Effective use of Ceres requires some familiarity with the basic components of a nonlinear least squares solver, so before we describe how to configure the solver, we will begin by taking a brief look at how some of the core optimization algorithms in Ceres work and the various linear solvers and preconditioners that power it.
$ F(x)=\left[f_1(x), \hdots, f_{m}(x)\right]^{\top}$ be a $m$-dimensional function of $x$. We are interested in solving the following optimization problem~\footnote{At the level of the non-linear solver, the block and residual structure is not relevant, therefore our discussion here is in terms of an optimization problem defined over a state vector of size $n$.},
Here, the Jacobian $J(x)$ of $F(x)$ is an $m\times n$ matrix, where $J_{ij}(x)=\partial_j f_i(x)$ and the gradient vector $g(x)=\nabla\frac{1}{2}\|F(x)\|^2= J(x)^\top F(x)$. Since the efficient global optimization of~\eqref{eq:nonlinsq} for general $F(x)$ is an intractable problem, we will have to settle for finding a local minimum.
The general strategy when solving non-linear optimization problems is to solve a sequence of approximations to the original problem~\cite{nocedal2000numerical}. At each iteration, the approximation is solved to determine a correction $\Delta x$ to the vector $x$. For non-linear least squares, an approximation can be constructed by using the linearization $F(x+\Delta x)\approx F(x)+ J(x)\Delta x$, which leads to the following linear least squares problem:
\begin{equation}
\min_{\Delta x}\frac{1}{2}\|J(x)\Delta x + F(x)\|^2
Here, $\mu$ is the trust region radius, $D(x)$ is some matrix used to define a metric on the domain of $F(x)$ and $\rho$ measures the quality of the step $\Delta x$, i.e., how well did the linear model predict the decrease in the value of the non-linear objective. The idea is to increase or decrease the radius of the trust region depending on how well the linearization predicts the behavior of the non-linear objective, which in turn is reflected in the value of $\rho$.
The key computational step in a trust-region algorithm is the solution of the constrained optimization problem
\begin{align}
\arg\min_{\Delta x}&\frac{1}{2}\|J(x)\Delta x + F(x)\|^2 \\
There are a number of different ways of solving this problem, each giving rise to a different concrete trust-region algorithm. Currently Ceres, implements two trust-region algorithms - Levenberg-Marquardt and Dogleg.
The Levenberg-Marquardt algorithm~\cite{levenberg1944method, marquardt1963algorithm} is the most popular algorithm for solving non-linear least squares problems. It was also the first trust region algorithm to be developed~\cite{levenberg1944method,marquardt1963algorithm}. Ceres implements an exact step~\cite{madsen2004methods} and an inexact step variant of the Levenberg-Marquardt algorithm~\cite{wright1985inexact,nash1990assessing}.
It can be shown, that the solution to~\eqref{eq:trp} can be obtained by solving an unconstrained optimization of the form
\begin{align}
\arg\min_{\Delta x}&\frac{1}{2}\|J(x)\Delta x + F(x)\|^2 +\lambda\|D(x)\Delta x\|^2
\end{align}
Where, $\lambda$ is a Lagrange multiplier that is inverse related to $\mu$. In Ceres, we solve for
\begin{align}
\arg\min_{\Delta x}&\frac{1}{2}\|J(x)\Delta x + F(x)\|^2 + \frac{1}{\mu}\|D(x)\Delta x\|^2
\label{eq:lsqr}
\end{align}
The matrix $D(x)$ is a non-negative diagonal matrix, typically the square root of the diagonal of the matrix $J(x)^\top J(x)$.
Before going further, let us make some notational simplifications. We will assume that the matrix $\sqrt{\mu} D$ has been concatenated at the bottom of the matrix $J$ and similarly a vector of zeros has been added to the bottom of the vector $f$ and the rest of our discussion will be in terms of $J$ and $f$, \ie the linear least squares problem.
\begin{align}
\min_{\Delta x}\frac{1}{2}\|J(x)\Delta x + f(x)\|^2 .
For all but the smallest problems the solution of~\eqref{eq:simple} in each iteration of the Levenberg-Marquardt algorithm is the dominant computational cost in Ceres. Ceres provides a number of different options for solving~\eqref{eq:simple}. There are two major classes of methods - factorization and iterative.
The factorization methods are based on computing an exact solution of~\eqref{eq:lsqr} using a Cholesky or a QR factorization and lead to an exact step Levenberg-Marquardt algorithm. But it is not clear if an exact solution of~\eqref{eq:lsqr} is necessary at each step of the LM algorithm to solve~\eqref{eq:nonlinsq}. In fact, we have already seen evidence that this may not be the case, as~\eqref{eq:lsqr} is itself a regularized version of~\eqref{eq:linearapprox}. Indeed, it is possible to construct non-linear optimization algorithms in which the linearized problem is solved approximately. These algorithms are known as inexact Newton or truncated Newton methods~\cite{nocedal2000numerical}.
An inexact Newton method requires two ingredients. First, a cheap method for approximately solving systems of linear equations. Typically an iterative linear solver like the Conjugate Gradients method is used for this purpose~\cite{nocedal2000numerical}. Second, a termination rule for the iterative solver. A typical termination rule is of the form
\begin{equation}
\|H(x) \Delta x + g(x)\|\leq\eta_k \|g(x)\|. \label{eq:inexact}
\end{equation}
Here, $k$ indicates the Levenberg-Marquardt iteration number and $0 < \eta_k <1$ is known as the forcing sequence. Wright \& Holt \cite{wright1985inexact} prove that a truncated Levenberg-Marquardt algorithm that uses an inexact Newton step based on~\eqref{eq:inexact} converges for any sequence $\eta_k \leq\eta_0 < 1$ and the rate of convergence depends on the choice of the forcing sequence $\eta_k$.
Ceres supports both exact and inexact step solution strategies. When the user chooses a factorization based linear solver, the exact step Levenberg-Marquardt algorithm is used. When the user chooses an iterative linear solver, the inexact step Levenberg-Marquardt algorithm is used.
Note that the vector $\Delta x^{\text{Gauss-Newton}}$ is the solution
to~\eqref{eq:linearapprox} and $\Delta x^{\text{Cauchy}}$ is the
vector that minimizes the linear approximation if we restrict
ourselves to moving along the direction of the gradient. Dogleg methods finds a vector $\Delta x$ defined by $\Delta
x^{\text{Gauss-Newton}}$ and $\Delta x^{\text{Cauchy}}$ that solves
the trust region problem. Ceres supports two
variants.
\texttt{TRADITIONAL\_DOGLEG} as described by Powell,
constructs two line segments using the Gauss-Newton and Cauchy vectors
and finds the point farthest along this line shaped like a dogleg
(hence the name) that is contained in the
trust-region. For more details on the exact reasoning and computations, please see Madsen et al~\cite{madsen2004methods}.
\texttt{SUBSPACE\_DOGLEG} is a more sophisticated method
that considers the entire two dimensional subspace spanned by these
two vectors and finds the point that minimizes the trust region
problem in this subspace\cite{byrd1988approximate}.
The key advantage of the Dogleg over Levenberg Marquardt is that if the step computation for a particular choice of $\mu$ does not result in sufficient decrease in the value of the objective function, Levenberg-Marquardt solves the linear approximation from scratch with a smaller value of $\mu$. Dogleg on the other hand, only needs to compute the interpolation between the Gauss-Newton and the Cauchy vectors, as neither of them depend on the value of $\mu$.
Recall that in both of the trust-region methods described above, the key computational cost is the solution of a linear least squares problem of the form
\begin{align}
\min_{\Delta x}\frac{1}{2}\|J(x)\Delta x + f(x)\|^2 .
\label{eq:simple2}
\end{align}
Let $H(x)= J(x)^\top J(x)$ and $g(x)=-J(x)^\top f(x)$. For notational convenience let us also drop the dependence on $x$. Then it is easy to see that solving~\eqref{eq:simple2} is equivalent to solving the {\em normal equations}
For small problems (a couple of hundred parameters and a few thousand residuals) with relatively dense Jacobians, \texttt{DENSE\_QR} is the method of choice~\cite{bjorck1996numerical}. Let $J = QR$ be the QR-decomposition of $J$, where $Q$ is an orthonormal matrix and $R$ is an upper triangular matrix~\cite{trefethen1997numerical}. Then it can be shown that the solution to~\eqref{eq:normal} is given by
Large non-linear least square problems are usually sparse. In such cases, using a dense QR factorization is inefficient. Let $H = R^\top R$ be the Cholesky factorization of the normal equations, where $R$ is an upper triangular matrix, then the solution to ~\eqref{eq:normal} is given by
While it is possible to use \texttt{SPARSE\_NORMAL\_CHOLESKY} to solve bundle adjustment problems, bundle adjustment problem have a special structure, and a more efficient scheme for solving~\eqref{eq:normal} can be constructed.
Suppose that the SfM problem consists of $p$ cameras and $q$ points and the variable vector $x$ has the block structure $x =[y_{1},\hdots,y_{p},z_{1},\hdots,z_{q}]$. Where, $y$ and $z$ correspond to camera and point parameters, respectively. Further, let the camera blocks be of size $c$ and the point blocks be of size $s$ (for most problems $c$ = $6$--$9$ and $s =3$). Ceres does not impose any constancy requirement on these block sizes, but choosing them to be constant simplifies the exposition.
A key characteristic of the bundle adjustment problem is that there is no term $f_{i}$ that includes two or more point blocks. This in turn implies that the matrix $H$ is of the form
\begin{equation}
H = \left[
\begin{matrix} B & E\\ E^\top& C
\end{matrix}
\right]\ ,
\label{eq:hblock}
\end{equation}
where, $B \in\reals^{pc\times pc}$ is a block sparse matrix with $p$ blocks of size $c\times c$ and $C \in\reals^{qs\times qs}$ is a block diagonal matrix with $q$ blocks of size $s\times s$. $E \in\reals^{pc\times qs}$ is a general block sparse matrix, with a block of size $c\times s$ for each observation. Let us now block partition $\Delta x =[\Delta y,\Delta z]$ and $g=[v,w]$ to restate~\eqref{eq:normal} as the block structured linear system
\begin{equation}
\left[
\begin{matrix} B & E\\ E^\top& C
\end{matrix}
\right]\left[
\begin{matrix}\Delta y \\\Delta z
\end{matrix}
\right]
=
\left[
\begin{matrix} v\\ w
\end{matrix}
\right]\ ,
\label{eq:linear2}
\end{equation}
and apply Gaussian elimination to it. As we noted above, $C$ is a block diagonal matrix, with small diagonal blocks of size $s\times s$.
Thus, calculating the inverse of $C$ by inverting each of these blocks is cheap. This allows us to eliminate $\Delta z$ by observing that $\Delta z = C^{-1}(w - E^\top\Delta y)$, giving us
\begin{equation}
\left[B - EC^{-1}E^\top\right]\Delta y = v - EC^{-1}w\ . \label{eq:schur}
\end{equation}
The matrix
\begin{equation}
S = B - EC^{-1}E^\top\ ,
\end{equation}
is the Schur complement of $C$ in $H$. It is also known as the {\em reduced camera matrix}, because the only variables participating in~\eqref{eq:schur} are the ones corresponding to the cameras. $S \in\reals^{pc\times pc}$ is a block structured symmetric positive definite matrix, with blocks of size $c\times c$. The block $S_{ij}$ corresponding to the pair of images $i$ and $j$ is non-zero if and only if the two images observe at least one common point.
Now, \eqref{eq:linear2}~can be solved by first forming $S$, solving for $\Delta y$, and then back-substituting $\Delta y$ to obtain the value of $\Delta z$.
Thus, the solution of what was an $n\times n$, $n=pc+qs$ linear system is reduced to the inversion of the block diagonal matrix $C$, a few matrix-matrix and matrix-vector multiplies, and the solution of block sparse $pc\times pc$ linear system~\eqref{eq:schur}. For almost all problems, the number of cameras is much smaller than the number of points, $p \ll q$, thus solving~\eqref{eq:schur} is significantly cheaper than solving~\eqref{eq:linear2}. This is the {\em Schur complement trick}~\cite{brown-58}.
This still leaves open the question of solving~\eqref{eq:schur}. The
method of choice for solving symmetric positive definite systems
exactly is via the Cholesky
factorization~\cite{trefethen1997numerical} and depending upon the
structure of the matrix, there are, in general, two options. The first
is direct factorization, where we store and factor $S$ as a dense
matrix~\cite{trefethen1997numerical}. This method has $O(p^2)$ space complexity and $O(p^3)$ time
complexity and is only practical for problems with up to a few hundred
cameras. Ceres implements this strategy as the \texttt{DENSE\_SCHUR} solver.
But, $S$ is typically a fairly sparse matrix, as most images
only see a small fraction of the scene. This leads us to the second
option: sparse direct methods. These methods store $S$ as a sparse
matrix, use row and column re-ordering algorithms to maximize the
sparsity of the Cholesky decomposition, and focus their compute effort
on the non-zero part of the factorization~\cite{chen2006acs}.
Sparse direct methods, depending on the exact sparsity structure of the Schur complement,
allow bundle adjustment algorithms to significantly scale up over those based on dense
factorization. Ceres implements this strategy as the \texttt{SPARSE\_SCHUR} solver.
For general sparse problems, if the problem is too large for \texttt{CHOLMOD} or a sparse linear algebra library is not linked into Ceres, another option is the \texttt{CGNR} solver. This solver uses the Conjugate Gradients solver on the {\em normal equations}, but without forming the normal equations explicitly. It exploits the relation
When the user chooses \texttt{ITERATIVE\_SCHUR} as the linear solver, Ceres automatically switches from the exact step algorithm to an inexact step algorithm.
Another option for bundle adjustment problems is to apply PCG to the reduced camera matrix $S$ instead of $H$. One reason to do this is that $S$ is a much smaller matrix than $H$, but more importantly, it can be shown that $\kappa(S)\leq\kappa(H)$. Ceres implements PCG on $S$ as the \texttt{ITERATIVE\_SCHUR} solver. When the user chooses \texttt{ITERATIVE\_SCHUR} as the linear solver, Ceres automatically switches from the exact step algorithm to an inexact step algorithm.
The cost of forming and storing the Schur complement $S$ can be prohibitive for large problems. Indeed, for an inexact Newton solver that computes $S$ and runs PCG on it, almost all of its time is spent in constructing $S$; the time spent inside the PCG algorithm is negligible in comparison. Because PCG only needs access to $S$ via its product with a vector, one way to evaluate $Sx$ is to observe that
Thus, we can run PCG on $S$ with the same computational effort per iteration as PCG on $H$, while reaping the benefits of a more powerful preconditioner. In fact, we do not even need to compute $H$, \eqref{eq:schurtrick1} can be implemented using just the columns of $J$.
Equation~\eqref{eq:schurtrick1} is closely related to {\em Domain Decomposition methods} for solving large linear systems that arise in structural engineering and partial differential equations. In the language of Domain Decomposition, each point in a bundle adjustment problem is a domain, and the cameras form the interface between these domains. The iterative solution of the Schur complement then falls within the sub-category of techniques known as Iterative Sub-structuring~\cite{saad2003iterative,mathew2008domain}.
The convergence rate of Conjugate Gradients for solving~\eqref{eq:normal} depends on the distribution of eigenvalues of $H$~\cite{saad2003iterative}. A useful upper bound is $\sqrt{\kappa(H)}$, where, $\kappa(H)$f is the condition number of the matrix $H$. For most bundle adjustment problems, $\kappa(H)$ is high and a direct application of Conjugate Gradients to~\eqref{eq:normal} results in extremely poor performance.
The solution to this problem is to replace~\eqref{eq:normal} with a {\em preconditioned} system. Given a linear system, $Ax =b$ and a preconditioner $M$ the preconditioned system is given by $M^{-1}Ax = M^{-1}b$. The resulting algorithm is known as Preconditioned Conjugate Gradients algorithm (PCG) and its worst case complexity now depends on the condition number of the {\em preconditioned} matrix $\kappa(M^{-1}A)$.
The computational cost of using a preconditioner $M$ is the cost of computing $M$ and evaluating the product $M^{-1}y$ for arbitrary vectors $y$. Thus, there are two competing factors to consider: How much of $H$'s structure is captured by $M$ so that the condition number $\kappa(HM^{-1})$ is low, and the computational cost of constructing and using $M$. The ideal preconditioner would be one for which $\kappa(M^{-1}A)=1$. $M=A$ achieves this, but it is not a practical choice, as applying this preconditioner would require solving a linear system equivalent to the unpreconditioned problem. It is usually the case that the more information $M$ has about $H$, the more expensive it is use. For example, Incomplete Cholesky factorization based preconditioners have much better convergence behavior than the Jacobi preconditioner, but are also much more expensive.
The simplest of all preconditioners is the diagonal or Jacobi preconditioner, \ie, $M=\operatorname{diag}(A)$, which for block structured matrices like $H$ can be generalized to the block Jacobi preconditioner.
For \texttt{ITERATIVE\_SCHUR} there are two obvious choices for block diagonal preconditioners for $S$. The block diagonal of the matrix $B$~\cite{mandel1990block} and the block diagonal $S$, \ie the block Jacobi preconditioner for $S$. Ceres's implements both of these preconditioners and refers to them as \texttt{JACOBI} and \texttt{SCHUR\_JACOBI} respectively.
For bundle adjustment problems arising in reconstruction from community photo collections, more effective preconditioners can be constructed by analyzing and exploiting the camera-point visibility structure of the scene~\cite{kushal2012}. Ceres implements the two visibility based preconditioners described by Kushal \& Agarwal as \texttt{CLUSTER\_JACOBI} and \texttt{CLUSTER\_TRIDIAGONAL}. These are fairly new preconditioners and Ceres' implementation of them is in its early stages and is not as mature as the other preconditioners described above.
