Applications of Lie Groups and Lie Algebra to Computer Vision: A Brief Survey
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2012 International Conference on Systems and Informatics (ICSAI 2012)
Applications of Lie Groups and Lie Algebra to
Computer Vision: A Brief Survey
Qiang Xu, Dengwu Ma
Department of Ordnance Science and Technology
Naval Aeronautical and Astronautical University
Yantai, China
Abstract—Recent years an extensive literature appears using the SE(3) [1, 3, 4, 5, 6], special Linear group SL(3) [7] and their
Lie groups theory to solve the problems of computer vision. Lie corresponding Lie algebra ga(2), so(3), se(3), sl(3).
groups theory is the natural representation of a space of
transformations. Lie algebra is the tangent space of Lie groups at One of the advantages of using Lie algebras is that it allows
the identity. From Lie groups to Lie algebra, we can establish a us to guide an object along the shortest orbit (geodesic), i.e. the
mapping from the multiplicative structure to an equivalent shortest path between two points of the Lie group manifold.
vector space representation, which makes correlation calculation Another advantage is that it is possible to make a natural
become rational and precise. Based on the linear structure of Lie parameterization of the transformation whether it is in 2D or
algebra, many statistical learning methods can be readily 3D space. And also the linear features of Lie algebra make it
applied. This survey briefly reviews the different approaches possible to build statistical model. In the past decade, most
about the use of Lie groups theory that have been developed by approaches fall roughly into one of two categories: (1) Using
research; introducing the mathematical background of Lie Lie algebra corresponding to Lie transformation groups as
groups theory corresponding to computer vision; describing the parameters to model and optimize, including object tracking [2,
main approaches in details according two categories. 8, 9, 10, 11], calculating mean [4, 12, 13, 14] and learning
parameters of transformation model [1, 6, 15, 16, 17], (2)
Keywords-Lie groups; Lie algebra; transformation matrix; Based the linear features of vector space of Lie algebra, build
computer vision;
probability model [3, 5, 18, 19] and clustering algorithm [20,
21, 22, 23] for pattern recognition.
I. INTRODUCTION
The rest of the paper is organized as follows:§2 presents
The transformation matrices play a crucial role in the the mathematical background of Lie groups and Lie algebra.
dynamical analysis of computer vision. All the transformation
§3 analysis all kinds of applications of Lie groups and Lie
matrices comprise the elements of an algebraic structure
known as Lie groups. The tangent space of Lie groups at the algebra in computer vision according to above categories.
identity is called the Lie algebra. The exponential map exp is a Finally gives the conclusion.
mapping from Lie algebra elements to Lie groups elements.
The logarithmic map log takes group elements onto the tangent II. MATHEMATICAL BACKGROUND
plane. Based the exponential maps, the “multiplication” of Lie
group elements become the “sum” of Lie algebra elements, A. Lie groups and Lie algebra
which make dynamical analysis easy. A Lie group G is an algebraic group with the structure of a
A Lie groups is a group in the category of smooth differentiable Riemannian manifold. In particular, the group is
manifolds. Beside the algebraic properties, they enjoy also characterized by a unique identity element e ∈ G and two
differential geometric properties. The most obvious group operations
construction is a Lie algebra which is the tangent space at the multiplication g1 g 2 : G × G → G , and
unit endowed with the Lie bracket between left-invariant
vector fields. Beside the structure theory there is also the wide inversion g −1 : G → G ，
field of representation theory. The most common examples of
Lie groups, and those which have the greatest application to which are differentiable mapping. The tangent space at the
computer vision, are the matrix groups. These are all identity is called the Lie algebra of the group. We denote as
subgroups of the general linear group GL( N; R ) the group of exp( ) and log( ), respectively, the exponential map and the
nonsingular N×N real matrices. The Lie algebra associated logarithm map at the identity e. These mapping at a generic
with GL( N; R ) is gl( N; R ), the set of all N×N real matrices. point X∈G can be computed using parallel transport
Most of literature about the applications of Lie groups in as:
computer vision have focused on Affine group GA(2) [1, 2],
Rotation group SO(3) [3, 4, 5, 6], special Euclidean group exp X ( A ) = X exp ( X −1 A ) , (1)
2024 978-1-4673-0199-2/12/$31.00 ©2012 IEEElog X ( B ) = X log ( X −1 B ) , (2) In the limit N → ∞ , this expression reduces to the matrix
exponential equation,
With A ∈ TX G and B ∈ G. The exponential and logarithm
I ( z ) = e zG I 0 (5)
maps of logarithm maps of a matrix are given by
where I0 is the initial or reference input. Thus, each of the
( −1)
n −1
∞ ∞
1 elements of our one-parameter Lie group can be written as T(z)
exp ( A ) = ∑ An log ( B ) = ∑ (B − I )
n
. (3)
n=0 n! n=0 n = ezG. The generator G of the Lie group is related to the
dT
When we look the transformation matrix as Lie group, the derivative of T(z) with respect to z : = GT .This leads to an
corresponding Lie algebra can be looked as infinitesimal dx
transformation matrix. Any finite transformation can then be alternate way of deriving (5). Consider the Taylor series
constructed by the repeated application, or “integration,” of expansion of a transformed input I(z) in terms of a previous
this infinitesimal transformation. Below We will analysis it in input I(0):
detail. Interested readers are referred to [24, 25] for an
dI ( 0 ) d 2 I (0) z 2
elaborated introduction. I ( z ) = I (0) + z+ + ", (6)
dz dz 2
Suppose we have a point (in general, a vector) I 0 , which is
an element in space F. Let T I 0 denote a transformation of the where z denotes the relative transformation between I(z) and I
dI
point I 0 to another point, say I1 . The transformation operator (0). Defining = GI for some operator matrix G, we can
dz
T is completely specified by its actions on all points in the
space F. Suppose T belongs to a family of operators Г. rewrite (6) as I ( z ) = e zG I ( 0 ) , which is the same as (5) with
Consider the case where T is a group, that is, there exists a I0= I(0). Thus, some previous approaches based on first order
mapping f : Г × Г → Г from pairs of transformations to Taylor series expansions can be viewed as special cases of the
another transformation such that (1) f is associative, (2) there Lie group–based generative model.
exists a unique identity transformation, and (3) for every T ∈ an alternative to the matrix representation of infinitesimal
Г, there exists a unique inverse transformation of T. We are generators is in terms of differential operators. We demonstrate
interested in transformation groups because most common how to solve the infinitesimal generators and differential
types of image transformation obey properties 1 to 3. For operators taking SO(3) for example.
example, it is easy to see that translation is associative
( T1 (T2 • T3 ) I 0 = (T1 • T2 )T3 I 0 ), with a unique identity (zero 1) Matrix Form of Generators
translation) and a unique inverse (the inverse of T is T -1). Consider first rotations about the x-axis by an angle ϕ1 :
Continuous transformations are those that can be made
infinitesimally small. Due to their favorable properties as ⎛1 0 0 ⎞
⎜ ⎟
described below, we will be especially concerned with R1 (ϕ1 ) = ⎜ 0 cos ϕ1 − sin ϕ1 ⎟ . (7)
continuous transformation groups or Lie groups. Continuity is ⎜ 0 sin ϕ cos ϕ1 ⎟⎠
associated with both the transformation operators T and the ⎝ 1
group Г. Each T ∈Г is assumed to implement a continuous
mapping from F → F . We focus on the case where T is
parameterized by a single real number z (multiple The corresponding infinitesimal generator is calculated:
transformations in an image can be handled by combining
⎛0 0 0 ⎞
several single-parameter transformations as discussed below). dR ⎜ ⎟
Then the group T is continuous if the function T(z) : R→ T is X1 = 1 ϕ1 = ⎜ 0 0 −1⎟ . (8)
dϕ1 ⎜0 1 0 ⎟
continuous; that is, any T ∈ Г is the image of some z ∈ R , ⎝ ⎠
and any continuous variation of z results in a continuous
variation of T. Let T(0) be equivalent to the identity For rotations about the x-axes by an angle ϕ3 , the rotation
transformation. Then as z → 0, the transformation T(z) gets matrix is:
arbitrarily close to identity. Its effect on I0 can be written as (to
first order in z) T(z)I0≈ (1 + zG)I0 for some matrix G, which ⎛ cos ϕ3 − sin ϕ3 0⎞
⎜ ⎟
is known as the generator (or operator) for the transformation R3 (ϕ3 ) = ⎜ sin ϕ3 cos ϕ3 0⎟ . (9)
group. A macroscopic transformation I1= I (z) = T (z)I0 can be ⎜ 0 0 1 ⎟⎠
produced by chaining together a number of these infinitesimal ⎝
transformations. For example, by dividing the parameter z into and the corresponding generator is:
N equal parts and performing each transformation in turn, we
obtain ⎛ 0 −1 0 ⎞
dR3 ⎜ ⎟
X3 = = ⎜1 0 0⎟ . (10)
I ( z ) = (1 + ( z N ) G ) I 0 .
N ϕ3
(4) d ϕ3 ⎜0 0 0⎟
⎝ ⎠
2025Finally, for rotations about the y-axis by an angle ϕ 2 , we transformations. A principal geodesic is one that accounts for
have the maximum variation in the set of transformations along the
path, analogous to principal component of a covariance matrix.
⎛ cos ϕ2 0 sin ϕ 2 ⎞ The intrinsic mean represents the “average” of a set of
⎜ ⎟
R2 (ϕ 2 ) = ⎜ 0 1 0 ⎟. (11) transformations, i.e. the transformation that minimizes the Lie
⎜ − sin ϕ 0 cos ϕ 2 ⎟⎠ distance to all the transformations in the set.
⎝ 2
and the generator is: 1) Baker-Campbell-Hausdorff formlua
If A and B are matrices, then
⎛ 0 0 1⎞
dR ⎜ ⎟ ⎧ 1 ⎫
X2 = 2 ϕ2 = ⎜ 0 0 0⎟ . (12) ⎪ A + B + 2 [ A, B ] ⎪
dϕ2 ⎜ −1 0 0 ⎟ ⎪ ⎪
⎝ ⎠ ⎪ 1 ⎪
exp ( A ) • exp ( B ) = exp ⎨+ ⎡⎣ A, [ A, B ]⎤⎦ ⎬ (16)
2) Operator Form of Generators ⎪ 12 ⎪
⎪ 1 ⎪
we first write the general rotation as an expansion to first ⎪ 12 ⎣ [
− ⎡ B , A, B ]⎦ ⎪
⎤ + "
order in each of the ϕi about the identity. This yields the ⎩ ⎭
transformation matrix: where “...” represents a series of more complicated
commutators involving A and B. In other words, if we can
⎛ x′ ⎞ ⎛ 1 −ϕ3 ϕ2 ⎞ ⎛ x ⎞ express two group elements as exponentials of lie algebra
⎜ ′⎟ ⎜ ⎟⎜ ⎟ elements, then we can express their product as an exponential
⎜ y ⎟ = ⎜ ϕ3 1 −ϕ1 ⎟ ⎜ y ⎟ , (13)
of a lie algebra element (because the algebra is closed under
⎜ z ′ ⎟ ⎜ −ϕ ϕ1 1 ⎟⎠ ⎜⎝ z ⎟⎠
⎝ ⎠ ⎝ 2 commutation), and furthermore it is possible to derive the
and expanding the right-hand side to first order in the ϕi group multiplication structure from the commutation relations.
yields the following expression: exp ( A ) • exp ( B ) ≈ exp ( A + B ) (17)
F ( x′, y ′, z ′ ) = F ( x, y, z ) We always use (17) for first-order approximation of (16).
⎛ ∂F ∂F ⎞ 2) Lie distance and principal geodesics
+⎜ y− z ⎟ ϕ1 The distances on the manifold are measured by the lengths
⎝ ∂z ∂y ⎠
of the curves connecting the points, and the minimum length
⎛ ∂F ∂F ⎞ (14) curve between two points is called the geodesic. From I there
+⎜ z− x ⎟ ϕ2
⎝ ∂x ∂z ⎠ exists a unique geodesic starting with vector m ∈ g. The
⎛ ∂F ∂F ⎞ exponential map, exp: g → G maps the vector m to the point
+⎜ x− y ⎟ ϕ3 . reached by this geodesic (for noncompact groups, the
⎝ ∂y ∂x ⎠ exponential map should be expp [7]). Let exp(m) = M, then
Since F is an arbitrary differentiable function, we can the length of the geodesic is given by ρ(I, M) = m . The inverse
identify the generators Xi of rotations about the coordinate axes mapping is given by log : G → g. Let M1 and M2 be two
from the coefficients of the ϕi , i.e., with the differential motion matrices, and let m1=log(M1) and m2= log(M2). Using
operators: Baker-Campbell-Hausdorff formula (16) which gives the
exponential identity for non-commutative Lie groups, a first
∂ ∂ order approximation to the geodesic distance between the two
X1 = y −z motion matrices, the geodesic distance between two group
∂z ∂y
elements is measured by
∂ ∂
X2 = x − y (15)
∂y ∂x ρ ( M 1 , M 2 ) = log ⎡⎣ M 1−1 M 2 ⎤⎦
∂ ∂ = log ⎡⎣ exp ( −m1 ) exp ( m2 ) ⎤⎦
X3 = x − y .
∂y ∂x (18)
⎣ (
= log ⎡ exp m2 − m1 + o ( ( m , m ) ) )⎤⎦
1 2
B. Relative caculation
The central elements of the Lie group and Lie algebra ≈ m2 − m1 .
framework are: multiplication; Lie distance; principal 3) Intrinsic mean
geodesics; and intrinsic mean; Multiplication is Baker-
Let { xi }i =1 be a set of points in a smooth Riemannian
N
Campbell-Hausdorff formlua that demonstrate how the
multplication of Lie groups matrix can be expressed by Lie manifold M. The Riemannian metric can then be used to obtain
algebra. Lie distance is a measure of the similarity of two the geometric distance d(x, y) between any two points x, y ∈
transformations. A geodesic is a 1-d subspace of M. The intrinsic mean of this set is defined as the point
transformations that is the shortest path between two x ∈ M for which the sum of squared distances
2026N N
∑d (x , x)
i =1
2
i (19) variational minimiser of the “deviation” ∑(X
i =1
i − X )2 .
However such a notion cannot be applied directly to elements
is minimized. It can be shown that a necessary and sufficient
of a group since a group manifold is not equivalent to a vector
condition for x to be the intrinsic mean is that space. For example, the arithmetic average of two rotation
N matrices is not necessarily a valid rotation matrix. If we view
∑ log ( x ) = 0 .
x i (20) elements of a group G as being embedded in a real, vector
i =1 space the sample average is denoted as the extrinsic average.
Here, the group is first embedded in a Euclidean space φ : G
III. APPLICATIONS → Rn which induces a metric on the space and the sample
Recent years, there are extensive literatures to take 1 N
average can be defined as φ ( X ) = ∑ φ ( X i ) . However this
advantage of the structure of Lie groups and Lie algebra to N i =1
handle problems in computer vision. They mainly fall into two sample average is not necessarily an element of the group G
kinds, optimizing problem and pattern recognition. and we need to project it onto the manifold G in an optimal
A. Optimizing Problem sense, X = Ρ(φ ( X ) ) . On the other hand if we define d (. , .) as
1) Object tracking the intrinsic distance between points on a manifold, then the
The tracking problem is a very important topic in the “true” intrinsic average can be defined as (19). Here, the
computer vision field. It is widely used in a variety of areas, intrinsic average is computed by measuring the Riemannian
such as surveillance visual servoing, medical applications, etc. distance between elements of the group and is automatically an
This problem has been addressed using very different element of the group G. In general, the intrinsic average is
approaches, being the most popular feature tracking and preferable over the extrinsic average but is often hard to
template tracking. In the template alignment methods, the sum compute due to the non-linearity of the Riemannian distance
of squared difference between the template and the image function d(. , .) and the need to parameterize the group
intensities was minimized as an iterative least squares problem. manifold G. However as we have seen in (18), for matrix Lie
The method requires computation of the image gradient, the groups the intrinsic average can be computed efficiently.
Jacobian and the Hessian for each iteration, which makes it In [4], the Lie-algebras of the Special Orthogonal and
slow. Variants of the method were proposed to overcome the Special Euclidean groups are used to define averages on the
difficulty. These methods estimate the additive updates to the Lie-group which in turn gives statistically meaningful, efficient
motion parameters via linearization. The methods of Lie and accurate algorithms for fusing motion information. In [13],
groups estimate the additive updates to the motion parameters they give precise definitions of different, properly invariant
via Lie algebra. notions of mean or average rotation. Each mean is associated
In contrast to the conventional approach, when using the with a metric in SO(3). The metric induced from the Frobenius
Lie algebra representation, each intermediate transform is also inner product gives rise to a mean rotation that is given by the
a rigid transformation, i.e. the elements of the sequence remain closest special orthogonal matrix to the usual arithmetic mean
within the subgroup. Generally, when the process is of the given rotation matrices. The mean rotation associated
parameterized using the Lie algebra representation, if the with the intrinsic metric on SO(3) is the Riemannian center of
initial and final transformations are within a subgroup, then mass of the given rotation matrices. The paper show that the
intermediate transformations along the path will also lie in that Riemannian mean rotation shares many common features with
subgroup. Moreover, the process defined using the Lie algebra the geometric mean of positive numbers and the geometric
is optimal in the sense that it corresponds to the shortest mean of positive Hermitian operators. They give some
geodesics connecting the two transforms on the affine examples with closed-form solutions of both notions of mean.
manifold. In [14], they propose distributed algorithms for estimating the
average pose of an object viewed by a localized network of
In [11] they use Lie groups’ theory for motion estimation. camera motes. To this effect, the paper proposes distributed
A few of the related papers are as follows. In [6], a mode averaging consensus algorithms on the group of 3-D rigid-
finding algorithm on Euclidean motion group was described body transformations, SE(3). They rigorously analyze the
for a multiple motion estimation problem. In [1], an addition convergence of the proposed algorithms, and show that naive
operation was defined on the Lie algebra for tracking an affine generalizations of Euclidean consensus algorithms fail to
snake. In [8], the additive updates were performed on the Lie converge to the correct solution.
algebra for template tracking.
3) learning parameters of transformation model
2) calculating mean
In computer vision, how to learn parameters of
In many applications, such as data fusion and motion transformation model accurately play an important role.
estimation, calculating mean is necessary.
In [16], they describe a new, unsupervised approach to
In a vector space, the sample mean or average of a set learning invariances based on Lie group theory. Unlike
1 N traditional approaches that sacrifice information about
{X1, · · · , XN} is given by X = ∑ X i which is the transformations to achieve invariance, the Lie group approach
N i =1
2027explicitly models the effects of transformations in images. As a Euclidean vector space. The medial parameters are not elements
result, estimates of transformations are available for other of a Euclidean space, and thus standard PCA is not applicable.
purposes, such as pose estimation and visuomotor control. In this paper , they develop the notion of a Gaussian distribution
Previous approaches based on first-order Taylor series on this Lie group. They then derive the maximum likelihood
expansions of images can be regarded as special cases of the estimates of the mean and the covariance of this distribution.
Lie group approach, which utilizes a matrix-exponential-based Analogous to principal component analysis of covariance in
generative model of images and can handle arbitrarily large Euclidean spaces, They define principal geodesic analysis on
transformations. They present an unsupervised expectation- Lie groups for the study of anatomical variability in medially-
maximization algorithm for learning Lie transformation defined objects.
operators directly from image data containing examples of
transformations. They then demonstrate that the algorithm can In [7], they represent geometric warps between planar
also recover novel transformation operators from natural image objects by projective Lie groups. Compared with the compact
sequences. We conclude by showing that the learned operators Lie group SO(n, R), the Riemannian exponential map on the
can be used to both generate and estimate transformations in noncompact Lie group SL(n, R) determined by a Riemannian
images, thereby providing a basis for achieving visual metric is usually different from the Lie group exponential map
determined by one-parameter subgroups. They compute the
invariance.
samples intrinsic means on the special linear group SL(3, R)
In [2], they address the parameters’ estimation of 2D and based on the Riemannian manifold optimum algorithm and
3D transformations. For the estimation they present a method propose the Lie group norm distribution. The test results of the
based on system identification theory, they named it the “A- planar object recognition in the simple background, which is
method”. The transformations are considered as elements of based on the full Bayes statistical rule, have shown that the
the Lie group GL(n) or one of its subgroups. They represent proposed algorithm with the intrinsic statistical property of the
the trasformations in terms of their Lie Algebra elements. The projective group may improve the rate of recognition
Lie algebra approach assures to follow the shortest path or effectively.
geodesic in the involved Lie group.
2) clustering algorithm
B. pattern recognition In [23], they introduce a 3-d representation of vehicles as a
One difficulty in developing statistical models of space of scale and orientation transformations that define the
parameterized geometric transformations is that their inherent shape of individual vehicle instances. This shape space forms a
group structure, which complicates the application of statistical group, where the similarity of different vehicle observations
methods that rely on underlying vector spaces. While one may can be evaluated using a distance measure defined by Lie
ignore the multiplicative nature of the underlying group group theory. A generic class of vehicles is represented by a
structure, treating transform matrices as if they lie in a linear set of curves on the Lie group manifold, called geodesics. The
space, this leads to undesired effects and complexities when classification of any given vehicle instance is achieved by
incorporating geometric constraints. Lie algebra theory finding the class with the smallest Lie distance between the
mitigates many of these issues. Its main utility is to construct geodesics and the vehicle shape. Vehicle recognition is carried
equivalent representation in a vector space while maintaining out on 3-d LIDAR point clouds. The performance of the Lie
connections to the geometric transform group. Thereby, we classifier is evaluated against two other approaches and found
acquire a vector representation of each transform to which to provide superior recognition performance, particularly with
many widely used statistical models can readily be applied respect to the ability to generalize from a small number of
labeled prototypes.
1) probability model
In [21] , they exploit the linear behavior of Lie group
In [19], a Lie algebraic representation of the transform elements in the tangent plane to the group manifold in order to
process is introduced, which maps the transformation group to implement a mean-shift algorithm for clustering
a vector space, and thus overcomes the difficulties due to the transformations. The modes correspond to global
group structure. Consequently, the statistical learning transformation of objects due to 3-d orientation.
techniques based on vector spaces can be readily applied.
Moreover, they discuss the intrinsic connections between the IV. CONCLUSION
Lie algebra and the Linear dynamical processes, showing that
the model induces spatially varying fields that can be estimated Lie groups theory is the natural representation of a space of
from local motions without continuous tracking. Following transformations. Lie algebra is the tangent space of Lie groups
this, they further develop a statistical framework to robustly at the identity. From Lie groups to Lie algebra, we can
learn the flow models from noisy and partially corrupted establish a mapping from the multiplicative structure to an
observations. equivalent vector space representation. when the process is
parameterized using the Lie algebra representation, if the
In [26], the Gaussian distribution is the basis for many initial and final transformations are within a subgroup, then
methods used in the statistical analysis of shape. One such intermediate transformations along the path will also lie in that
method is principal component analysis, which has proven to be subgroup. Moreover, the process defined using the Lie algebra
a powerful technique for describing the geometric variability of is optimal in the sense that it corresponds to the shortest
a population of objects. The Gaussian framework is well geodesics connecting the two transforms on the affine
understood when the data being studied are elements of a
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