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/build* | ||
/doc* | ||
*.pyc | ||
*.DS_Store | ||
/examples/Data/dubrovnik-3-7-pre-rewritten.txt | ||
/examples/Data/pose2example-rewritten.txt | ||
/examples/Data/pose3example-rewritten.txt | ||
/.settings/ | ||
*.txt.user | ||
*.txt.user.6d59f0c |
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/org.eclipse.cdt.codan.core.prefs |
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eclipse.preferences.version=1 | ||
environment/project/cdt.managedbuild.toolchain.gnu.macosx.base.1359703544/PATH/delimiter=\: | ||
environment/project/cdt.managedbuild.toolchain.gnu.macosx.base.1359703544/PATH/operation=append | ||
environment/project/cdt.managedbuild.toolchain.gnu.macosx.base.1359703544/PATH/value=$PATH\:/opt/local/bin | ||
environment/project/cdt.managedbuild.toolchain.gnu.macosx.base.1359703544/append=true | ||
environment/project/cdt.managedbuild.toolchain.gnu.macosx.base.1359703544/appendContributed=true |
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GTSAM Concepts | ||
============== | ||
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As discussed in [Generic Programming Techniques](http://www.boost.org/community/generic_programming.html), concepts define | ||
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* associated types | ||
* valid expressions, like functions and values | ||
* invariants | ||
* complexity guarantees | ||
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Below we discuss the most important concepts use in GTSAM, and after that we discuss how they are implemented/used/enforced. | ||
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Manifold | ||
-------- | ||
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To optimize over continuous types, we assume they are manifolds. This is central to GTSAM and hence discussed in some more detail below. | ||
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[Manifolds](http://en.wikipedia.org/wiki/Manifold#Charts.2C_atlases.2C_and_transition_maps) and [charts](http://en.wikipedia.org/wiki/Manifold#Charts.2C_atlases.2C_and_transition_maps) are intimately linked concepts. We are only interested here in [differentiable manifolds](http://en.wikipedia.org/wiki/Differentiable_manifold#Definition), continuous spaces that can be locally approximated *at any point* using a local vector space, called the [tangent space](http://en.wikipedia.org/wiki/Tangent_space). A *chart* is an invertible map from the manifold to that tangent space. | ||
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In GTSAM, all properties and operations needed to use a type must be defined through template specialization of the struct `gtsam::traits`. Concept checks are used to check that all required functions are implemented. | ||
In detail, we ask the following are defined in the traits object: | ||
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* values: | ||
* `enum { dimension = D};`, an enum that indicates the dimensionality *n* of the manifold. In Eigen-fashion, we also support manifolds whose dimenionality is only defined at runtime, by specifying the value -1. | ||
* types: | ||
* `TangentVector`, type that lives in tangent space. This will almost always be an `Eigen::Matrix<double,n,1>`. | ||
* `ChartJacobian`, a typedef for `OptionalJacobian<dimension, dimension>`. | ||
* `ManifoldType`, a pointer back to the type. | ||
* `structure_category`, a tag type that defines what requirements the type fulfills, and therefore what requirements this traits class must fulfill. It should be defined to be one of the following: | ||
* `gtsam::traits::manifold_tag` -- Everything in this list is expected | ||
* `gtsam::traits::group_tag` -- The functions defined under **Groups** below. | ||
* `gtsam::traits::lie_group_tag` -- Everything in this list is expected, plus the functions defined under **Groups**, and **Lie Groups** below. | ||
* `gtsam::traits::vector_space_tag` -- Everything in this list is expected, plus the functions defined under **Groups**, and **Lie Groups** below. | ||
* valid expressions: | ||
* `size_t dim = traits<T>::getDimension(p);` static function should be defined. This is mostly useful if the size is not known at compile time. | ||
* `v = traits<T>::Local(p,q)`, the chart, from manifold to tangent space, think of it as *q (-) p*, where *p* and *q* are elements of the manifold and the result, *v* is an element of the vector space. | ||
* `p = traits<T>::Retract(p,v)`, the inverse chart, from tangent space to manifold, think of it as *p (+) v*, where *p* is an element of the manifold and the result, *v* is an element of the vector space. | ||
* invariants | ||
* `Retract(p, Local(p,q)) == q` | ||
* `Local(p, Retract(p, v)) == v` | ||
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Group | ||
----- | ||
A [group]("http://en.wikipedia.org/wiki/Group_(mathematics)"") should be well known from grade school :-), and provides a type with a composition operation that is closed, associative, has an identity element, and an inverse for each element. The following should be added to the traits class for a group: | ||
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* valid expressions: | ||
* `r = traits<T>::Compose(p,q)`, where *p*, *q*, and *r* are elements of the manifold. | ||
* `q = traits<T>::Inverse(p)`, where *p* and*q* are elements of the manifold. | ||
* `r = traits<T>::Between(p,q)`, where *p*, *q*, and *r* are elements of the manifold. | ||
* static members: | ||
* `traits<T>::Identity`, a static const member that represents the group's identity element. | ||
* invariants: | ||
* `Compose(p,Inverse(p)) == Identity` | ||
* `Compose(p,Between(p,q)) == q` | ||
* `Between(p,q) == Compose(Inverse(p),q)` | ||
The `gtsam::group::traits` namespace defines the following: | ||
* values: | ||
* `traits<T>::Identity` -- The identity element for this group stored as a static const. | ||
* `traits<T>::group_flavor` -- the flavor of this group's `compose()` operator, either: | ||
* `gtsam::traits::group_multiplicative_tag` for multiplicative operator syntax ,or | ||
* `gtsam::traits::group_additive_tag` for additive operator syntax. | ||
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We do *not* at this time support more than one composition operator per type. Although mathematically possible, it is hardly ever needed, and the machinery to support it would be burdensome and counter-intuitive. | ||
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Also, a type should provide either multiplication or addition operators depending on the flavor of the operation. To distinguish between the two, we will use a tag (see below). | ||
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Lie Group | ||
--------- | ||
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A Lie group is both a manifold *and* a group. Hence, a LIE_GROUP type should implements both MANIFOLD and GROUP concepts. | ||
However, we now also need to be able to evaluate the derivatives of compose and inverse. | ||
Hence, we have the following extra valid static functions defined in the struct `gtsam::traits<T>`: | ||
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* `r = traits<T>::Compose(p,q,Hq,Hp)` | ||
* `q = traits<T>::Inverse(p,Hp)` | ||
* `r = traits<T>::Between(p,q,Hq,H2p)` | ||
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where above the *H* arguments stand for optional Jacobian arguments. | ||
That makes it possible to create factors implementing priors (PriorFactor) or relations between two instances of a Lie group type (BetweenFactor). | ||
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In addition, a Lie group has a Lie algebra, which affords two extra valid expressions: | ||
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* `v = traits<T>::Logmap(p,Hp)`, the log map, with optional Jacobian | ||
* `p = traits<T>::Expmap(v,Hv)`, the exponential map, with optional Jacobian | ||
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Note that in the Lie group case, the usual valid expressions for Retract and Local can be generated automatically, e.g. | ||
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``` | ||
T Retract(p,v,Hp,Hv) { | ||
T q = Expmap(v,Hqv); | ||
T r = Compose(p,q,Hrp,Hrq); | ||
Hv = Hrq * Hqv; // chain rule | ||
return r; | ||
} | ||
``` | ||
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For Lie groups, the `exponential map` above is the most obvious mapping: it | ||
associates straight lines in the tangent space with geodesics on the manifold | ||
(and it's inverse, the log map). However, there are two cases in which we deviate from this: | ||
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However, the exponential map is unnecessarily expensive for use in optimization. Hence, in GTSAM there is the option to provide a cheaper chart by means of the `ChartAtOrigin` struct in a class. This is done for *SE(2)*, *SO(3)* and *SE(3)* (see `Pose2`, `Rot3`, `Pose3`) | ||
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Most Lie groups we care about are *Matrix groups*, continuous sub-groups of *GL(n)*, the group of *n x n* invertible matrices. In this case, a lot of the derivatives calculations needed can be standardized, and this is done by the `LieGroup` superclass. You only need to provide an `AdjointMap` method. | ||
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Vector Space | ||
------------ | ||
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While vector spaces are in principle also manifolds, it is overkill to think about charts etc. Really, we should simply think about vector addition and subtraction. I.e.where | ||
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* `Identity == 0` | ||
* `Inverse(p) == -p` | ||
* `Compose(p,q) == p+q` | ||
* `Between(p,q) == q-p` | ||
* `Local(q) == p-q` | ||
* `Retract(v) == p+v` | ||
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This considerably simplifies certain operations. A `VectorSpace` superclass is available to implement the traits. Types that are vector space models include `Matrix`, `Vector`, any fixed or dynamic Eigen Matrix, `Point2`, and `Point3`. | ||
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Testable | ||
-------- | ||
Unit tests heavily depend on the following two functions being defined for all types that need to be tested: | ||
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* valid expressions: | ||
* `Print(p,s)` where s is an optional string | ||
* `Equals(p,q,tol)` where tol is an optional (double) tolerance | ||
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Implementation | ||
============== | ||
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GTSAM Types start with Uppercase, e.g., `gtsam::Point2`, and are models of the | ||
TESTABLE, MANIFOLD, GROUP, LIE_GROUP, and VECTOR_SPACE concepts. | ||
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`gtsam::traits` is our way to associate these concepts with types, | ||
and we also define a limited number of `gtsam::tags` to select the correct implementation | ||
of certain functions at compile time (tag dispatching). | ||
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Traits | ||
------ | ||
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However, a base class is not a good way to implement/check the other concepts, as we would like these | ||
to apply equally well to types that are outside GTSAM control, e.g., `Eigen::VectorXd`. This is where | ||
[traits](http://www.boost.org/doc/libs/1_57_0/libs/type_traits/doc/html/boost_typetraits/background.html) come in. | ||
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We use Eigen-style or STL-style traits, that define *many* properties at once. | ||
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Note that not everything that makes a concept is defined by traits. Valid expressions such as traits<T>::Compose are | ||
defined simply as static functions within the traits class. | ||
Finally, for GTSAM types, it is perfectly acceptable (and even desired) to define associated types as internal types, | ||
rather than having to use traits internally. | ||
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Concept Checks | ||
-------------- | ||
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Boost provides a nice way to check whether a given type satisfies a concept. For example, the following | ||
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BOOST_CONCEPT_ASSERT(IsVectorSpace<Point2>) | ||
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asserts that Point2 indeed is a model for the VectorSpace concept. | ||
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Future Concepts | ||
=============== | ||
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Group Action | ||
------------ | ||
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Group actions are concepts in and of themselves that can be concept checked (see below). | ||
In particular, a group can *act* on another space. | ||
For example, the [cyclic group of order 6](http://en.wikipedia.org/wiki/Cyclic_group) can rotate 2D vectors around the origin: | ||
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q = R(i)*p | ||
where R(i) = R(60)^i, where R(60) rotates by 60 degrees | ||
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Hence, we formalize by the following extension of the concept: | ||
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* valid expressions: | ||
* `q = traits<T>::Act(g,p)`, for some instance, *p*, of a space *S*, that can be acted upon by the group element *g* to produce *q* in *S*. | ||
* `q = traits<T>::Act(g,p,Hp)`, if the space acted upon is a continuous differentiable manifold. * | ||
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In the latter case, if *S* is an n-dimensional manifold, *Hp* is an output argument that should be | ||
filled with the *nxn* Jacobian matrix of the action with respect to a change in *p*. It typically depends | ||
on the group element *g*, but in most common example will *not* depend on the value of *p*. For example, in | ||
the cyclic group example above, we simply have | ||
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Hp = R(i) | ||
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Note there is no derivative of the action with respect to a change in g. That will only be defined | ||
for Lie groups, which we introduce now. | ||
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Lie Group Action | ||
---------------- | ||
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When a Lie group acts on a space, we have two derivatives to care about: | ||
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* `gtasm::manifold::traits<T>::act(g,p,Hg,Hp)`, if the space acted upon is a continuous differentiable manifold. | ||
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An example is a *similarity transform* in 3D, which can act on 3D space, like | ||
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q = s*R*p + t | ||
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Note that again the derivative in *p*, *Hp* is simply *s R*, which depends on *g* but not on *p*. | ||
The derivative in *g*, *Hg*, is in general more complex. | ||
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For now, we won't care about Lie groups acting on non-manifolds. |
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GTSAM was made possible by the efforts of many collaborators at Georgia Tech | ||
GTSAM was made possible by the efforts of many collaborators at Georgia Tech, listed below with their current afffiliation, if they left Tech: | ||
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Doru Balcan | ||
Chris Beall | ||
Alex Cunningham | ||
Alireza Fathi | ||
Eohan George | ||
Viorela Ila | ||
Yong-Dian Jian | ||
Michael Kaess | ||
Kai Ni | ||
Carlos Nieto | ||
Duy-Nguyen | ||
Manohar Paluri | ||
Christian Potthast | ||
Richard Roberts | ||
Grant Schindler | ||
* Sungtae An | ||
* Doru Balcan, Bank of America | ||
* Chris Beall | ||
* Luca Carlone | ||
* Alex Cunningham, U Michigan | ||
* Jing Dong | ||
* Alireza Fathi, Stanford | ||
* Eohan George | ||
* Alex Hagiopol | ||
* Viorela Ila, Czeck Republic | ||
* Vadim Indelman, the Technion | ||
* David Jensen, GTRI | ||
* Yong-Dian Jian, Baidu | ||
* Michael Kaess, Carnegie Mellon | ||
* Zhaoyang Lv | ||
* Andrew Melim, Oculus Rift | ||
* Kai Ni, Baidu | ||
* Carlos Nieto | ||
* Duy-Nguyen Ta | ||
* Manohar Paluri, Facebook | ||
* Christian Potthast, USC | ||
* Richard Roberts, Google X | ||
* Grant Schindler, Consultant | ||
* Natesh Srinivasan | ||
* Alex Trevor | ||
* Stephen Williams, BossaNova | ||
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at ETH, Zurich | ||
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* Paul Furgale | ||
* Mike Bosse | ||
* Hannes Sommer | ||
* Thomas Schneider | ||
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at Uni Zurich: | ||
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* Christian Forster | ||
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Many thanks for your hard work!!!! | ||
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Frank Dellaert |
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