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KURT KONOLIGE: SPARSE SPARSE BUNDLE ADJUSTMENT 1

Sparse Sparse Bundle Adjustment

Kurt Konolige http://www.willowgarage.com/~konolige

Willow Garage 68 Willow Road Menlo Park, USA

Abstract

Sparse Bundle Adjustment (SBA) is a method for simultaneously optimizing a set of camera poses and visible points. It exploits the sparse primary structure of the problem, where connections exist just between points and cameras. In this paper, we implement an efficient version of SBA for systems where the secondary structure (relations among cameras) is also sparse. The method, which we call Sparse SBA (sSBA), integrates an efficient method for setting up the linear subproblem with recent advances in direct sparse Cholesky solvers. sSBA outperforms the current SBA standard implementation on datasets with sparse secondary structure by at least an order of magnitude, while also being more efficient on dense datasets.

1 Introduction Sparse Bundle Adjustment (SBA) is the standard method for optimizing a structure-from- motion problem in computer vision. With the success of Photosynth and similar systems for stitching together large collections of images [16], attention has turned to the problem of making SBA more efficient. There are two different types of large-scale systems:

• Photosynth-type systems focus on reconstruction from a large number of images con- centrated in a small area; we call these object-centered.

• Visual mapping systems [1, 3, 9] cover a more extended area with fewer images, and real-time performance is often important (see Figure 1).

These types are at two ends of a spectrum: object-centered systems produce dense relations between cameras, while visual mapping systems are much sparser, with cameras in a local neighborhood sharing common points (Guilbert et al. call these “sparse systems” [8]). In this paper, we are interested in fast SBA methods for the latter case, where it is possible to exploit the sparse secondary structure (camera to camera relations) of the problem.

Nonlinear optimization in SBA typically proceeds by iteration: form a linear subproblem around the current solution, solve it, and repeat until convergence. For large problems, the computational bottleneck is usually the solution of the linear subproblem, which can grow as the cube of the number of cameras. The fill-in of the linear problem is directly tied to the camera-point structure of the problem: if each camera only sees features in a small neighborhood of other cameras, the number of non-zero elements grows only linearly or nearly linearly with the number of cameras.

c© 2010. The copyright of this document resides with its authors. It may be distributed unchanged freely in print or electronic forms.

BMVC 2010 doi:10.5244/C.24.102

Citation Citation {Snavely, Seitz, and Szeliski} 2008

Citation Citation {Agrawal and Konolige} 2008

Citation Citation {Cummins and Newman} 2009

Citation Citation {Klein and Murray} 2007

Citation Citation {Guilbert, Bartoli, and Heyden} 2006

2 KURT KONOLIGE: SPARSE SPARSE BUNDLE ADJUSTMENT

Figure 1: An overhead view showing of the New College mapping dataset [15], with 2.2k views (cyan) and 290k points (red). Grid lines are at 10 m. At the end, sSBA computes each iteration in 6 seconds.

In this paper we present an engineering approach to efficiently solving SBA problems with sparse linear subproblems. Our approach is to exploit recent fast direct Cholesky de- composition methods to solve the linear subproblem. These methods use a compressed rep- resentation of large sparse matrices, and we present a method for efficiently handling the block data structures of SBA to take advantage of this representation. The end result is a sys- tem, which we call Sparse SBA or sSBA, that outperforms the current reference system for SBA from Lourakis and Argyros [11] by an order of magnitude on sparse seconary-structure problems, and uses far less space on large problems. For example, Figure 1 shows a recon- struction of a New College dataset [15] that contains 2200 views and 290k points, and is solved in about 6 seconds per iteration at the end. Interestingly, sSBA also outperforms [11] on problems with dense secondary structure, where the setup computation often dominates, although by a lesser margin.

2 Related Work

The standard reference for SBA is the monograph of Triggs et al. [17]. This work explores many of the mathematical and computational aspects of SBA, including various methods for solving the nonlinear optimization problem at the heart of SBA. In this paper, we use Levenberg-Marquardt [14], which is a standard algorithm for solving unconstrained nonlin- ear optimization. Alternative solvers for SBA include preconditioned conjugate gradient [2] or Powell’s dog-leg solver [12].

The LM implementation of SBA repeatedly solves a large linear subproblem whose LHS matrix is positive definite. Recent work in direct sparse Cholesky solvers [4] has yielded algorithms that are very efficient for large problems, and we use these methods here.

Most current applications that incorporate SBA use an open-source version developed by Lourakis and Argyros [11], which we call laSBA. For example, the open source Bundler program originally used for PhotoSynth [16], an application for stitching together tourist photos, uses laSBA as its optimization engine. An exception is the work of Klein and Murray [9], which has an SBA engine. We have tested this system and found it slower than laSBA,

Citation Citation {Smith, Baldwin, Churchill, Paul, and Newman} 2009

Citation Citation {Lourakis and Argyros} 2009

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Citation Citation {Lourakis and Argyros} 2009

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Citation Citation {Marquardt} 1963

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Citation Citation {Davis} 2006

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KURT KONOLIGE: SPARSE SPARSE BUNDLE ADJUSTMENT 3

so laSBA will be our reference implementation. In the context of mapping systems, there are references to using sparse solvers in SBA, e.g., Guilbert et al. [8] mention sparse QR decomposition and supernodal Cholesky methods. However, there is no explicit algorithm for setting up the sparse linear problem, which we have found to be an important bottleneck.

3 SBA Basics This section summarizes the basic formulation of Sparse Bundle Adjustment used in the paper. For the most part it follows the excellent exposition of Engels et al. [5], and the reader can consult this paper for derivations.

3.1 Error Formulation Sparse Bundle Adjustment (SBA) is a method of nonlinear optimization among camera frames (ci) and points (p j). Each camera frame consists of a translation ti and rotation Ri giving the position and orientation of the frame in global coordinates. For any such pair, the measured projection of p j on the camera frame is called z̄i j. The calculated feature value comes from the projection equation:

g(ci, p j)≡ R>i (ti− p j) h(ci, p j)≡ gx,y(ci, p j)/gz(ci, p j)

(1)

The function g transforms the point p j into ci’s coordinate system, and h projects it to a normalized image plane.

The error function associated with a projection is the difference between the calculated and measured projection. The total error is the sum over all projections.

ei j ≡ h(ci, p j)− z̄i j E(c,p)≡∑

i j e>i jΛi jei j (2)

Λi j is the precision matrix (inverse covariance) of the feature measurement. In the case of SBA, it is often assumed to be isotropic (diagonal) and on the order of a pixel, making it the identity matrix. For simplicity we drop it from the rest of the exposition; the system could be easily modified to accommodate it.

3.2 Levenberg-Marquardt System The optimal placement of c,p is found by minimizing the total error in Equation 2. A stan- dard method for solving this problem is to iterate a linearized solution around the current values of c,p. The linear solution is found by second-order Taylor expansion around c,p, and an approximation of the second-order derivative matrix (the Hessian) by Jacobian prod- ucts (the normal or Gauss-Newton approximation).

The resultant linear system is formed by stacking the variables c,p into a vector x, and the error functions into a vector e. Let

J≡ ∂e ∂x

H≡ J>J (3)

Citation Citation {Guilbert, Bartoli, and Heyden} 2006

Citation Citation {Engels, Stewénius, and Nister} 2006

4 KURT KONOLIGE: SPARSE SPARSE BUNDLE ADJUSTMENT

The linear system is: H∆x =−J>e (4)

In general, solving this system is not guaranteed to produce a step ∆x that decreases the error. The Levenberg-Marquardt (LM) method augments H by adding λ diag(H), where λ is a small positive multiplier. Larger λ produces a gradient descent system that will converge (but slowly). There are various strategies for manipulating λ , initially using gradient descent and then the faster Newton-Euler method at the end.

3.3 Primary Structure The size of the matrix H is dominated by ||p||, which in typical problems is several orders of magnitude larger than ||c||. But we can take advantage of the structure of the Jacobians to reduce Equation 4 to just the variables c. If we organize Equation 4 so that cameras and points are clustered, it has a characteristic structure:[

J&