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首页SBA:高效灵活的通用稀疏束调整软件包
SBA:高效灵活的通用稀疏束调整软件包
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更新于2024-07-15
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SBA是一个专为通用稀疏束调整设计的软件包,由Manolis I. Lourakis和Antonisa Argyros开发,隶属于希腊的研究与技术基础组织。束调整在计算机视觉中的结构与运动估计算法中扮演着关键角色,它是一个大型的、非线性最小二乘问题,用于获取最优的特征匹配参数。然而,由于涉及大量参数,通用最小二乘方法在应用于束调整时,会带来显著的计算和内存消耗。 SBA软件特别关注于稀疏性这一特性,即在参数之间的相互作用不强时,Jacobian矩阵呈现为稀疏。这意味着可以通过利用这个稀疏性来显著减少计算成本,提高效率。该软件包采用C/C++编程语言实现,旨在提供高度有效和灵活的参数化方式,适用于广泛的场景分析任务,如运动估计、形状分析、立体视觉以及时间变化图像处理和跟踪。 SBA的核心优势在于其针对实际问题的高度适应性,用户可以根据具体应用的需求,选择不同的参数模型和优化策略,这使得它在处理大规模数据集和复杂场景时,能够保持高效的性能。算法设计和分析方面,SBA提供了清晰的数学框架和优化策略,以便研究人员和开发者深入理解和扩展其功能。 SBA是一个强大的工具,它将稀疏优化与高效软件工程相结合,对于那些寻求在计算机视觉领域进行精确结构和运动估计,并希望降低计算负担的用户来说,是不可或缺的资源。通过使用SBA,开发者可以专注于解决实际问题,而不必过分关注底层的优化细节,从而推动了计算机视觉领域的研究和实践发展。
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2:6
•
M. I. A. Lourakis and A. A. Argyros
is iterative: initiated at the starting point p
0
, it produces a series of vectors
p
1
, p
2
, ..., that converge toward a local minimizer p
+
for f . Hence, at each
iteration, it is required to find the step δ
p
that minimizes the quantity ||x −
f (p + δ
p
)||≈||x − f (p) − Jδ
p
||=|| − Jδ
p
||. The sought δ
p
is thus the solution
to a linear least-squares problem: the minimum is attained when Jδ
p
− is
orthogonal to the column space of J. This leads to J
T
(Jδ
p
− ) = 0, which yields
δ
p
as the solution of the so-called normal equations [Golub and van Loan 1996]:
J
T
Jδ
p
= J
T
. (3)
Matrix J
T
J in the above equation is the first order approximation to the Hessian
of
1
2
T
[Nocedal and Wright 1999] and δ
p
is the Gauss-Newton step. Note also
that J
T
corresponds to the steepest descent direction since the gradient of
1
2
T
is −J
T
. The LM method actually solves a slight variation of Equation (3),
known as the augmented normal equations
Nδ
p
= J
T
, with N ≡ J
T
J + μI, μ>0. (4)
The strategy of altering the diagonal elements of J
T
J is called damping and
μ is referred to as the damping term. If the updated parameter vector p + δ
p
with δ
p
computed from Equation (4) leads to a reduction in the error
T
, the
update is accepted and the process repeats with a decreased damping term.
Otherwise, the damping term is increased, the augmented normal equations
are solved again, and the process iterates until a value of δ
p
that decreases
the error is found. The process of repeatedly solving Equation (4) for different
values of the damping term until an acceptable update to the parameter vector
is found corresponds to one iteration of the LM algorithm.
In LM, the damping term is adjusted at each iteration to assure a reduction in
the error. If the damping is set to a large value, matrix N in Equation (4) is nearly
diagonal and the LM update step δ
p
is skewed toward the steepest descent
direction J
T
. Moreover, the magnitude of δ
p
is reduced in this case, ensuring
that excessively large Gauss-Newton steps are not taken. Damping also handles
situations where the Jacobian is rank deficient and J
T
J is therefore singular
[Dennis and Schnabel 1996; Lampton 1997]. The damping term can be chosen
so that matrix N in Equation (4) is safely nonsingular and, therefore, positive
definite, thus ensuring that the δ
p
computed from it is a descent direction. If
the damping is small, the LM step approximates the Newton minimizer of the
local quadratic model m(δ
p
) =
1
2
T
− (J
T
)
T
δ
p
+
1
2
δ
T
p
J
T
Jδ
p
of
1
2
T
about p (cf.
Equation (3)), which is exact in the case of a fully linear problem [Kelley 1999].
LM is adaptive because it controls its own damping: it raises the damping if a
step fails to reduce
T
; otherwise it reduces the damping. By doing so, LM is
capable of alternating between a slow descent approach when being far from the
minimum and a fast convergence when being at the minimum’s neighborhood
[Lampton 1997]. An efficient updating strategy for the damping term that is
also used in this work has been described in Nielsen [1999]. The LM algorithm
terminates when any of the following conditions is met:
—the magnitude of the gradient drops below a threshold ε
1
;
—the relative magnitude of δ
p
drops below a threshold involving a parameter ε
2
;
ACM Transactions on Mathematical Software, Vol. 36, No. 1, Article 2, Publication date: March 2009.
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