| // Ceres Solver - A fast non-linear least squares minimizer |
| // Copyright 2019 Google Inc. All rights reserved. |
| // http://ceres-solver.org/ |
| // |
| // Redistribution and use in source and binary forms, with or without |
| // modification, are permitted provided that the following conditions are met: |
| // |
| // * Redistributions of source code must retain the above copyright notice, |
| // this list of conditions and the following disclaimer. |
| // * Redistributions in binary form must reproduce the above copyright notice, |
| // this list of conditions and the following disclaimer in the documentation |
| // and/or other materials provided with the distribution. |
| // * Neither the name of Google Inc. nor the names of its contributors may be |
| // used to endorse or promote products derived from this software without |
| // specific prior written permission. |
| // |
| // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" |
| // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE |
| // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE |
| // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE |
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| // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) |
| // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE |
| // POSSIBILITY OF SUCH DAMAGE. |
| // |
| // Author: sameeragarwal@google.com (Sameer Agarwal) |
| |
| #ifndef CERES_PUBLIC_COVARIANCE_H_ |
| #define CERES_PUBLIC_COVARIANCE_H_ |
| |
| #include <memory> |
| #include <utility> |
| #include <vector> |
| |
| #include "ceres/internal/config.h" |
| #include "ceres/internal/disable_warnings.h" |
| #include "ceres/internal/export.h" |
| #include "ceres/types.h" |
| |
| namespace ceres { |
| |
| class Problem; |
| |
| namespace internal { |
| class CovarianceImpl; |
| } // namespace internal |
| |
| // WARNING |
| // ======= |
| // It is very easy to use this class incorrectly without understanding |
| // the underlying mathematics. Please read and understand the |
| // documentation completely before attempting to use it. |
| // |
| // |
| // This class allows the user to evaluate the covariance for a |
| // non-linear least squares problem and provides random access to its |
| // blocks |
| // |
| // Background |
| // ========== |
| // One way to assess the quality of the solution returned by a |
| // non-linear least squares solver is to analyze the covariance of the |
| // solution. |
| // |
| // Let us consider the non-linear regression problem |
| // |
| // y = f(x) + N(0, I) |
| // |
| // i.e., the observation y is a random non-linear function of the |
| // independent variable x with mean f(x) and identity covariance. Then |
| // the maximum likelihood estimate of x given observations y is the |
| // solution to the non-linear least squares problem: |
| // |
| // x* = arg min_x |f(x) - y|^2 |
| // |
| // And the covariance of x* is given by |
| // |
| // C(x*) = inverse[J'(x*)J(x*)] |
| // |
| // Here J(x*) is the Jacobian of f at x*. The above formula assumes |
| // that J(x*) has full column rank. |
| // |
| // If J(x*) is rank deficient, then the covariance matrix C(x*) is |
| // also rank deficient and is given by |
| // |
| // C(x*) = pseudoinverse[J'(x*)J(x*)] |
| // |
| // Note that in the above, we assumed that the covariance |
| // matrix for y was identity. This is an important assumption. If this |
| // is not the case and we have |
| // |
| // y = f(x) + N(0, S) |
| // |
| // Where S is a positive semi-definite matrix denoting the covariance |
| // of y, then the maximum likelihood problem to be solved is |
| // |
| // x* = arg min_x f'(x) inverse[S] f(x) |
| // |
| // and the corresponding covariance estimate of x* is given by |
| // |
| // C(x*) = inverse[J'(x*) inverse[S] J(x*)] |
| // |
| // So, if it is the case that the observations being fitted to have a |
| // covariance matrix not equal to identity, then it is the user's |
| // responsibility that the corresponding cost functions are correctly |
| // scaled, e.g. in the above case the cost function for this problem |
| // should evaluate S^{-1/2} f(x) instead of just f(x), where S^{-1/2} |
| // is the inverse square root of the covariance matrix S. |
| // |
| // This class allows the user to evaluate the covariance for a |
| // non-linear least squares problem and provides random access to its |
| // blocks. The computation assumes that the CostFunctions compute |
| // residuals such that their covariance is identity. |
| // |
| // Since the computation of the covariance matrix requires computing |
| // the inverse of a potentially large matrix, this can involve a |
| // rather large amount of time and memory. However, it is usually the |
| // case that the user is only interested in a small part of the |
| // covariance matrix. Quite often just the block diagonal. This class |
| // allows the user to specify the parts of the covariance matrix that |
| // she is interested in and then uses this information to only compute |
| // and store those parts of the covariance matrix. |
| // |
| // Rank of the Jacobian |
| // -------------------- |
| // As we noted above, if the jacobian is rank deficient, then the |
| // inverse of J'J is not defined and instead a pseudo inverse needs to |
| // be computed. |
| // |
| // The rank deficiency in J can be structural -- columns which are |
| // always known to be zero or numerical -- depending on the exact |
| // values in the Jacobian. |
| // |
| // Structural rank deficiency occurs when the problem contains |
| // parameter blocks that are constant. This class correctly handles |
| // structural rank deficiency like that. |
| // |
| // Numerical rank deficiency, where the rank of the matrix cannot be |
| // predicted by its sparsity structure and requires looking at its |
| // numerical values is more complicated. Here again there are two |
| // cases. |
| // |
| // a. The rank deficiency arises from overparameterization. e.g., a |
| // four dimensional quaternion used to parameterize SO(3), which is |
| // a three dimensional manifold. In cases like this, the user should |
| // use an appropriate LocalParameterization/Manifold. Not only will this lead |
| // to better numerical behaviour of the Solver, it will also expose |
| // the rank deficiency to the Covariance object so that it can |
| // handle it correctly. |
| // |
| // b. More general numerical rank deficiency in the Jacobian |
| // requires the computation of the so called Singular Value |
| // Decomposition (SVD) of J'J. We do not know how to do this for |
| // large sparse matrices efficiently. For small and moderate sized |
| // problems this is done using dense linear algebra. |
| // |
| // Gauge Invariance |
| // ---------------- |
| // In structure from motion (3D reconstruction) problems, the |
| // reconstruction is ambiguous up to a similarity transform. This is |
| // known as a Gauge Ambiguity. Handling Gauges correctly requires the |
| // use of SVD or custom inversion algorithms. For small problems the |
| // user can use the dense algorithm. For more details see |
| // |
| // Ken-ichi Kanatani, Daniel D. Morris: Gauges and gauge |
| // transformations for uncertainty description of geometric structure |
| // with indeterminacy. IEEE Transactions on Information Theory 47(5): |
| // 2017-2028 (2001) |
| // |
| // Example Usage |
| // ============= |
| // |
| // double x[3]; |
| // double y[2]; |
| // |
| // Problem problem; |
| // problem.AddParameterBlock(x, 3); |
| // problem.AddParameterBlock(y, 2); |
| // <Build Problem> |
| // <Solve Problem> |
| // |
| // Covariance::Options options; |
| // Covariance covariance(options); |
| // |
| // std::vector<std::pair<const double*, const double*>> covariance_blocks; |
| // covariance_blocks.push_back(make_pair(x, x)); |
| // covariance_blocks.push_back(make_pair(y, y)); |
| // covariance_blocks.push_back(make_pair(x, y)); |
| // |
| // CHECK(covariance.Compute(covariance_blocks, &problem)); |
| // |
| // double covariance_xx[3 * 3]; |
| // double covariance_yy[2 * 2]; |
| // double covariance_xy[3 * 2]; |
| // covariance.GetCovarianceBlock(x, x, covariance_xx) |
| // covariance.GetCovarianceBlock(y, y, covariance_yy) |
| // covariance.GetCovarianceBlock(x, y, covariance_xy) |
| // |
| class CERES_EXPORT Covariance { |
| public: |
| struct CERES_EXPORT Options { |
| // Sparse linear algebra library to use when a sparse matrix |
| // factorization is being used to compute the covariance matrix. |
| // |
| // Currently this only applies to SPARSE_QR. |
| SparseLinearAlgebraLibraryType sparse_linear_algebra_library_type = |
| #if !defined(CERES_NO_SUITESPARSE) |
| SUITE_SPARSE; |
| #else |
| // Eigen's QR factorization is always available. |
| EIGEN_SPARSE; |
| #endif |
| |
| // Ceres supports two different algorithms for covariance |
| // estimation, which represent different tradeoffs in speed, |
| // accuracy and reliability. |
| // |
| // 1. DENSE_SVD uses Eigen's JacobiSVD to perform the |
| // computations. It computes the singular value decomposition |
| // |
| // U * D * V' = J |
| // |
| // and then uses it to compute the pseudo inverse of J'J as |
| // |
| // pseudoinverse[J'J] = V * pseudoinverse[D^2] * V' |
| // |
| // It is an accurate but slow method and should only be used |
| // for small to moderate sized problems. It can handle |
| // full-rank as well as rank deficient Jacobians. |
| // |
| // 2. SPARSE_QR uses the sparse QR factorization algorithm |
| // to compute the decomposition |
| // |
| // Q * R = J |
| // |
| // [J'J]^-1 = [R'*R]^-1 |
| // |
| // SPARSE_QR is not capable of computing the covariance if the |
| // Jacobian is rank deficient. Depending on the value of |
| // Covariance::Options::sparse_linear_algebra_library_type, either |
| // Eigen's Sparse QR factorization algorithm will be used or |
| // SuiteSparse's high performance SuiteSparseQR algorithm will be |
| // used. |
| CovarianceAlgorithmType algorithm_type = SPARSE_QR; |
| |
| // During QR factorization, if a column with Euclidean norm less |
| // than column_pivot_threshold is encountered it is treated as |
| // zero. |
| // |
| // If column_pivot_threshold < 0, then an automatic default value |
| // of 20*(m+n)*eps*sqrt(max(diag(J’*J))) is used. Here m and n are |
| // the number of rows and columns of the Jacobian (J) |
| // respectively. |
| // |
| // This is an advanced option meant for users who know enough |
| // about their Jacobian matrices that they can determine a value |
| // better than the default. |
| double column_pivot_threshold = -1; |
| |
| // If the Jacobian matrix is near singular, then inverting J'J |
| // will result in unreliable results, e.g, if |
| // |
| // J = [1.0 1.0 ] |
| // [1.0 1.0000001 ] |
| // |
| // which is essentially a rank deficient matrix, we have |
| // |
| // inv(J'J) = [ 2.0471e+14 -2.0471e+14] |
| // [-2.0471e+14 2.0471e+14] |
| // |
| // This is not a useful result. Therefore, by default |
| // Covariance::Compute will return false if a rank deficient |
| // Jacobian is encountered. How rank deficiency is detected |
| // depends on the algorithm being used. |
| // |
| // 1. DENSE_SVD |
| // |
| // min_sigma / max_sigma < sqrt(min_reciprocal_condition_number) |
| // |
| // where min_sigma and max_sigma are the minimum and maxiumum |
| // singular values of J respectively. |
| // |
| // 2. SPARSE_QR |
| // |
| // rank(J) < num_col(J) |
| // |
| // Here rank(J) is the estimate of the rank of J returned by the |
| // sparse QR factorization algorithm. It is a fairly reliable |
| // indication of rank deficiency. |
| // |
| double min_reciprocal_condition_number = 1e-14; |
| |
| // When using DENSE_SVD, the user has more control in dealing with |
| // singular and near singular covariance matrices. |
| // |
| // As mentioned above, when the covariance matrix is near |
| // singular, instead of computing the inverse of J'J, the |
| // Moore-Penrose pseudoinverse of J'J should be computed. |
| // |
| // If J'J has the eigen decomposition (lambda_i, e_i), where |
| // lambda_i is the i^th eigenvalue and e_i is the corresponding |
| // eigenvector, then the inverse of J'J is |
| // |
| // inverse[J'J] = sum_i e_i e_i' / lambda_i |
| // |
| // and computing the pseudo inverse involves dropping terms from |
| // this sum that correspond to small eigenvalues. |
| // |
| // How terms are dropped is controlled by |
| // min_reciprocal_condition_number and null_space_rank. |
| // |
| // If null_space_rank is non-negative, then the smallest |
| // null_space_rank eigenvalue/eigenvectors are dropped |
| // irrespective of the magnitude of lambda_i. If the ratio of the |
| // smallest non-zero eigenvalue to the largest eigenvalue in the |
| // truncated matrix is still below |
| // min_reciprocal_condition_number, then the Covariance::Compute() |
| // will fail and return false. |
| // |
| // Setting null_space_rank = -1 drops all terms for which |
| // |
| // lambda_i / lambda_max < min_reciprocal_condition_number. |
| // |
| // This option has no effect on the SUITE_SPARSE_QR and |
| // EIGEN_SPARSE_QR algorithms. |
| int null_space_rank = 0; |
| |
| int num_threads = 1; |
| |
| // Even though the residual blocks in the problem may contain loss |
| // functions, setting apply_loss_function to false will turn off |
| // the application of the loss function to the output of the cost |
| // function and in turn its effect on the covariance. |
| // |
| // TODO(sameergaarwal): Expand this based on Jim's experiments. |
| bool apply_loss_function = true; |
| }; |
| |
| explicit Covariance(const Options& options); |
| ~Covariance(); |
| |
| // Compute a part of the covariance matrix. |
| // |
| // The vector covariance_blocks, indexes into the covariance matrix |
| // block-wise using pairs of parameter blocks. This allows the |
| // covariance estimation algorithm to only compute and store these |
| // blocks. |
| // |
| // Since the covariance matrix is symmetric, if the user passes |
| // (block1, block2), then GetCovarianceBlock can be called with |
| // block1, block2 as well as block2, block1. |
| // |
| // covariance_blocks cannot contain duplicates. Bad things will |
| // happen if they do. |
| // |
| // Note that the list of covariance_blocks is only used to determine |
| // what parts of the covariance matrix are computed. The full |
| // Jacobian is used to do the computation, i.e. they do not have an |
| // impact on what part of the Jacobian is used for computation. |
| // |
| // The return value indicates the success or failure of the |
| // covariance computation. Please see the documentation for |
| // Covariance::Options for more on the conditions under which this |
| // function returns false. |
| bool Compute(const std::vector<std::pair<const double*, const double*>>& |
| covariance_blocks, |
| Problem* problem); |
| |
| // Compute a part of the covariance matrix. |
| // |
| // The vector parameter_blocks contains the parameter blocks that |
| // are used for computing the covariance matrix. From this vector |
| // all covariance pairs are generated. This allows the covariance |
| // estimation algorithm to only compute and store these blocks. |
| // |
| // parameter_blocks cannot contain duplicates. Bad things will |
| // happen if they do. |
| // |
| // Note that the list of covariance_blocks is only used to determine |
| // what parts of the covariance matrix are computed. The full |
| // Jacobian is used to do the computation, i.e. they do not have an |
| // impact on what part of the Jacobian is used for computation. |
| // |
| // The return value indicates the success or failure of the |
| // covariance computation. Please see the documentation for |
| // Covariance::Options for more on the conditions under which this |
| // function returns false. |
| bool Compute(const std::vector<const double*>& parameter_blocks, |
| Problem* problem); |
| |
| // Return the block of the cross-covariance matrix corresponding to |
| // parameter_block1 and parameter_block2. |
| // |
| // Compute must be called before the first call to |
| // GetCovarianceBlock and the pair <parameter_block1, |
| // parameter_block2> OR the pair <parameter_block2, |
| // parameter_block1> must have been present in the vector |
| // covariance_blocks when Compute was called. Otherwise |
| // GetCovarianceBlock will return false. |
| // |
| // covariance_block must point to a memory location that can store a |
| // parameter_block1_size x parameter_block2_size matrix. The |
| // returned covariance will be a row-major matrix. |
| bool GetCovarianceBlock(const double* parameter_block1, |
| const double* parameter_block2, |
| double* covariance_block) const; |
| |
| // Return the block of the cross-covariance matrix corresponding to |
| // parameter_block1 and parameter_block2. |
| // Returns cross-covariance in the tangent space if a local |
| // parameterization is associated with either parameter block; |
| // else returns cross-covariance in the ambient space. |
| // |
| // Compute must be called before the first call to |
| // GetCovarianceBlock and the pair <parameter_block1, |
| // parameter_block2> OR the pair <parameter_block2, |
| // parameter_block1> must have been present in the vector |
| // covariance_blocks when Compute was called. Otherwise |
| // GetCovarianceBlock will return false. |
| // |
| // covariance_block must point to a memory location that can store a |
| // parameter_block1_local_size x parameter_block2_local_size matrix. The |
| // returned covariance will be a row-major matrix. |
| bool GetCovarianceBlockInTangentSpace(const double* parameter_block1, |
| const double* parameter_block2, |
| double* covariance_block) const; |
| |
| // Return the covariance matrix corresponding to all parameter_blocks. |
| // |
| // Compute must be called before calling GetCovarianceMatrix and all |
| // parameter_blocks must have been present in the vector |
| // parameter_blocks when Compute was called. Otherwise |
| // GetCovarianceMatrix returns false. |
| // |
| // covariance_matrix must point to a memory location that can store |
| // the size of the covariance matrix. The covariance matrix will be |
| // a square matrix whose row and column count is equal to the sum of |
| // the sizes of the individual parameter blocks. The covariance |
| // matrix will be a row-major matrix. |
| bool GetCovarianceMatrix(const std::vector<const double*>& parameter_blocks, |
| double* covariance_matrix) const; |
| |
| // Return the covariance matrix corresponding to parameter_blocks |
| // in the tangent space if a local parameterization is associated |
| // with one of the parameter blocks else returns the covariance |
| // matrix in the ambient space. |
| // |
| // Compute must be called before calling GetCovarianceMatrix and all |
| // parameter_blocks must have been present in the vector |
| // parameters_blocks when Compute was called. Otherwise |
| // GetCovarianceMatrix returns false. |
| // |
| // covariance_matrix must point to a memory location that can store |
| // the size of the covariance matrix. The covariance matrix will be |
| // a square matrix whose row and column count is equal to the sum of |
| // the sizes of the tangent spaces of the individual parameter |
| // blocks. The covariance matrix will be a row-major matrix. |
| bool GetCovarianceMatrixInTangentSpace( |
| const std::vector<const double*>& parameter_blocks, |
| double* covariance_matrix) const; |
| |
| private: |
| std::unique_ptr<internal::CovarianceImpl> impl_; |
| }; |
| |
| } // namespace ceres |
| |
| #include "ceres/internal/reenable_warnings.h" |
| |
| #endif // CERES_PUBLIC_COVARIANCE_H_ |