfred schild invented a way of doing parallel transport of those vectors. Schilds ladder project gutenberg selfpublishing ebooks. We propose here a di erent construction for the parallel transport of vectors based on geodesics parallelograms.
Parallel transport methods in medical image analysis are currently based on ladders like schild s and pole ladders 6,7 or on variations of jacobi fields 14,8. Schilds ladder parallel transport procedure for an arbitrary. Discrete ladders for parallel transport in transformation. The modern notion of tangent vectors in mathematics involves a tangent bundle, which is to say a separate tangent space at. Parallel transport of deformations in shape space of elastic surfaces qian xie1, sebastian kurtek2, huiling le3. See also basic introduction to the mathematics of curved spacetime. But that demands that you use straight measuring rods geodesics. In geometries with a symmetric connection it fulfills its goal toexpress connection and parallel transport of any vector in terms of geodesics ofsuch. Efficient parallel transport of deformations in time.
Schilds ladder for the parallel transport of di eomorphic deformations parameterized by tangent velocity elds, based on the construction of a geodesic parallelogram on a manifold. In curved space, parallel transport around a loop will generally produce a. Schild s ladder may be however inefficient for transporting longitudinal deformations. In this last class, pole ladder is a simplification of schild s ladder for the parallel transport along geodesics that was shown to be particularly simple and numerically stable in lie groups. Schild s ladder is a master piece of hard science fiction and of the skill and style of greg egan. These methods use schilds ladder to approximate parallel transport along a curve by taking small steps in the associated tangent space at each time 6.
The end goal of both argumetns is to demonstrate the existence and unqiueness of this process we call parallel transport, and to build up some intuition. We analyze the schilds ladder parallel transport procedure for an arbitraryconnection. In this paper, we propose a domain adaptation method using the analytic expression of pt on the cone manifold of spd matrices. Schilds ladder approximate parallel transport mvirt ronny. Schilds ladder parallel transport procedure for an arbitrary connection springerlink. Parallel transport of a vector around a closed loop from a to n to b and back to a on the sphere. Parallel transport can be discretely approximated by schilds ladder, which takes finite steps along a curve, and approximates levicivita parallelogramoids by approximate parallelograms. Parallel transport and geodesics february 24, 20 1 parallel transport beforede. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Schilds ladder is a 2002 science fiction novel by australian author greg egan. As parallel transport supplies a local realization of the connection, it also supplies a local realization of the curvature known as holonomy. The existing methods for computing parallel transport require either the computation of riemannian logarithms, such as schild s ladder, or the christoffel symbols.
Parallel transport on the cone manifold of spd matrices for. In the theory of general relativity, and differential geometry more generally, schilds ladder is a firstorder method for approximating parallel transport of a vector. Inria efficient parallel transport of deformations in time. In this work, we propose the schilds ladder framework as an effective method to transport longitudinal deformations in time series of images in a common space using diffeomorphic registration. The title, schild s ladder, is taken from a method in differential geometry which is used to approximate the parallel transport of a vector along a curve. Journal of mathematical imaging and vision, springer verlag, 20, 50 12, pp. In this last class, pole ladder is a simpli cation of schild s ladder for the parallel transport along geodesics that was shown to be particularly simple. Pole ladder is a modi cation of schilds ladder for the parallel transport along geodesic curves which is based on the observation that this geodesic can be taken as one of the diagonals of the geodesic parallelogram.
The proposed method rely on a solid mathematical background and can be easily used in the lddmm and svf registration settings. To perform parallel transport, we use the schild s ladder algorithm 5, 6. This novel is perhaps the hardest science fiction ever published by egan, filled with nontrivial mathematics and theoretical physics. Schilds ladder for the parallel transport of diffeomorphic deformations parame terized by tangent velocity fields, based on the construction of a. A fanning scheme for the parallel transport along geodesics. Differential geometry of surfaces riemannian connection and parallel transport parallel transport. Chose a point p 2 on the curve separated by p 0 by the value of the parameters the values. Schild s ladder for the parallel transport of deformations in time series of images. Recently, it was shown that for connected, complete, symmetric riemannian manifold, the pole ladder coincides with the parallel trans port along geodesics 50. The logarithm is rarely given in closed form, and therefore expensive to compute, whereas the christoffel symbols are in general hard and costly to compute. We demonstrate that the procedure, while it can be performed forany connection, in fact is only capable of detecting the symmetric part of thisconnection. This is not an action packed futuristic romp that would make the perfect summer blockbuster movie. Parallel transport of deformations in shape space of elastic.
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