If you reached this page while looking for a general-purpose Levenberg-Marquardt C/C++ implementation, please have a look at sparseLM and levmar, which are respectively aimed at problems with sparse and dense Jacobians.


This site concerns sba, a C/C++ package for generic sparse bundle adjustment that is distributed under the GNU General Public License (GPL). Bundle Adjustment (BA) is almost invariably used as the last step of every feature-based multiple view reconstruction vision algorithm to obtain optimal 3D structure and motion (i.e. camera matrix) parameter estimates. Provided with initial estimates, BA simultaneously refines motion and structure by minimizing the reprojection error between the observed and predicted image points. The minimization is typically carried out with the aid of the Levenberg-Marquardt (LM) algorithm. However, due to the large number of unknowns contributing to the minimized reprojection error, a general purpose implementation of the LM algorithm (such as MINPACK's lmder) incurs high computational costs when applied to the minimization problem defined in the context of BA.

Fortunately, the lack of interaction among parameters for different 3D points and cameras results in the underlying normal equations exhibiting a sparse block structure (click here for an example). sba exploits this sparseness by employing a tailored sparse variant of the LM algorithm that leads to considerable computational gains. sba is generic in the sense that it grants the user full control over the definition of the parameters describing cameras and 3D structure. Therefore, it can support virtually any manifestation/parameterization of the multiple view reconstruction problem such as arbitrary projective cameras, partially or fully intrinsically calibrated cameras, exterior orientation (i.e. pose) estimation from fixed 3D points, refinement of intrinsic parameters, etc. All the user has to do to adapt sba to any such problem is to supply it with appropriate routines for computing the estimated image projections and their Jacobian for the problem and parameterization at hand. Routines for computing analytic Jacobians can be either coded by hand, generated with a tool supporting symbolic differentiation (e.g. maple), or obtained using automatic differentiation techniques. There is also the alternative of approximating Jacobians with the aid of finite differences. Additionally, sba includes routines for checking the consistency of user-supplied Jacobians. To the best of our knowledge, sba is the first and currently the only software package of its kind to be released as free software.


sba has been implemented with special emphasis on flexibility and performance efficiency. It is, in essence, a specialized LM variant that is tailored to fit the “arrowhead” type of sparseness commonly encountered in SfM problems. To achieve this, sba employs a scheme that partitions the normal matrix into distinct camera and structure blocks and solves the (sparse) normal equations by employing the sparse Schur complement of the points submatrix. By adopting this scheme, sba can effectively deal with very large reconstruction problems. Further details regarding the theory behind sba can be found in the documentation.
As an indication of its efficiency, it is noted here that one of the moderately-sized test problems to which sba has been applied involved 54 cameras and 5207 3D points that gave rise to 24609 image projections. The corresponding minimization problem depended on 15999 variables and was solved by sba in about 7 sec using unoptimized BLAS on an Intel P4@1.8 GHz running Linux. Without a sparse implementation of BA, a problem of this size would simply be intractable.

sba relies on LAPACK for all linear algebra operations arising in the course of the LM algorithm. If LAPACK (or an equivalent vendor optimized library such as Intel's MKL, AMD's AMCL, Sun's performance library, IBM's ESSL, SGI's SCSL, NAG, ...), is not available, it is suggested to install CLAPACK, the f2c'ed version of LAPACK. Under MSWin, there is the additional option of downloading precompiled 32 bit LAPACK/BLAS libraries from here. For better performance, ATLAS or GotoBLAS, respectively the automatically tuned BLAS implementation and K. Goto's high-performance BLAS, are recommended. Apart from LAPACK & BLAS though, sba requires no other third party libraries. A MEX-file interface for using sba from within matlab is also available. It will probably work with GNU octave as well but this has not been tested.


Detailed descriptions of the theory behind sba can be found in the correspondind ACM TOMS paper (bibtex entry) or the (somewhat outdated) 2004 ICS/FORTH Technical Report #340 entitled The Design and Implementation of a Generic Sparse Bundle Adjustment Software Package Based on the Levenberg-Marquardt Algorithm. An overview presentation of sba and BA can be found in these slides.

Sparse BA is also discussed in Appendix 6 of Hartley & Zisserman's book.

Available Functions

User-callable functions offered by sba obey the following naming convention: All functions are prefixed by sba_; the string following this prefix specifies whether the function implements full or partial (i.e., motion or structure only) BA. To cater for different user needs, expert and simple drivers to sparse bundle adjustment have been developed; expert drivers are distinguished by the existence of the _x suffix in a function's name. More specifically, sba includes the functions below:

  • sba_motstr_levmar(), sba_motstr_levmar_x():
    Resp. simple and expert driver for full motion and structure BA.
  • sba_mot_levmar(), sba_mot_levmar_x():
    Resp. simple and expert driver for motion only BA. Strictly speaking, this is not BA since structure is kept unmodified. However, this function is very useful when dealing with problems involving camera resectioning, i.e. pose estimation from known 3D-2D correspondences.
  • sba_str_levmar(), sba_str_levmar_x():
    Resp. simple and expert driver for structure only BA. Again, this is not real BA since motion is kept unmodified. This function can, for example, be useful when dealing with intersection problems, i.e. reconstructing 3D points seen in a set of extrinsically calibrated images.

The exact prototype of sba_motstr_levmar_x() is given next for reference, along with a description of each of its arguments. I and O in these descriptions denote input and output arguments, respectively. Motion and structure only BA routines have very similar arguments. Consult Madsen et al's lecture notes for details on the roles of opts and info arguments. Argument adata is intended to help avoid direct use of globals in the routines computing the image projection function and its Jacobian. A structure containing pointers to appropriate data structures can be set up and a pointer to it can be passed to the BA function which then passes it uninterpreted to each call of the user-supplied routines. More accurate argument descriptions can be found in the documentation, whereas Madsen et al's lecture notes provide more details on the LM algorithm implemented by sba. Also included in sba's distribution is a sample program (i.e., eucsbademo.c), that provides an example of using sba for various flavors of Euclidean BA.


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Examples of practical uses of sba are included in the demo pages on camera tracking and localization of unordered panoramic images.

Contact Address

If you find sba useful in your research or have any comments/questions/suggestions, please contact me at

Be warned that although I try to reply to most messages, I might take long to do so.
In case that you use sba in your published work, please include a reference to this paper: [ bibtex entry ].

hits since Thu Nov 10 15:23:31 EET 2005