Modeling Nonlinear Least Squares¶
Introduction¶
Ceres solver consists of two distinct parts. A modeling API which provides a rich set of tools to construct an optimization problem one term at a time and a solver API that controls the minimization algorithm. This chapter is devoted to the task of modeling optimization problems using Ceres. Solving Nonlinear Least Squares discusses the various ways in which an optimization problem can be solved using Ceres.
Ceres solves robustified bounds constrained nonlinear least squares problems of the form:
In Ceres parlance, the expression
\(\rho_i\left(\left\f_i\left(x_{i_1},...,x_{i_k}\right)\right\^2\right)\)
is known as a residual block, where \(f_i(\cdot)\) is a
CostFunction
that depends on the parameter blocks
\(\left\{x_{i_1},... , x_{i_k}\right\}\).
In most optimization problems small groups of scalars occur together. For example the three components of a translation vector and the four components of the quaternion that define the pose of a camera. We refer to such a group of scalars as a parameter block. Of course a parameter block can be just a single scalar too.
\(\rho_i\) is a LossFunction
. A LossFunction
is
a scalar valued function that is used to reduce the influence of
outliers on the solution of nonlinear least squares problems.
\(l_j\) and \(u_j\) are lower and upper bounds on the parameter block \(x_j\).
As a special case, when \(\rho_i(x) = x\), i.e., the identity function, and \(l_j = \infty\) and \(u_j = \infty\) we get the more familiar unconstrained nonlinear least squares problem.
CostFunction
¶
For each term in the objective function, a CostFunction
is
responsible for computing a vector of residuals and if asked a vector
of Jacobian matrices, i.e., given \(\left[x_{i_1}, ... ,
x_{i_k}\right]\), compute the vector
\(f_i\left(x_{i_1},...,x_{i_k}\right)\) and the matrices
\[J_{ij} = \frac{\partial}{\partial x_{i_j}}f_i\left(x_{i_1},...,x_{i_k}\right),\quad \forall j \in \{1, \ldots, k\}\]

class
CostFunction
¶ class CostFunction { public: virtual bool Evaluate(double const* const* parameters, double* residuals, double** jacobians) = 0; const vector<int32>& parameter_block_sizes(); int num_residuals() const; protected: vector<int32>* mutable_parameter_block_sizes(); void set_num_residuals(int num_residuals); };
The signature of the CostFunction
(number and sizes of input
parameter blocks and number of outputs) is stored in
CostFunction::parameter_block_sizes_
and
CostFunction::num_residuals_
respectively. User code
inheriting from this class is expected to set these two members with
the corresponding accessors. This information will be verified by the
Problem
when added with Problem::AddResidualBlock()
.

bool
CostFunction::
Evaluate
(double const *const *parameters, double *residuals, double **jacobians)¶ Compute the residual vector and the Jacobian matrices.
parameters
is an array of pointers to arrays containing the various parameter blocks.parameters
has the same number of elements asCostFunction::parameter_block_sizes_
and the parameter blocks are in the same order asCostFunction::parameter_block_sizes_
.residuals
is an array of sizenum_residuals_
.jacobians
is an array of sizeCostFunction::parameter_block_sizes_
containing pointers to storage for Jacobian matrices corresponding to each parameter block. The Jacobian matrices are in the same order asCostFunction::parameter_block_sizes_
.jacobians[i]
is an array that containsCostFunction::num_residuals_
xCostFunction::parameter_block_sizes_
[i]
elements. Each Jacobian matrix is stored in rowmajor order, i.e.,jacobians[i][r * parameter_block_size_[i] + c]
= \(\frac{\partial residual[r]}{\partial parameters[i][c]}\)If
jacobians
isNULL
, then no derivatives are returned; this is the case when computing cost only. Ifjacobians[i]
isNULL
, then the Jacobian matrix corresponding to the \(i^{\textrm{th}}\) parameter block must not be returned, this is the case when a parameter block is marked constant.NOTE The return value indicates whether the computation of the residuals and/or jacobians was successful or not.
This can be used to communicate numerical failures in Jacobian computations for instance.
SizedCostFunction
¶

class
SizedCostFunction
¶ If the size of the parameter blocks and the size of the residual vector is known at compile time (this is the common case),
SizeCostFunction
can be used where these values can be specified as template parameters and the user only needs to implementCostFunction::Evaluate()
.template<int kNumResiduals, int N0 = 0, int N1 = 0, int N2 = 0, int N3 = 0, int N4 = 0, int N5 = 0, int N6 = 0, int N7 = 0, int N8 = 0, int N9 = 0> class SizedCostFunction : public CostFunction { public: virtual bool Evaluate(double const* const* parameters, double* residuals, double** jacobians) const = 0; };
AutoDiffCostFunction
¶

class
AutoDiffCostFunction
¶ Defining a
CostFunction
or aSizedCostFunction
can be a tedious and error prone especially when computing derivatives. To this end Ceres provides automatic differentiation.template <typename CostFunctor, int kNumResiduals, // Number of residuals, or ceres::DYNAMIC. int N0, // Number of parameters in block 0. int N1 = 0, // Number of parameters in block 1. int N2 = 0, // Number of parameters in block 2. int N3 = 0, // Number of parameters in block 3. int N4 = 0, // Number of parameters in block 4. int N5 = 0, // Number of parameters in block 5. int N6 = 0, // Number of parameters in block 6. int N7 = 0, // Number of parameters in block 7. int N8 = 0, // Number of parameters in block 8. int N9 = 0> // Number of parameters in block 9. class AutoDiffCostFunction : public SizedCostFunction<kNumResiduals, N0, N1, N2, N3, N4, N5, N6, N7, N8, N9> { public: explicit AutoDiffCostFunction(CostFunctor* functor); // Ignore the template parameter kNumResiduals and use // num_residuals instead. AutoDiffCostFunction(CostFunctor* functor, int num_residuals); };
To get an auto differentiated cost function, you must define a class with a templated
operator()
(a functor) that computes the cost function in terms of the template parameterT
. The autodiff framework substitutes appropriateJet
objects forT
in order to compute the derivative when necessary, but this is hidden, and you should write the function as ifT
were a scalar type (e.g. a doubleprecision floating point number).The function must write the computed value in the last argument (the only non
const
one) and return true to indicate success.For example, consider a scalar error \(e = k  x^\top y\), where both \(x\) and \(y\) are twodimensional vector parameters and \(k\) is a constant. The form of this error, which is the difference between a constant and an expression, is a common pattern in least squares problems. For example, the value \(x^\top y\) might be the model expectation for a series of measurements, where there is an instance of the cost function for each measurement \(k\).
The actual cost added to the total problem is \(e^2\), or \((k  x^\top y)^2\); however, the squaring is implicitly done by the optimization framework.
To write an autodifferentiable cost function for the above model, first define the object
class MyScalarCostFunctor { MyScalarCostFunctor(double k): k_(k) {} template <typename T> bool operator()(const T* const x , const T* const y, T* e) const { e[0] = k_  x[0] * y[0]  x[1] * y[1]; return true; } private: double k_; };
Note that in the declaration of
operator()
the input parametersx
andy
come first, and are passed as const pointers to arrays ofT
. If there were three input parameters, then the third input parameter would come aftery
. The output is always the last parameter, and is also a pointer to an array. In the example above,e
is a scalar, so onlye[0]
is set.Then given this class definition, the auto differentiated cost function for it can be constructed as follows.
CostFunction* cost_function = new AutoDiffCostFunction<MyScalarCostFunctor, 1, 2, 2>( new MyScalarCostFunctor(1.0)); ^ ^ ^    Dimension of residual +   Dimension of x +  Dimension of y +
In this example, there is usually an instance for each measurement of
k
.In the instantiation above, the template parameters following
MyScalarCostFunction
,<1, 2, 2>
describe the functor as computing a 1dimensional output from two arguments, both 2dimensional.AutoDiffCostFunction
also supports cost functions with a runtimedetermined number of residuals. For example:CostFunction* cost_function = new AutoDiffCostFunction<MyScalarCostFunctor, DYNAMIC, 2, 2>( new CostFunctorWithDynamicNumResiduals(1.0), ^ ^ ^ runtime_number_of_residuals); <+            Actual number of residuals +    Indicate dynamic number of residuals +   Dimension of x +  Dimension of y +
The framework can currently accommodate cost functions of up to 10 independent variables, and there is no limit on the dimensionality of each of them.
WARNING 1 A common beginner’s error when first using
AutoDiffCostFunction
is to get the sizing wrong. In particular, there is a tendency to set the template parameters to (dimension of residual, number of parameters) instead of passing a dimension parameter for every parameter block. In the example above, that would be<MyScalarCostFunction, 1, 2>
, which is missing the 2 as the last template argument.
DynamicAutoDiffCostFunction
¶

class
DynamicAutoDiffCostFunction
¶ AutoDiffCostFunction
requires that the number of parameter blocks and their sizes be known at compile time. It also has an upper limit of 10 parameter blocks. In a number of applications, this is not enough e.g., Bezier curve fitting, Neural Network training etc.template <typename CostFunctor, int Stride = 4> class DynamicAutoDiffCostFunction : public CostFunction { };
In such cases
DynamicAutoDiffCostFunction
can be used. LikeAutoDiffCostFunction
the user must define a templated functor, but the signature of the functor differs slightly. The expected interface for the cost functors is:struct MyCostFunctor { template<typename T> bool operator()(T const* const* parameters, T* residuals) const { } }
Since the sizing of the parameters is done at runtime, you must also specify the sizes after creating the dynamic autodiff cost function. For example:
DynamicAutoDiffCostFunction<MyCostFunctor, 4>* cost_function = new DynamicAutoDiffCostFunction<MyCostFunctor, 4>( new MyCostFunctor()); cost_function>AddParameterBlock(5); cost_function>AddParameterBlock(10); cost_function>SetNumResiduals(21);
Under the hood, the implementation evaluates the cost function multiple times, computing a small set of the derivatives (four by default, controlled by the
Stride
template parameter) with each pass. There is a performance tradeoff with the size of the passes; Smaller sizes are more cache efficient but result in larger number of passes, and larger stride lengths can destroy cachelocality while reducing the number of passes over the cost function. The optimal value depends on the number and sizes of the various parameter blocks.As a rule of thumb, try using
AutoDiffCostFunction
before you useDynamicAutoDiffCostFunction
.
NumericDiffCostFunction
¶

class
NumericDiffCostFunction
¶ In some cases, its not possible to define a templated cost functor, for example when the evaluation of the residual involves a call to a library function that you do not have control over. In such a situation, numerical differentiation can be used.
Note
TODO(sameeragarwal): Add documentation for the constructor and for NumericDiffOptions. Update DynamicNumericDiffOptions in a similar manner.
template <typename CostFunctor, NumericDiffMethodType method = CENTRAL, int kNumResiduals, // Number of residuals, or ceres::DYNAMIC. int N0, // Number of parameters in block 0. int N1 = 0, // Number of parameters in block 1. int N2 = 0, // Number of parameters in block 2. int N3 = 0, // Number of parameters in block 3. int N4 = 0, // Number of parameters in block 4. int N5 = 0, // Number of parameters in block 5. int N6 = 0, // Number of parameters in block 6. int N7 = 0, // Number of parameters in block 7. int N8 = 0, // Number of parameters in block 8. int N9 = 0> // Number of parameters in block 9. class NumericDiffCostFunction : public SizedCostFunction<kNumResiduals, N0, N1, N2, N3, N4, N5, N6, N7, N8, N9> { };
To get a numerically differentiated
CostFunction
, you must define a class with aoperator()
(a functor) that computes the residuals. The functor must write the computed value in the last argument (the only nonconst
one) and returntrue
to indicate success. Please seeCostFunction
for details on how the return value may be used to impose simple constraints on the parameter block. e.g., an object of the formstruct ScalarFunctor { public: bool operator()(const double* const x1, const double* const x2, double* residuals) const; }
For example, consider a scalar error \(e = k  x'y\), where both \(x\) and \(y\) are twodimensional column vector parameters, the prime sign indicates transposition, and \(k\) is a constant. The form of this error, which is the difference between a constant and an expression, is a common pattern in least squares problems. For example, the value \(x'y\) might be the model expectation for a series of measurements, where there is an instance of the cost function for each measurement \(k\).
To write an numericallydifferentiable class:CostFunction for the above model, first define the object
class MyScalarCostFunctor { MyScalarCostFunctor(double k): k_(k) {} bool operator()(const double* const x, const double* const y, double* residuals) const { residuals[0] = k_  x[0] * y[0] + x[1] * y[1]; return true; } private: double k_; };
Note that in the declaration of
operator()
the input parametersx
andy
come first, and are passed as const pointers to arrays ofdouble
s. If there were three input parameters, then the third input parameter would come aftery
. The output is always the last parameter, and is also a pointer to an array. In the example above, the residual is a scalar, so onlyresiduals[0]
is set.Then given this class definition, the numerically differentiated
CostFunction
with central differences used for computing the derivative can be constructed as follows.CostFunction* cost_function = new NumericDiffCostFunction<MyScalarCostFunctor, CENTRAL, 1, 2, 2>( new MyScalarCostFunctor(1.0)); ^ ^ ^ ^     Finite Differencing Scheme +    Dimension of residual +   Dimension of x +  Dimension of y +
In this example, there is usually an instance for each measurement of k.
In the instantiation above, the template parameters following
MyScalarCostFunctor
,1, 2, 2
, describe the functor as computing a 1dimensional output from two arguments, both 2dimensional.NumericDiffCostFunction also supports cost functions with a runtimedetermined number of residuals. For example:
CostFunction* cost_function = new NumericDiffCostFunction<MyScalarCostFunctor, CENTRAL, DYNAMIC, 2, 2>( new CostFunctorWithDynamicNumResiduals(1.0), ^ ^ ^ TAKE_OWNERSHIP,    runtime_number_of_residuals); <+            Actual number of residuals +    Indicate dynamic number of residuals +   Dimension of x +  Dimension of y +
The framework can currently accommodate cost functions of up to 10 independent variables, and there is no limit on the dimensionality of each of them.
There are three available numeric differentiation schemes in ceressolver:
The
FORWARD
difference method, which approximates \(f'(x)\) by computing \(\frac{f(x+h)f(x)}{h}\), computes the cost function one additional time at \(x+h\). It is the fastest but least accurate method.The
CENTRAL
difference method is more accurate at the cost of twice as many function evaluations than forward difference, estimating \(f'(x)\) by computing \(\frac{f(x+h)f(xh)}{2h}\).The
RIDDERS
difference method[Ridders]_ is an adaptive scheme that estimates derivatives by performing multiple central differences at varying scales. Specifically, the algorithm starts at a certain \(h\) and as the derivative is estimated, this step size decreases. To conserve function evaluations and estimate the derivative error, the method performs Richardson extrapolations between the tested step sizes. The algorithm exhibits considerably higher accuracy, but does so by additional evaluations of the cost function.Consider using
CENTRAL
differences to begin with. Based on the results, either try forward difference to improve performance or Ridders’ method to improve accuracy.WARNING A common beginner’s error when first using
NumericDiffCostFunction
is to get the sizing wrong. In particular, there is a tendency to set the template parameters to (dimension of residual, number of parameters) instead of passing a dimension parameter for every parameter. In the example above, that would be<MyScalarCostFunctor, 1, 2>
, which is missing the last2
argument. Please be careful when setting the size parameters.
Numeric Differentiation & LocalParameterization¶
If your cost function depends on a parameter block that must lie on a manifold and the functor cannot be evaluated for values of that parameter block not on the manifold then you may have problems numerically differentiating such functors.
This is because numeric differentiation in Ceres is performed by perturbing the individual coordinates of the parameter blocks that a cost functor depends on. In doing so, we assume that the parameter blocks live in an Euclidean space and ignore the structure of manifold that they live As a result some of the perturbations may not lie on the manifold corresponding to the parameter block.
For example consider a four dimensional parameter block that is interpreted as a unit Quaternion. Perturbing the coordinates of this parameter block will violate the unit norm property of the parameter block.
Fixing this problem requires that
NumericDiffCostFunction
be aware of theLocalParameterization
associated with each parameter block and only generate perturbations in the local tangent space of each parameter block.For now this is not considered to be a serious enough problem to warrant changing the
NumericDiffCostFunction
API. Further, in most cases it is relatively straightforward to project a point off the manifold back onto the manifold before using it in the functor. For example in case of the Quaternion, normalizing the 4vector before using it does the trick.Alternate Interface
For a variety of reasons, including compatibility with legacy code,
NumericDiffCostFunction
can also takeCostFunction
objects as input. The following describes how.To get a numerically differentiated cost function, define a subclass of
CostFunction
such that theCostFunction::Evaluate()
function ignores thejacobians
parameter. The numeric differentiation wrapper will fill in the jacobian parameter if necessary by repeatedly calling theCostFunction::Evaluate()
with small changes to the appropriate parameters, and computing the slope. For performance, the numeric differentiation wrapper class is templated on the concrete cost function, even though it could be implemented only in terms of theCostFunction
interface.The numerically differentiated version of a cost function for a cost function can be constructed as follows:
CostFunction* cost_function = new NumericDiffCostFunction<MyCostFunction, CENTRAL, 1, 4, 8>( new MyCostFunction(...), TAKE_OWNERSHIP);where
MyCostFunction
has 1 residual and 2 parameter blocks with sizes 4 and 8 respectively. Look at the tests for a more detailed example.
DynamicNumericDiffCostFunction
¶

class
DynamicNumericDiffCostFunction
¶ Like
AutoDiffCostFunction
NumericDiffCostFunction
requires that the number of parameter blocks and their sizes be known at compile time. It also has an upper limit of 10 parameter blocks. In a number of applications, this is not enough.template <typename CostFunctor, NumericDiffMethodType method = CENTRAL> class DynamicNumericDiffCostFunction : public CostFunction { };
In such cases when numeric differentiation is desired,
DynamicNumericDiffCostFunction
can be used.Like
NumericDiffCostFunction
the user must define a functor, but the signature of the functor differs slightly. The expected interface for the cost functors is:struct MyCostFunctor { bool operator()(double const* const* parameters, double* residuals) const { } }
Since the sizing of the parameters is done at runtime, you must also specify the sizes after creating the dynamic numeric diff cost function. For example:
DynamicNumericDiffCostFunction<MyCostFunctor>* cost_function = new DynamicNumericDiffCostFunction<MyCostFunctor>(new MyCostFunctor); cost_function>AddParameterBlock(5); cost_function>AddParameterBlock(10); cost_function>SetNumResiduals(21);
As a rule of thumb, try using
NumericDiffCostFunction
before you useDynamicNumericDiffCostFunction
.WARNING The same caution about mixing local parameterizations with numeric differentiation applies as is the case with
NumericDiffCostFunction
.
CostFunctionToFunctor
¶

class
CostFunctionToFunctor
¶ CostFunctionToFunctor
is an adapter class that allows users to useCostFunction
objects in templated functors which are to be used for automatic differentiation. This allows the user to seamlessly mix analytic, numeric and automatic differentiation.For example, let us assume that
class IntrinsicProjection : public SizedCostFunction<2, 5, 3> { public: IntrinsicProjection(const double* observation); virtual bool Evaluate(double const* const* parameters, double* residuals, double** jacobians) const; };
is a
CostFunction
that implements the projection of a point in its local coordinate system onto its image plane and subtracts it from the observed point projection. It can compute its residual and either via analytic or numerical differentiation can compute its jacobians.Now we would like to compose the action of this
CostFunction
with the action of camera extrinsics, i.e., rotation and translation. Say we have a templated functiontemplate<typename T> void RotateAndTranslatePoint(const T* rotation, const T* translation, const T* point, T* result);
Then we can now do the following,
struct CameraProjection { CameraProjection(double* observation) : intrinsic_projection_(new IntrinsicProjection(observation)) { } template <typename T> bool operator()(const T* rotation, const T* translation, const T* intrinsics, const T* point, T* residual) const { T transformed_point[3]; RotateAndTranslatePoint(rotation, translation, point, transformed_point); // Note that we call intrinsic_projection_, just like it was // any other templated functor. return intrinsic_projection_(intrinsics, transformed_point, residual); } private: CostFunctionToFunctor<2,5,3> intrinsic_projection_; };
Note that
CostFunctionToFunctor
takes ownership of theCostFunction
that was passed in to the constructor.In the above example, we assumed that
IntrinsicProjection
is aCostFunction
capable of evaluating its value and its derivatives. Suppose, if that were not the case andIntrinsicProjection
was defined as follows:struct IntrinsicProjection IntrinsicProjection(const double* observation) { observation_[0] = observation[0]; observation_[1] = observation[1]; } bool operator()(const double* calibration, const double* point, double* residuals) { double projection[2]; ThirdPartyProjectionFunction(calibration, point, projection); residuals[0] = observation_[0]  projection[0]; residuals[1] = observation_[1]  projection[1]; return true; } double observation_[2]; };
Here
ThirdPartyProjectionFunction
is some third party library function that we have no control over. So this function can compute its value and we would like to use numeric differentiation to compute its derivatives. In this case we can use a combination ofNumericDiffCostFunction
andCostFunctionToFunctor
to get the job done.struct CameraProjection { CameraProjection(double* observation) intrinsic_projection_( new NumericDiffCostFunction<IntrinsicProjection, CENTRAL, 2, 5, 3>( new IntrinsicProjection(observation)) { } template <typename T> bool operator()(const T* rotation, const T* translation, const T* intrinsics, const T* point, T* residuals) const { T transformed_point[3]; RotateAndTranslatePoint(rotation, translation, point, transformed_point); return intrinsic_projection_(intrinsics, transformed_point, residual); } private: CostFunctionToFunctor<2,5,3> intrinsic_projection_; };
DynamicCostFunctionToFunctor
¶

class
DynamicCostFunctionToFunctor
¶ DynamicCostFunctionToFunctor
provides the same functionality asCostFunctionToFunctor
for cases where the number and size of the parameter vectors and residuals are not known at compiletime. The API provided byDynamicCostFunctionToFunctor
matches what would be expected byDynamicAutoDiffCostFunction
, i.e. it provides a templated functor of this form:template<typename T> bool operator()(T const* const* parameters, T* residuals) const;
Similar to the example given for
CostFunctionToFunctor
, let us assume thatclass IntrinsicProjection : public CostFunction { public: IntrinsicProjection(const double* observation); virtual bool Evaluate(double const* const* parameters, double* residuals, double** jacobians) const; };
is a
CostFunction
that projects a point in its local coordinate system onto its image plane and subtracts it from the observed point projection.Using this
CostFunction
in a templated functor would then look like this:struct CameraProjection { CameraProjection(double* observation) : intrinsic_projection_(new IntrinsicProjection(observation)) { } template <typename T> bool operator()(T const* const* parameters, T* residual) const { const T* rotation = parameters[0]; const T* translation = parameters[1]; const T* intrinsics = parameters[2]; const T* point = parameters[3]; T transformed_point[3]; RotateAndTranslatePoint(rotation, translation, point, transformed_point); const T* projection_parameters[2]; projection_parameters[0] = intrinsics; projection_parameters[1] = transformed_point; return intrinsic_projection_(projection_parameters, residual); } private: DynamicCostFunctionToFunctor intrinsic_projection_; };
Like
CostFunctionToFunctor
,DynamicCostFunctionToFunctor
takes ownership of theCostFunction
that was passed in to the constructor.
ConditionedCostFunction
¶

class
ConditionedCostFunction
¶ This class allows you to apply different conditioning to the residual values of a wrapped cost function. An example where this is useful is where you have an existing cost function that produces N values, but you want the total cost to be something other than just the sum of these squared values  maybe you want to apply a different scaling to some values, to change their contribution to the cost.
Usage:
// my_cost_function produces N residuals CostFunction* my_cost_function = ... CHECK_EQ(N, my_cost_function>num_residuals()); vector<CostFunction*> conditioners; // Make N 1x1 cost functions (1 parameter, 1 residual) CostFunction* f_1 = ... conditioners.push_back(f_1); CostFunction* f_N = ... conditioners.push_back(f_N); ConditionedCostFunction* ccf = new ConditionedCostFunction(my_cost_function, conditioners);
Now
ccf
‘sresidual[i]
(i=0..N1) will be passed though the \(i^{\text{th}}\) conditioner.ccf_residual[i] = f_i(my_cost_function_residual[i])
and the Jacobian will be affected appropriately.
GradientChecker
¶

class
GradientChecker
¶ This class compares the Jacobians returned by a cost function against derivatives estimated using finite differencing. It is meant as a tool for unit testing, giving you more finegrained control than the check_gradients option in the solver options.
The condition enforced is that
\[\forall{i,j}: \frac{J_{ij}  J'_{ij}}{max_{ij}(J_{ij}  J'_{ij})} < r\]where \(J_{ij}\) is the jacobian as computed by the supplied cost function (by the user) multiplied by the local parameterization Jacobian, \(J'_{ij}\) is the jacobian as computed by finite differences, multiplied by the local parameterization Jacobian as well, and \(r\) is the relative precision.
Usage:
// my_cost_function takes two parameter blocks. The first has a local // parameterization associated with it. CostFunction* my_cost_function = ... LocalParameterization* my_parameterization = ... NumericDiffOptions numeric_diff_options; std::vector<LocalParameterization*> local_parameterizations; local_parameterizations.push_back(my_parameterization); local_parameterizations.push_back(NULL); std::vector parameter1; std::vector parameter2; // Fill parameter 1 & 2 with test data... std::vector<double*> parameter_blocks; parameter_blocks.push_back(parameter1.data()); parameter_blocks.push_back(parameter2.data()); GradientChecker gradient_checker(my_cost_function, local_parameterizations, numeric_diff_options); GradientCheckResults results; if (!gradient_checker.Probe(parameter_blocks.data(), 1e9, &results) { LOG(ERROR) << "An error has occurred:\n" << results.error_log; }
NormalPrior
¶

class
NormalPrior
¶ class NormalPrior: public CostFunction { public: // Check that the number of rows in the vector b are the same as the // number of columns in the matrix A, crash otherwise. NormalPrior(const Matrix& A, const Vector& b); virtual bool Evaluate(double const* const* parameters, double* residuals, double** jacobians) const; };
Implements a cost function of the form
\[cost(x) = A(x  b)^2\]where, the matrix \(A\) and the vector \(b\) are fixed and \(x\) is the variable. In case the user is interested in implementing a cost function of the form
\[cost(x) = (x  \mu)^T S^{1} (x  \mu)\]where, \(\mu\) is a vector and \(S\) is a covariance matrix, then, \(A = S^{1/2}\), i.e the matrix \(A\) is the square root of the inverse of the covariance, also known as the stiffness matrix. There are however no restrictions on the shape of \(A\). It is free to be rectangular, which would be the case if the covariance matrix \(S\) is rank deficient.
LossFunction
¶

class
LossFunction
¶ For least squares problems where the minimization may encounter input terms that contain outliers, that is, completely bogus measurements, it is important to use a loss function that reduces their influence.
Consider a structure from motion problem. The unknowns are 3D points and camera parameters, and the measurements are image coordinates describing the expected reprojected position for a point in a camera. For example, we want to model the geometry of a street scene with fire hydrants and cars, observed by a moving camera with unknown parameters, and the only 3D points we care about are the pointy tippytops of the fire hydrants. Our magic image processing algorithm, which is responsible for producing the measurements that are input to Ceres, has found and matched all such tippytops in all image frames, except that in one of the frame it mistook a car’s headlight for a hydrant. If we didn’t do anything special the residual for the erroneous measurement will result in the entire solution getting pulled away from the optimum to reduce the large error that would otherwise be attributed to the wrong measurement.
Using a robust loss function, the cost for large residuals is reduced. In the example above, this leads to outlier terms getting downweighted so they do not overly influence the final solution.
class LossFunction { public: virtual void Evaluate(double s, double out[3]) const = 0; };
The key method is
LossFunction::Evaluate()
, which given a nonnegative scalars
, computes\[out = \begin{bmatrix}\rho(s), & \rho'(s), & \rho''(s)\end{bmatrix}\]Here the convention is that the contribution of a term to the cost function is given by \(\frac{1}{2}\rho(s)\), where \(s =\f_i\^2\). Calling the method with a negative value of \(s\) is an error and the implementations are not required to handle that case.
Most sane choices of \(\rho\) satisfy:
\[\begin{split}\rho(0) &= 0\\ \rho'(0) &= 1\\ \rho'(s) &< 1 \text{ in the outlier region}\\ \rho''(s) &< 0 \text{ in the outlier region}\end{split}\]so that they mimic the squared cost for small residuals.
Scaling
Given one robustifier \(\rho(s)\) one can change the length scale at which robustification takes place, by adding a scale factor \(a > 0\) which gives us \(\rho(s,a) = a^2 \rho(s / a^2)\) and the first and second derivatives as \(\rho'(s / a^2)\) and \((1 / a^2) \rho''(s / a^2)\) respectively.
The reason for the appearance of squaring is that \(a\) is in the units of the residual vector norm whereas \(s\) is a squared norm. For applications it is more convenient to specify \(a\) than its square.
Instances¶
Ceres includes a number of predefined loss functions. For simplicity
we described their unscaled versions. The figure below illustrates
their shape graphically. More details can be found in
include/ceres/loss_function.h
.

class
TrivialLoss
¶  \[\rho(s) = s\]

class
HuberLoss
¶  \[\begin{split}\rho(s) = \begin{cases} s & s \le 1\\ 2 \sqrt{s}  1 & s > 1 \end{cases}\end{split}\]

class
SoftLOneLoss
¶  \[\rho(s) = 2 (\sqrt{1+s}  1)\]

class
CauchyLoss
¶  \[\rho(s) = \log(1 + s)\]

class
ArctanLoss
¶  \[\rho(s) = \arctan(s)\]

class
TolerantLoss
¶  \[\rho(s,a,b) = b \log(1 + e^{(s  a) / b})  b \log(1 + e^{a / b})\]

class
ComposedLoss
¶ Given two loss functions
f
andg
, implements the loss functionh(s) = f(g(s))
.class ComposedLoss : public LossFunction { public: explicit ComposedLoss(const LossFunction* f, Ownership ownership_f, const LossFunction* g, Ownership ownership_g); };

class
ScaledLoss
¶ Sometimes you want to simply scale the output value of the robustifier. For example, you might want to weight different error terms differently (e.g., weight pixel reprojection errors differently from terrain errors).
Given a loss function \(\rho(s)\) and a scalar \(a\),
ScaledLoss
implements the function \(a \rho(s)\).Since we treat a
NULL
Loss function as the Identity loss function, \(rho\) =NULL
: is a valid input and will result in the input being scaled by \(a\). This provides a simple way of implementing a scaled ResidualBlock.

class
LossFunctionWrapper
¶ Sometimes after the optimization problem has been constructed, we wish to mutate the scale of the loss function. For example, when performing estimation from data which has substantial outliers, convergence can be improved by starting out with a large scale, optimizing the problem and then reducing the scale. This can have better convergence behavior than just using a loss function with a small scale.
This templated class allows the user to implement a loss function whose scale can be mutated after an optimization problem has been constructed, e.g,
Problem problem; // Add parameter blocks CostFunction* cost_function = new AutoDiffCostFunction < UW_Camera_Mapper, 2, 9, 3>( new UW_Camera_Mapper(feature_x, feature_y)); LossFunctionWrapper* loss_function(new HuberLoss(1.0), TAKE_OWNERSHIP); problem.AddResidualBlock(cost_function, loss_function, parameters); Solver::Options options; Solver::Summary summary; Solve(options, &problem, &summary); loss_function>Reset(new HuberLoss(1.0), TAKE_OWNERSHIP); Solve(options, &problem, &summary);
Theory¶
Let us consider a problem with a single problem and a single parameter block.
Then, the robustified gradient and the GaussNewton Hessian are
where the terms involving the second derivatives of \(f(x)\) have been ignored. Note that \(H(x)\) is indefinite if \(\rho''f(x)^\top f(x) + \frac{1}{2}\rho' < 0\). If this is not the case, then its possible to reweight the residual and the Jacobian matrix such that the corresponding linear least squares problem for the robustified GaussNewton step.
Let \(\alpha\) be a root of
Then, define the rescaled residual and Jacobian as
In the case \(2 \rho''\left\f(x)\right\^2 + \rho' \lesssim 0\), we limit \(\alpha \le 1 \epsilon\) for some small \(\epsilon\). For more details see [Triggs].
With this simple rescaling, one can use any Jacobian based nonlinear least squares algorithm to robustified nonlinear least squares problems.
LocalParameterization
¶

class
LocalParameterization
¶ class LocalParameterization { public: virtual ~LocalParameterization() {} virtual bool Plus(const double* x, const double* delta, double* x_plus_delta) const = 0; virtual bool ComputeJacobian(const double* x, double* jacobian) const = 0; virtual bool MultiplyByJacobian(const double* x, const int num_rows, const double* global_matrix, double* local_matrix) const; virtual int GlobalSize() const = 0; virtual int LocalSize() const = 0; };
Sometimes the parameters \(x\) can overparameterize a problem. In that case it is desirable to choose a parameterization to remove the null directions of the cost. More generally, if \(x\) lies on a manifold of a smaller dimension than the ambient space that it is embedded in, then it is numerically and computationally more effective to optimize it using a parameterization that lives in the tangent space of that manifold at each point.
For example, a sphere in three dimensions is a two dimensional manifold, embedded in a three dimensional space. At each point on the sphere, the plane tangent to it defines a two dimensional tangent space. For a cost function defined on this sphere, given a point \(x\), moving in the direction normal to the sphere at that point is not useful. Thus a better way to parameterize a point on a sphere is to optimize over two dimensional vector \(\Delta x\) in the tangent space at the point on the sphere point and then “move” to the point \(x + \Delta x\), where the move operation involves projecting back onto the sphere. Doing so removes a redundant dimension from the optimization, making it numerically more robust and efficient.
More generally we can define a function
\[x' = \boxplus(x, \Delta x),\]where \(x'\) has the same size as \(x\), and \(\Delta x\) is of size less than or equal to \(x\). The function \(\boxplus\), generalizes the definition of vector addition. Thus it satisfies the identity
\[\boxplus(x, 0) = x,\quad \forall x.\]Instances of
LocalParameterization
implement the \(\boxplus\) operation and its derivative with respect to \(\Delta x\) at \(\Delta x = 0\).

int
LocalParameterization::
GlobalSize
()¶ The dimension of the ambient space in which the parameter block \(x\) lives.

int
LocalParameterization::
LocalSize
()¶ The size of the tangent space that \(\Delta x\) lives in.

bool
LocalParameterization::
Plus
(const double *x, const double *delta, double *x_plus_delta) const¶ LocalParameterization::Plus()
implements \(\boxplus(x,\Delta x)\).

bool
LocalParameterization::
ComputeJacobian
(const double *x, double *jacobian) const¶ Computes the Jacobian matrix
\[J = \left . \frac{\partial }{\partial \Delta x} \boxplus(x,\Delta x)\right_{\Delta x = 0}\]in row major form.

bool
MultiplyByJacobian
(const double *x, const int num_rows, const double *global_matrix, double *local_matrix) const¶ local_matrix = global_matrix * jacobian
global_matrix is a num_rows x GlobalSize row major matrix. local_matrix is a num_rows x LocalSize row major matrix. jacobian is the matrix returned by
LocalParameterization::ComputeJacobian()
at \(x\).This is only used by GradientProblem. For most normal uses, it is okay to use the default implementation.
Instances¶

class
IdentityParameterization
¶ A trivial version of \(\boxplus\) is when \(\Delta x\) is of the same size as \(x\) and
\[\boxplus(x, \Delta x) = x + \Delta x\]

class
SubsetParameterization
¶ A more interesting case if \(x\) is a two dimensional vector, and the user wishes to hold the first coordinate constant. Then, \(\Delta x\) is a scalar and \(\boxplus\) is defined as
\[\begin{split}\boxplus(x, \Delta x) = x + \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \Delta x\end{split}\]SubsetParameterization
generalizes this construction to hold any part of a parameter block constant.

class
QuaternionParameterization
¶ Another example that occurs commonly in Structure from Motion problems is when camera rotations are parameterized using a quaternion. There, it is useful only to make updates orthogonal to that 4vector defining the quaternion. One way to do this is to let \(\Delta x\) be a 3 dimensional vector and define \(\boxplus\) to be
(3)\[\boxplus(x, \Delta x) = \left[ \cos(\Delta x), \frac{\sin\left(\Delta x\right)}{\Delta x} \Delta x \right] * x\]The multiplication between the two 4vectors on the right hand side is the standard quaternion product.
QuaternionParameterization
is an implementation of (3).

class
EigenQuaternionParameterization
¶ Eigen uses a different internal memory layout for the elements of the quaternion than what is commonly used. Specifically, Eigen stores the elements in memory as [x, y, z, w] where the real part is last whereas it is typically stored first. Note, when creating an Eigen quaternion through the constructor the elements are accepted in w, x, y, z order. Since Ceres operates on parameter blocks which are raw double pointers this difference is important and requires a different parameterization.
EigenQuaternionParameterization
uses the same update asQuaternionParameterization
but takes into account Eigen’s internal memory element ordering.

class
HomogeneousVectorParameterization
¶ In computer vision, homogeneous vectors are commonly used to represent entities in projective geometry such as points in projective space. One example where it is useful to use this overparameterization is in representing points whose triangulation is illconditioned. Here it is advantageous to use homogeneous vectors, instead of an Euclidean vector, because it can represent points at infinity.
When using homogeneous vectors it is useful to only make updates orthogonal to that \(n\)vector defining the homogeneous vector [HartleyZisserman]. One way to do this is to let \(\Delta x\) be a \(n1\) dimensional vector and define \(\boxplus\) to be
\[\boxplus(x, \Delta x) = \left[ \frac{\sin\left(0.5 \Delta x\right)}{\Delta x} \Delta x, \cos(0.5 \Delta x) \right] * x\]The multiplication between the two vectors on the right hand side is defined as an operator which applies the update orthogonal to \(x\) to remain on the sphere. Note, it is assumed that last element of \(x\) is the scalar component of the homogeneous vector.

class
ProductParameterization
¶ Consider an optimization problem over the space of rigid transformations \(SE(3)\), which is the Cartesian product of \(SO(3)\) and \(\mathbb{R}^3\). Suppose you are using Quaternions to represent the rotation, Ceres ships with a local parameterization for that and \(\mathbb{R}^3\) requires no, or
IdentityParameterization
parameterization. So how do we construct a local parameterization for a parameter block a rigid transformation?In cases, where a parameter block is the Cartesian product of a number of manifolds and you have the local parameterization of the individual manifolds available,
ProductParameterization
can be used to construct a local parameterization of the cartesian product. For the case of the rigid transformation, where say you have a parameter block of size 7, where the first four entries represent the rotation as a quaternion, a local parameterization can be constructed asProductParameterization se3_param(new QuaternionParameterization(), new IdentityTransformation(3));
AutoDiffLocalParameterization
¶

class
AutoDiffLocalParameterization
¶ AutoDiffLocalParameterization
does forLocalParameterization
whatAutoDiffCostFunction
does forCostFunction
. It allows the user to define a templated functor that implements theLocalParameterization::Plus()
operation and it uses automatic differentiation to implement the computation of the Jacobian.To get an auto differentiated local parameterization, you must define a class with a templated operator() (a functor) that computes
\[x' = \boxplus(x, \Delta x),\]For example, Quaternions have a three dimensional local parameterization. Its plus operation can be implemented as (taken from internal/ceres/autodiff_local_parameterization_test.cc )
struct QuaternionPlus { template<typename T> bool operator()(const T* x, const T* delta, T* x_plus_delta) const { const T squared_norm_delta = delta[0] * delta[0] + delta[1] * delta[1] + delta[2] * delta[2]; T q_delta[4]; if (squared_norm_delta > 0.0) { T norm_delta = sqrt(squared_norm_delta); const T sin_delta_by_delta = sin(norm_delta) / norm_delta; q_delta[0] = cos(norm_delta); q_delta[1] = sin_delta_by_delta * delta[0]; q_delta[2] = sin_delta_by_delta * delta[1]; q_delta[3] = sin_delta_by_delta * delta[2]; } else { // We do not just use q_delta = [1,0,0,0] here because that is a // constant and when used for automatic differentiation will // lead to a zero derivative. Instead we take a first order // approximation and evaluate it at zero. q_delta[0] = T(1.0); q_delta[1] = delta[0]; q_delta[2] = delta[1]; q_delta[3] = delta[2]; } Quaternionproduct(q_delta, x, x_plus_delta); return true; } };
Given this struct, the auto differentiated local parameterization can now be constructed as
LocalParameterization* local_parameterization = new AutoDiffLocalParameterization<QuaternionPlus, 4, 3>;   Global Size +  Local Size +
Problem
¶

class
Problem
¶ Problem
holds the robustified bounds constrained nonlinear least squares problem (1). To create a least squares problem, use theProblem::AddResidualBlock()
andProblem::AddParameterBlock()
methods.For example a problem containing 3 parameter blocks of sizes 3, 4 and 5 respectively and two residual blocks of size 2 and 6:
double x1[] = { 1.0, 2.0, 3.0 }; double x2[] = { 1.0, 2.0, 3.0, 5.0 }; double x3[] = { 1.0, 2.0, 3.0, 6.0, 7.0 }; Problem problem; problem.AddResidualBlock(new MyUnaryCostFunction(...), x1); problem.AddResidualBlock(new MyBinaryCostFunction(...), x2, x3);
Problem::AddResidualBlock()
as the name implies, adds a residual block to the problem. It adds aCostFunction
, an optionalLossFunction
and connects theCostFunction
to a set of parameter block.The cost function carries with it information about the sizes of the parameter blocks it expects. The function checks that these match the sizes of the parameter blocks listed in
parameter_blocks
. The program aborts if a mismatch is detected.loss_function
can beNULL
, in which case the cost of the term is just the squared norm of the residuals.The user has the option of explicitly adding the parameter blocks using
Problem::AddParameterBlock()
. This causes additional correctness checking; however,Problem::AddResidualBlock()
implicitly adds the parameter blocks if they are not present, so callingProblem::AddParameterBlock()
explicitly is not required.Problem::AddParameterBlock()
explicitly adds a parameter block to theProblem
. Optionally it allows the user to associate aLocalParameterization
object with the parameter block too. Repeated calls with the same arguments are ignored. Repeated calls with the same double pointer but a different size results in undefined behavior.You can set any parameter block to be constant using
Problem::SetParameterBlockConstant()
and undo this usingSetParameterBlockVariable()
.In fact you can set any number of parameter blocks to be constant, and Ceres is smart enough to figure out what part of the problem you have constructed depends on the parameter blocks that are free to change and only spends time solving it. So for example if you constructed a problem with a million parameter blocks and 2 million residual blocks, but then set all but one parameter blocks to be constant and say only 10 residual blocks depend on this one nonconstant parameter block. Then the computational effort Ceres spends in solving this problem will be the same if you had defined a problem with one parameter block and 10 residual blocks.
Ownership
Problem
by default takes ownership of thecost_function
,loss_function
andlocal_parameterization
pointers. These objects remain live for the life of theProblem
. If the user wishes to keep control over the destruction of these objects, then they can do this by setting the corresponding enums in theProblem::Options
struct.Note that even though the Problem takes ownership of
cost_function
andloss_function
, it does not preclude the user from reusing them in another residual block. The destructor takes care to call delete on eachcost_function
orloss_function
pointer only once, regardless of how many residual blocks refer to them.

ResidualBlockId
Problem::
AddResidualBlock
(CostFunction *cost_function, LossFunction *loss_function, const vector<double *> parameter_blocks)¶ Add a residual block to the overall cost function. The cost function carries with it information about the sizes of the parameter blocks it expects. The function checks that these match the sizes of the parameter blocks listed in parameter_blocks. The program aborts if a mismatch is detected. loss_function can be NULL, in which case the cost of the term is just the squared norm of the residuals.
The user has the option of explicitly adding the parameter blocks using AddParameterBlock. This causes additional correctness checking; however, AddResidualBlock implicitly adds the parameter blocks if they are not present, so calling AddParameterBlock explicitly is not required.
The Problem object by default takes ownership of the cost_function and loss_function pointers. These objects remain live for the life of the Problem object. If the user wishes to keep control over the destruction of these objects, then they can do this by setting the corresponding enums in the Options struct.
Note: Even though the Problem takes ownership of cost_function and loss_function, it does not preclude the user from reusing them in another residual block. The destructor takes care to call delete on each cost_function or loss_function pointer only once, regardless of how many residual blocks refer to them.
Example usage:
double x1[] = {1.0, 2.0, 3.0}; double x2[] = {1.0, 2.0, 5.0, 6.0}; double x3[] = {3.0, 6.0, 2.0, 5.0, 1.0}; Problem problem; problem.AddResidualBlock(new MyUnaryCostFunction(...), NULL, x1); problem.AddResidualBlock(new MyBinaryCostFunction(...), NULL, x2, x1);

void
Problem::
AddParameterBlock
(double *values, int size, LocalParameterization *local_parameterization)¶ Add a parameter block with appropriate size to the problem. Repeated calls with the same arguments are ignored. Repeated calls with the same double pointer but a different size results in undefined behavior.

void
Problem::
AddParameterBlock
(double *values, int size)¶ Add a parameter block with appropriate size and parameterization to the problem. Repeated calls with the same arguments are ignored. Repeated calls with the same double pointer but a different size results in undefined behavior.

void
Problem::
RemoveResidualBlock
(ResidualBlockId residual_block)¶ Remove a residual block from the problem. Any parameters that the residual block depends on are not removed. The cost and loss functions for the residual block will not get deleted immediately; won’t happen until the problem itself is deleted. If Problem::Options::enable_fast_removal is true, then the removal is fast (almost constant time). Otherwise, removing a residual block will incur a scan of the entire Problem object to verify that the residual_block represents a valid residual in the problem.
WARNING: Removing a residual or parameter block will destroy the implicit ordering, rendering the jacobian or residuals returned from the solver uninterpretable. If you depend on the evaluated jacobian, do not use remove! This may change in a future release. Hold the indicated parameter block constant during optimization.

void
Problem::
RemoveParameterBlock
(double *values)¶ Remove a parameter block from the problem. The parameterization of the parameter block, if it exists, will persist until the deletion of the problem (similar to cost/loss functions in residual block removal). Any residual blocks that depend on the parameter are also removed, as described above in RemoveResidualBlock(). If Problem::Options::enable_fast_removal is true, then the removal is fast (almost constant time). Otherwise, removing a parameter block will incur a scan of the entire Problem object.
WARNING: Removing a residual or parameter block will destroy the implicit ordering, rendering the jacobian or residuals returned from the solver uninterpretable. If you depend on the evaluated jacobian, do not use remove! This may change in a future release.

void
Problem::
SetParameterBlockConstant
(double *values)¶ Hold the indicated parameter block constant during optimization.

void
Problem::
SetParameterBlockVariable
(double *values)¶ Allow the indicated parameter to vary during optimization.

void
Problem::
SetParameterization
(double *values, LocalParameterization *local_parameterization)¶ Set the local parameterization for one of the parameter blocks. The local_parameterization is owned by the Problem by default. It is acceptable to set the same parameterization for multiple parameters; the destructor is careful to delete local parameterizations only once. The local parameterization can only be set once per parameter, and cannot be changed once set.

LocalParameterization *
Problem::
GetParameterization
(double *values) const¶ Get the local parameterization object associated with this parameter block. If there is no parameterization object associated then NULL is returned

void
Problem::
SetParameterLowerBound
(double *values, int index, double lower_bound)¶ Set the lower bound for the parameter at position index in the parameter block corresponding to values. By default the lower bound is \(\infty\).

void
Problem::
SetParameterUpperBound
(double *values, int index, double upper_bound)¶ Set the upper bound for the parameter at position index in the parameter block corresponding to values. By default the value is \(\infty\).

int
Problem::
NumParameterBlocks
() const¶ Number of parameter blocks in the problem. Always equals parameter_blocks().size() and parameter_block_sizes().size().

int
Problem::
NumParameters
() const¶ The size of the parameter vector obtained by summing over the sizes of all the parameter blocks.

int
Problem::
NumResidualBlocks
() const¶ Number of residual blocks in the problem. Always equals residual_blocks().size().

int
Problem::
NumResiduals
() const¶ The size of the residual vector obtained by summing over the sizes of all of the residual blocks.

int
Problem::
ParameterBlockSize
(const double *values) const¶ The size of the parameter block.

int
Problem::
ParameterBlockLocalSize
(const double *values) const¶ The size of local parameterization for the parameter block. If there is no local parameterization associated with this parameter block, then
ParameterBlockLocalSize
=ParameterBlockSize
.

bool
Problem::
HasParameterBlock
(const double *values) const¶ Is the given parameter block present in the problem or not?

void
Problem::
GetParameterBlocks
(vector<double *> *parameter_blocks) const¶ Fills the passed
parameter_blocks
vector with pointers to the parameter blocks currently in the problem. After this call,parameter_block.size() == NumParameterBlocks
.

void
Problem::
GetResidualBlocks
(vector<ResidualBlockId> *residual_blocks) const¶ Fills the passed residual_blocks vector with pointers to the residual blocks currently in the problem. After this call, residual_blocks.size() == NumResidualBlocks.

void
Problem::
GetParameterBlocksForResidualBlock
(const ResidualBlockId residual_block, vector<double *> *parameter_blocks) const¶ Get all the parameter blocks that depend on the given residual block.

void
Problem::
GetResidualBlocksForParameterBlock
(const double *values, vector<ResidualBlockId> *residual_blocks) const¶ Get all the residual blocks that depend on the given parameter block.
If Problem::Options::enable_fast_removal is true, then getting the residual blocks is fast and depends only on the number of residual blocks. Otherwise, getting the residual blocks for a parameter block will incur a scan of the entire
Problem
object.

const CostFunction *
GetCostFunctionForResidualBlock
(const ResidualBlockId residual_block) const¶ Get the
CostFunction
for the given residual block.

const LossFunction *
GetLossFunctionForResidualBlock
(const ResidualBlockId residual_block) const¶ Get the
LossFunction
for the given residual block.

bool
Problem::
Evaluate
(const Problem::EvaluateOptions &options, double *cost, vector<double> *residuals, vector<double> *gradient, CRSMatrix *jacobian)¶ Evaluate a
Problem
. Any of the output pointers can be NULL. Which residual blocks and parameter blocks are used is controlled by theProblem::EvaluateOptions
struct below.Note
The evaluation will use the values stored in the memory locations pointed to by the parameter block pointers used at the time of the construction of the problem, for example in the following code:
Problem problem; double x = 1; problem.Add(new MyCostFunction, NULL, &x); double cost = 0.0; problem.Evaluate(Problem::EvaluateOptions(), &cost, NULL, NULL, NULL);
The cost is evaluated at x = 1. If you wish to evaluate the problem at x = 2, then
x = 2; problem.Evaluate(Problem::EvaluateOptions(), &cost, NULL, NULL, NULL);
is the way to do so.
Note
If no local parameterizations are used, then the size of the gradient vector is the sum of the sizes of all the parameter blocks. If a parameter block has a local parameterization, then it contributes “LocalSize” entries to the gradient vector.
Note
This function cannot be called while the problem is being solved, for example it cannot be called from an
IterationCallback
at the end of an iteration during a solve.

class
Problem::
EvaluateOptions
¶ Options struct that is used to control
Problem::Evaluate()
.

vector<double *>
Problem::EvaluateOptions::
parameter_blocks
¶ The set of parameter blocks for which evaluation should be performed. This vector determines the order in which parameter blocks occur in the gradient vector and in the columns of the jacobian matrix. If parameter_blocks is empty, then it is assumed to be equal to a vector containing ALL the parameter blocks. Generally speaking the ordering of the parameter blocks in this case depends on the order in which they were added to the problem and whether or not the user removed any parameter blocks.
NOTE This vector should contain the same pointers as the ones used to add parameter blocks to the Problem. These parameter block should NOT point to new memory locations. Bad things will happen if you do.

vector<ResidualBlockId>
Problem::EvaluateOptions::
residual_blocks
¶ The set of residual blocks for which evaluation should be performed. This vector determines the order in which the residuals occur, and how the rows of the jacobian are ordered. If residual_blocks is empty, then it is assumed to be equal to the vector containing all the parameter blocks.
rotation.h
¶
Many applications of Ceres Solver involve optimization problems where some of the variables correspond to rotations. To ease the pain of work with the various representations of rotations (angleaxis, quaternion and matrix) we provide a handy set of templated functions. These functions are templated so that the user can use them within Ceres Solver’s automatic differentiation framework.

template<typename
T
> 
void
AngleAxisToQuaternion
(T const *angle_axis, T *quaternion)¶ Convert a value in combined axisangle representation to a quaternion.
The value
angle_axis
is a triple whose norm is an angle in radians, and whose direction is aligned with the axis of rotation, andquaternion
is a 4tuple that will contain the resulting quaternion.

template<typename
T
> 
void
QuaternionToAngleAxis
(T const *quaternion, T *angle_axis)¶ Convert a quaternion to the equivalent combined axisangle representation.
The value
quaternion
must be a unit quaternion  it is not normalized first, andangle_axis
will be filled with a value whose norm is the angle of rotation in radians, and whose direction is the axis of rotation.

template<typename
T
, introw_stride
, intcol_stride
> 
void
RotationMatrixToAngleAxis
(const MatrixAdapter<const T, row_stride, col_stride> &R, T *angle_axis)¶

template<typename
T
, introw_stride
, intcol_stride
> 
void
AngleAxisToRotationMatrix
(T const *angle_axis, const MatrixAdapter<T, row_stride, col_stride> &R)¶

template<typename
T
> 
void
AngleAxisToRotationMatrix
(T const *angle_axis, T *R)¶ Conversions between 3x3 rotation matrix with given column and row strides and axisangle rotation representations. The functions that take a pointer to T instead of a MatrixAdapter assume a column major representation with unit row stride and a column stride of 3.

template<typename
T
, introw_stride
, intcol_stride
> 
void
EulerAnglesToRotationMatrix
(const T *euler, const MatrixAdapter<T, row_stride, col_stride> &R)¶

template<typename
T
> 
void
EulerAnglesToRotationMatrix
(const T *euler, int row_stride, T *R)¶ Conversions between 3x3 rotation matrix with given column and row strides and Euler angle (in degrees) rotation representations.
The {pitch,roll,yaw} Euler angles are rotations around the {x,y,z} axes, respectively. They are applied in that same order, so the total rotation R is Rz * Ry * Rx.
The function that takes a pointer to T as the rotation matrix assumes a row major representation with unit column stride and a row stride of 3. The additional parameter row_stride is required to be 3.

template<typename
T
, introw_stride
, intcol_stride
> 
void
QuaternionToScaledRotation
(const T q[4], const MatrixAdapter<T, row_stride, col_stride> &R)¶

template<typename
T
> 
void
QuaternionToScaledRotation
(const T q[4], T R[3 * 3])¶ Convert a 4vector to a 3x3 scaled rotation matrix.
The choice of rotation is such that the quaternion \(\begin{bmatrix} 1 &0 &0 &0\end{bmatrix}\) goes to an identity matrix and for small \(a, b, c\) the quaternion \(\begin{bmatrix}1 &a &b &c\end{bmatrix}\) goes to the matrix
\[\begin{split}I + 2 \begin{bmatrix} 0 & c & b \\ c & 0 & a\\ b & a & 0 \end{bmatrix} + O(q^2)\end{split}\]which corresponds to a Rodrigues approximation, the last matrix being the crossproduct matrix of \(\begin{bmatrix} a& b& c\end{bmatrix}\). Together with the property that \(R(q1 * q2) = R(q1) * R(q2)\) this uniquely defines the mapping from \(q\) to \(R\).
In the function that accepts a pointer to T instead of a MatrixAdapter, the rotation matrix
R
is a rowmajor matrix with unit column stride and a row stride of 3.No normalization of the quaternion is performed, i.e. \(R = \q\^2 Q\), where \(Q\) is an orthonormal matrix such that \(\det(Q) = 1\) and \(Q*Q' = I\).

template<typename
T
> 
void
QuaternionToRotation
(const T q[4], const MatrixAdapter<T, row_stride, col_stride> &R)¶

template<typename
T
> 
void
QuaternionToRotation
(const T q[4], T R[3 * 3])¶ Same as above except that the rotation matrix is normalized by the Frobenius norm, so that \(R R' = I\) (and \(\det(R) = 1\)).

template<typename
T
> 
void
UnitQuaternionRotatePoint
(const T q[4], const T pt[3], T result[3])¶ Rotates a point pt by a quaternion q:
\[\text{result} = R(q) \text{pt}\]Assumes the quaternion is unit norm. If you pass in a quaternion with \(q^2 = 2\) then you WILL NOT get back 2 times the result you get for a unit quaternion.

template<typename
T
> 
void
QuaternionRotatePoint
(const T q[4], const T pt[3], T result[3])¶ With this function you do not need to assume that \(q\) has unit norm. It does assume that the norm is nonzero.

template<typename
T
> 
void
QuaternionProduct
(const T z[4], const T w[4], T zw[4])¶  \[zw = z * w\]
where \(*\) is the Quaternion product between 4vectors.
Cubic Interpolation¶
Optimization problems often involve functions that are given in the form of a table of values, for example an image. Evaluating these functions and their derivatives requires interpolating these values. Interpolating tabulated functions is a vast area of research and there are a lot of libraries which implement a variety of interpolation schemes. However, using them within the automatic differentiation framework in Ceres is quite painful. To this end, Ceres provides the ability to interpolate one dimensional and two dimensional tabular functions.
The one dimensional interpolation is based on the Cubic Hermite Spline, also known as the CatmullRom Spline. This produces a first order differentiable interpolating function. The two dimensional interpolation scheme is a generalization of the one dimensional scheme where the interpolating function is assumed to be separable in the two dimensions,
More details of the construction can be found Linear Methods for Image Interpolation by Pascal Getreuer.

class
CubicInterpolator
¶
Given as input an infinite one dimensional grid, which provides the following interface.
struct Grid1D {
enum { DATA_DIMENSION = 2; };
void GetValue(int n, double* f) const;
};
Where, GetValue
gives us the value of a function \(f\)
(possibly vector valued) for any integer \(n\) and the enum
DATA_DIMENSION
indicates the dimensionality of the function being
interpolated. For example if you are interpolating rotations in
axisangle format over time, then DATA_DIMENSION = 3
.
CubicInterpolator
uses Cubic Hermite splines to produce a
smooth approximation to it that can be used to evaluate the
\(f(x)\) and \(f'(x)\) at any point on the real number
line. For example, the following code interpolates an array of four
numbers.
const double data[] = {1.0, 2.0, 5.0, 6.0};
Grid1D<double, 1> array(x, 0, 4);
CubicInterpolator interpolator(array);
double f, dfdx;
interpolator.Evaluate(1.5, &f, &dfdx);
In the above code we use Grid1D
a templated helper class that
allows easy interfacing between C++
arrays and
CubicInterpolator
.
Grid1D
supports vector valued functions where the various
coordinates of the function can be interleaved or stacked. It also
allows the use of any numeric type as input, as long as it can be
safely cast to a double.

class
BiCubicInterpolator
¶
Given as input an infinite two dimensional grid, which provides the following interface:
struct Grid2D {
enum { DATA_DIMENSION = 2 };
void GetValue(int row, int col, double* f) const;
};
Where, GetValue
gives us the value of a function \(f\)
(possibly vector valued) for any pair of integers row
and
col
and the enum DATA_DIMENSION
indicates the
dimensionality of the function being interpolated. For example if you
are interpolating a color image with three channels (Red, Green &
Blue), then DATA_DIMENSION = 3
.
BiCubicInterpolator
uses the cubic convolution interpolation
algorithm of R. Keys [Keys], to produce a smooth approximation to it
that can be used to evaluate the \(f(r,c)\), \(\frac{\partial
f(r,c)}{\partial r}\) and \(\frac{\partial f(r,c)}{\partial c}\) at
any any point in the real plane.
For example the following code interpolates a two dimensional array.
const double data[] = {1.0, 3.0, 1.0, 4.0,
3.6, 2.1, 4.2, 2.0,
2.0, 1.0, 3.1, 5.2};
Grid2D<double, 1> array(data, 0, 3, 0, 4);
BiCubicInterpolator interpolator(array);
double f, dfdr, dfdc;
interpolator.Evaluate(1.2, 2.5, &f, &dfdr, &dfdc);
In the above code, the templated helper class Grid2D
is used to
make a C++
array look like a two dimensional table to
BiCubicInterpolator
.
Grid2D
supports row or column major layouts. It also supports
vector valued functions where the individual coordinates of the
function may be interleaved or stacked. It also allows the use of any
numeric type as input, as long as it can be safely cast to double.