# Modeling¶

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 discusses the various ways in which an optimization problem can be solved using Ceres.

Ceres solves robustified bounds constrained non-linear least squares problems of the form:

(1)$\begin{split}\min_{\mathbf{x}} &\quad \frac{1}{2}\sum_{i} \rho_i\left(\left\|f_i\left(x_{i_1}, ... ,x_{i_k}\right)\right\|^2\right) \\ \text{s.t.} &\quad l_j \le x_j \le u_j\end{split}$

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 non-linear 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 non-linear least squares problem.

(2)$\frac{1}{2}\sum_{i} \left\|f_i\left(x_{i_1}, ... ,x_{i_k}\right)\right\|^2.$

## 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 as CostFunction::parameter_block_sizes_ and the parameter blocks are in the same order as CostFunction::parameter_block_sizes_.

residuals is an array of size num_residuals_.

jacobians is an array of size CostFunction::parameter_block_sizes_ containing pointers to storage for Jacobian matrices corresponding to each parameter block. The Jacobian matrices are in the same order as CostFunction::parameter_block_sizes_. jacobians[i] is an array that contains CostFunction::num_residuals_ x CostFunction::parameter_block_sizes_ [i] elements. Each Jacobian matrix is stored in row-major order, i.e., jacobians[i][r * parameter_block_size_[i] + c] = $$\frac{\partial residual[r]}{\partial parameters[i][c]}$$

If jacobians is NULL, then no derivatives are returned; this is the case when computing cost only. If jacobians[i] is NULL, 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 implement CostFunction::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 a SizedCostFunction 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
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 parameter T. The autodiff framework substitutes appropriate Jet objects for T in order to compute the derivative when necessary, but this is hidden, and you should write the function as if T were a scalar type (e.g. a double-precision 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 two-dimensional 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 auto-differentiable 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] = T(k_) - x[0] * y[0] - x[1] * y[1];
return true;
}

private:
double k_;
};


Note that in the declaration of operator() the input parameters x and y come first, and are passed as const pointers to arrays of T. If there were three input parameters, then the third input parameter would come after y. The output is always the last parameter, and is also a pointer to an array. In the example above, e is a scalar, so only e[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 1-dimensional output from two arguments, both 2-dimensional.

AutoDiffCostFunction also supports cost functions with a runtime-determined 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 Since the functor will get instantiated with different types for T, you must convert from other numeric types to T before mixing computations with other variables of type T. In the example above, this is seen where instead of using k_ directly, k_ is wrapped with T(k_).

WARNING 2 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. Like AutoDiffCostFunction 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 MyCostFunctor());
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 cache-locality 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 use DynamicAutoDiffCostFunction.

## 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.

template <typename CostFunctor,
NumericDiffMethod 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 a operator() (a functor) that computes the residuals. The functor must write the computed value in the last argument (the only non-const one) and return true to indicate success. Please see CostFunction for details on how the return value may be used to impose simple constraints on the parameter block. e.g., an object of the form

struct 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 two-dimensional 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 numerically-differentiable 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 parameters x and y come first, and are passed as const pointers to arrays of double s. If there were three input parameters, then the third input parameter would come after y. 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 only residuals[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 1-dimensional output from two arguments, both 2-dimensional.

NumericDiffCostFunction also supports cost functions with a runtime-determined 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.

The CENTRAL difference method is considerably more accurate at the cost of twice as many function evaluations than forward difference. Consider using central differences begin with, and only after that works, trying forward difference to improve performance.

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 last 2 argument. Please be careful when setting the size parameters.

Alternate Interface

For a variety of reason, including compatibility with legacy code, NumericDiffCostFunction can also take CostFunction objects as input. The following describes how.

To get a numerically differentiated cost function, define a subclass of CostFunction such that the CostFunction::Evaluate() function ignores the jacobians parameter. The numeric differentiation wrapper will fill in the jacobian parameter if necessary by repeatedly calling the CostFunction::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 the CostFunction 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, NumericDiffMethod 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 MyCostFunctor());
cost_function.SetNumResiduals(21);


As a rule of thumb, try using NumericDiffCostFunction before you use DynamicNumericDiffCostFunction.

## NumericDiffFunctor¶

class NumericDiffFunctor

Sometimes parts of a cost function can be differentiated automatically or analytically but others require numeric differentiation. NumericDiffFunctor is a wrapper class that takes a variadic functor evaluating a function, numerically differentiates it and makes it available as a templated functor so that it can be easily used as part of Ceres’ automatic differentiation framework.

For example, let us assume that

struct IntrinsicProjection
IntrinsicProjection(const double* observations);
bool operator()(const double* calibration,
const double* point,
double* residuals);
};


is a functor that implements the projection of a point in its local coordinate system onto its image plane and subtracts it from the observed point projection.

Now we would like to compose the action of this functor with the action of camera extrinsics, i.e., rotation and translation, which is given by the following templated function

template<typename T>
void RotateAndTranslatePoint(const T* rotation,
const T* translation,
const T* point,
T* result);


To compose the extrinsics and intrinsics, we can construct a CameraProjection functor as follows.

struct CameraProjection {
typedef NumericDiffFunctor<IntrinsicProjection, CENTRAL, 2, 5, 3>
IntrinsicProjectionFunctor;

CameraProjection(double* observation) {
intrinsic_projection_.reset(
new IntrinsicProjectionFunctor(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:
scoped_ptr<IntrinsicProjectionFunctor> intrinsic_projection_;
};


Here, we made the choice of using CENTRAL differences to compute the jacobian of IntrinsicProjection.

Now, we are ready to construct an automatically differentiated cost function as

CostFunction* cost_function =
new AutoDiffCostFunction<CameraProjection, 2, 3, 3, 5>(
new CameraProjection(observations));


cost_function now seamlessly integrates automatic differentiation of RotateAndTranslatePoint with a numerically differentiated version of IntrinsicProjection.

## CostFunctionToFunctor¶

class CostFunctionToFunctor

Just like NumericDiffFunctor allows numeric differentiation to be mixed with automatic differentiation, CostFunctionToFunctor provides an even more general mechanism. CostFunctionToFunctor is an adapter class that allows users to use CostFunction 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* observations);
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 function

template<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_.reset(
new CostFunctionToFunctor<2, 5, 3>(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:
scoped_ptr<CostFunctionToFunctor<2,5,3> > intrinsic_projection_;
};


## 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 ‘s residual[i] (i=0..N-1) 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.

## 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 tippy-tops 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 tippy-tops 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 down-weighted 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 non-negative scalar s, computes

$\begin{split}out = \begin{bmatrix}\rho(s), & \rho'(s), & \rho''(s)\end{bmatrix}\end{split}$

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.

Shape of the various common loss functions.

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 and g, implements the loss function h(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 the 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;

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);

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.

$\min_x \frac{1}{2}\rho(f^2(x))$

Then, the robustified gradient and the Gauss-Newton Hessian are

$\begin{split}g(x) &= \rho'J^\top(x)f(x)\\ H(x) &= J^\top(x)\left(\rho' + 2 \rho''f(x)f^\top(x)\right)J(x)\end{split}$

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 re-weight the residual and the Jacobian matrix such that the corresponding linear least squares problem for the robustified Gauss-Newton step.

Let $$\alpha$$ be a root of

$\frac{1}{2}\alpha^2 - \alpha - \frac{\rho''}{\rho'}\|f(x)\|^2 = 0.$

Then, define the rescaled residual and Jacobian as

$\begin{split}\tilde{f}(x) &= \frac{\sqrt{\rho'}}{1 - \alpha} f(x)\\ \tilde{J}(x) &= \sqrt{\rho'}\left(1 - \alpha \frac{f(x)f^\top(x)}{\left\|f(x)\right\|^2} \right)J(x)\end{split}$

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 non-linear least squares algorithm to robustified non-linear 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 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 LocalParamterization::LocaLocalSize()

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.

### 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 4-vector 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 4-vectors on the right hand side is the standard quaternion product. QuaternionParameterization is an implementation of (3).

## AutoDiffLocalParameterization¶

class AutoDiffLocalParameterization

AutoDiffLocalParameterization does for LocalParameterization what AutoDiffCostFunction does for CostFunction. It allows the user to define a templated functor that implements the LocalParameterization::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. It’s plus operation can be implemented as (taken from internal/ceres/auto_diff_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 > T(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;
}
};


Then 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 -------------------+


WARNING: Since the functor will get instantiated with different types for T, you must to convert from other numeric types to T before mixing computations with other variables of type T. In the example above, this is seen where instead of using k_ directly, k_ is wrapped with T(k_).

## Problem¶

class Problem

Problem holds the robustified bounds constrained non-linear least squares problem (1). To create a least squares problem, use the Problem::AddResidualBlock() and Problem::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() as the name implies, adds a residual block to the problem. It adds a CostFunction, an optional LossFunction and connects the CostFunction 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 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 Problem::AddParameterBlock(). This causes additional correctness checking; however, Problem::AddResidualBlock() implicitly adds the parameter blocks if they are not present, so calling Problem::AddParameterBlock() explicitly is not required.

Problem::AddParameterBlock() explicitly adds a parameter block to the Problem. Optionally it allows the user to associate a LocalParameterization 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 using SetParameterBlockVariable().

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 non-constant 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 the cost_function, loss_function and local_parameterization pointers. These objects remain live for the life of the Problem. 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 Problem::Options struct.

Note that even though the Problem takes ownership of cost_function and loss_function, it does not preclude the user from re-using 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.

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 re-using 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;


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.

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.

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 (angle-axis, 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.

void AngleAxisToQuaternion<T>(T const* angle_axis, T* quaternion)

Convert a value in combined axis-angle 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, and quaternion is a 4-tuple that will contain the resulting quaternion.

void QuaternionToAngleAxis<T>(T const* quaternion, T* angle_axis)

Convert a quaternion to the equivalent combined axis-angle representation.

The value quaternion must be a unit quaternion - it is not normalized first, and angle_axis will be filled with a value whose norm is the angle of rotation in radians, and whose direction is the axis of rotation.

void RotationMatrixToAngleAxis<T, row_stride, col_stride>(const MatrixAdapter<const T, row_stride, col_stride>& R, T* angle_axis)
void AngleAxisToRotationMatrix<T, row_stride, col_stride>(T const* angle_axis, const MatrixAdapter<T, row_stride, col_stride>& R)
void RotationMatrixToAngleAxis<T>(T const* R, T* angle_axis)
void AngleAxisToRotationMatrix<T>(T const* angle_axis, T* R)

Conversions between 3x3 rotation matrix with given column and row strides and axis-angle 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.

void EulerAnglesToRotationMatrix<T, row_stride, col_stride>(const T* euler, const MatrixAdapter<T, row_stride, col_stride>& R)
void EulerAnglesToRotationMatrix<T>(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.

void QuaternionToScaledRotation<T, row_stride, col_stride>(const T q[4], const MatrixAdapter<T, row_stride, col_stride>& R)
void QuaternionToScaledRotation<T>(const T q[4], T R[3 * 3])

Convert a 4-vector 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 cross-product 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 row-major 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 $$Q*Q' = I$$.

void QuaternionToRotation<T>(const T q[4], const MatrixAdapter<T, row_stride, col_stride>& R)
void QuaternionToRotation<T>(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 $$\det(R) = 1$$).

void UnitQuaternionRotatePoint<T>(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.

void QuaternionRotatePoint<T>(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 non-zero.

void QuaternionProduct<T>(const T z[4], const T w[4], T zw[4])
$zw = z * w$

where $$*$$ is the Quaternion product between 4-vectors.

void CrossProduct<T>(const T x[3], const T y[3], T x_cross_y[3])
$\text{x_cross_y} = x \times y$
void AngleAxisRotatePoint<T>(const T angle_axis[3], const T pt[3], T result[3])
$y = R(\text{angle_axis}) x$