Sparse covariance estimation.
Add a Covariance object to the API.
Given a Problem object and a set of parameter block pairs the
Covariance object computes a sparse covariance matrix corresponding
to those block pairs and provides random access to them.
Constant parameter blocks and parameter blocks with local parameterizations
are correctly handled.
Sparse and dense implementations are provided. With the dense implementation
rank deficient Jacobians can also be handled.
Parts of the code are threaded using OpenMP if available.
Change-Id: I5b49583b3d79579df3e0f334c22567acb23ed4ad
diff --git a/internal/ceres/covariance_test.cc b/internal/ceres/covariance_test.cc
new file mode 100644
index 0000000..b74efda
--- /dev/null
+++ b/internal/ceres/covariance_test.cc
@@ -0,0 +1,647 @@
+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 2013 Google Inc. All rights reserved.
+// http://code.google.com/p/ceres-solver/
+//
+// 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
+// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+// 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)
+
+#include "ceres/covariance.h"
+
+#include <algorithm>
+#include <cmath>
+#include "ceres/compressed_row_sparse_matrix.h"
+#include "ceres/cost_function.h"
+#include "ceres/covariance_impl.h"
+#include "ceres/local_parameterization.h"
+#include "ceres/map_util.h"
+#include "ceres/problem_impl.h"
+#include "gtest/gtest.h"
+
+namespace ceres {
+namespace internal {
+
+TEST(CovarianceImpl, ComputeCovarianceSparsity) {
+ double parameters[10];
+
+ double* block1 = parameters;
+ double* block2 = block1 + 1;
+ double* block3 = block2 + 2;
+ double* block4 = block3 + 3;
+
+ ProblemImpl problem;
+
+ // Add in random order
+ problem.AddParameterBlock(block1, 1);
+ problem.AddParameterBlock(block4, 4);
+ problem.AddParameterBlock(block3, 3);
+ problem.AddParameterBlock(block2, 2);
+
+ // Sparsity pattern
+ //
+ // x 0 0 0 0 0 x x x x
+ // 0 x x x x x 0 0 0 0
+ // 0 x x x x x 0 0 0 0
+ // 0 0 0 x x x 0 0 0 0
+ // 0 0 0 x x x 0 0 0 0
+ // 0 0 0 x x x 0 0 0 0
+ // 0 0 0 0 0 0 x x x x
+ // 0 0 0 0 0 0 x x x x
+ // 0 0 0 0 0 0 x x x x
+ // 0 0 0 0 0 0 x x x x
+
+ int expected_rows[] = {0, 5, 10, 15, 18, 21, 24, 28, 32, 36, 40};
+ int expected_cols[] = {0, 6, 7, 8, 9,
+ 1, 2, 3, 4, 5,
+ 1, 2, 3, 4, 5,
+ 3, 4, 5,
+ 3, 4, 5,
+ 3, 4, 5,
+ 6, 7, 8, 9,
+ 6, 7, 8, 9,
+ 6, 7, 8, 9,
+ 6, 7, 8, 9};
+
+
+ vector<pair<const double*, const double*> > covariance_blocks;
+ covariance_blocks.push_back(make_pair(block1, block1));
+ covariance_blocks.push_back(make_pair(block4, block4));
+ covariance_blocks.push_back(make_pair(block2, block2));
+ covariance_blocks.push_back(make_pair(block3, block3));
+ covariance_blocks.push_back(make_pair(block2, block3));
+ covariance_blocks.push_back(make_pair(block4, block1)); // reversed
+
+ Covariance::Options options;
+ CovarianceImpl covariance_impl(options);
+ EXPECT_TRUE(covariance_impl
+ .ComputeCovarianceSparsity(covariance_blocks, &problem));
+
+ const CompressedRowSparseMatrix* crsm = covariance_impl.covariance_matrix();
+
+ EXPECT_EQ(crsm->num_rows(), 10);
+ EXPECT_EQ(crsm->num_cols(), 10);
+ EXPECT_EQ(crsm->num_nonzeros(), 40);
+
+ const int* rows = crsm->rows();
+ for (int r = 0; r < crsm->num_rows() + 1; ++r) {
+ EXPECT_EQ(rows[r], expected_rows[r])
+ << r << " "
+ << rows[r] << " "
+ << expected_rows[r];
+ }
+
+ const int* cols = crsm->cols();
+ for (int c = 0; c < crsm->num_nonzeros(); ++c) {
+ EXPECT_EQ(cols[c], expected_cols[c])
+ << c << " "
+ << cols[c] << " "
+ << expected_cols[c];
+ }
+}
+
+
+class UnaryCostFunction: public CostFunction {
+ public:
+ UnaryCostFunction(const int num_residuals,
+ const int16 parameter_block_size,
+ const double* jacobian)
+ : jacobian_(jacobian, jacobian + num_residuals * parameter_block_size) {
+ set_num_residuals(num_residuals);
+ mutable_parameter_block_sizes()->push_back(parameter_block_size);
+ }
+
+ virtual bool Evaluate(double const* const* parameters,
+ double* residuals,
+ double** jacobians) const {
+ for (int i = 0; i < num_residuals(); ++i) {
+ residuals[i] = 1;
+ }
+
+ if (jacobians == NULL) {
+ return true;
+ }
+
+ if (jacobians[0] != NULL) {
+ copy(jacobian_.begin(), jacobian_.end(), jacobians[0]);
+ }
+
+ return true;
+ }
+
+ private:
+ vector<double> jacobian_;
+};
+
+
+class BinaryCostFunction: public CostFunction {
+ public:
+ BinaryCostFunction(const int num_residuals,
+ const int16 parameter_block1_size,
+ const int16 parameter_block2_size,
+ const double* jacobian1,
+ const double* jacobian2)
+ : jacobian1_(jacobian1,
+ jacobian1 + num_residuals * parameter_block1_size),
+ jacobian2_(jacobian2,
+ jacobian2 + num_residuals * parameter_block2_size) {
+ set_num_residuals(num_residuals);
+ mutable_parameter_block_sizes()->push_back(parameter_block1_size);
+ mutable_parameter_block_sizes()->push_back(parameter_block2_size);
+ }
+
+ virtual bool Evaluate(double const* const* parameters,
+ double* residuals,
+ double** jacobians) const {
+ for (int i = 0; i < num_residuals(); ++i) {
+ residuals[i] = 2;
+ }
+
+ if (jacobians == NULL) {
+ return true;
+ }
+
+ if (jacobians[0] != NULL) {
+ copy(jacobian1_.begin(), jacobian1_.end(), jacobians[0]);
+ }
+
+ if (jacobians[1] != NULL) {
+ copy(jacobian2_.begin(), jacobian2_.end(), jacobians[1]);
+ }
+
+ return true;
+ }
+
+ private:
+ vector<double> jacobian1_;
+ vector<double> jacobian2_;
+};
+
+// x_plus_delta = delta * x;
+class PolynomialParameterization : public LocalParameterization {
+ public:
+ virtual ~PolynomialParameterization() {}
+
+ virtual bool Plus(const double* x,
+ const double* delta,
+ double* x_plus_delta) const {
+ x_plus_delta[0] = delta[0] * x[0];
+ x_plus_delta[1] = delta[0] * x[1];
+ return true;
+ }
+
+ virtual bool ComputeJacobian(const double* x, double* jacobian) const {
+ jacobian[0] = x[0];
+ jacobian[1] = x[1];
+ return true;
+ }
+
+ virtual int GlobalSize() const { return 2; }
+ virtual int LocalSize() const { return 1; }
+};
+
+class CovarianceTest : public ::testing::Test {
+ protected:
+ virtual void SetUp() {
+ double* x = parameters_;
+ double* y = x + 2;
+ double* z = y + 3;
+
+ x[0] = 1;
+ x[1] = 1;
+ y[0] = 2;
+ y[1] = 2;
+ y[2] = 2;
+ z[0] = 3;
+
+ {
+ double jacobian[] = { 1.0, 0.0, 0.0, 1.0};
+ problem_.AddResidualBlock(new UnaryCostFunction(2, 2, jacobian), NULL, x);
+ }
+
+ {
+ double jacobian[] = { 2.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 2.0 };
+ problem_.AddResidualBlock(new UnaryCostFunction(3, 3, jacobian), NULL, y);
+ }
+
+ {
+ double jacobian = 5.0;
+ problem_.AddResidualBlock(new UnaryCostFunction(1, 1, &jacobian), NULL, z);
+ }
+
+ {
+ double jacobian1[] = { 1.0, 2.0, 3.0 };
+ double jacobian2[] = { -5.0, -6.0 };
+ problem_.AddResidualBlock(
+ new BinaryCostFunction(1, 3, 2, jacobian1, jacobian2),
+ NULL,
+ y,
+ x);
+ }
+
+ {
+ double jacobian1[] = {2.0 };
+ double jacobian2[] = { 3.0, -2.0 };
+ problem_.AddResidualBlock(
+ new BinaryCostFunction(1, 1, 2, jacobian1, jacobian2),
+ NULL,
+ z,
+ x);
+ }
+
+ all_covariance_blocks_.push_back(make_pair(x, x));
+ all_covariance_blocks_.push_back(make_pair(y, y));
+ all_covariance_blocks_.push_back(make_pair(z, z));
+ all_covariance_blocks_.push_back(make_pair(x, y));
+ all_covariance_blocks_.push_back(make_pair(x, z));
+ all_covariance_blocks_.push_back(make_pair(y, z));
+
+ column_bounds_[x] = make_pair(0, 2);
+ column_bounds_[y] = make_pair(2, 5);
+ column_bounds_[z] = make_pair(5, 6);
+ }
+
+ void ComputeAndCompareCovarianceBlocks(const Covariance::Options& options,
+ const double* expected_covariance) {
+ // Generate all possible combination of block pairs and check if the
+ // covariance computation is correct.
+ for (int i = 1; i <= 64; ++i) {
+ vector<pair<const double*, const double*> > covariance_blocks;
+ if (i & 1) {
+ covariance_blocks.push_back(all_covariance_blocks_[0]);
+ }
+
+ if (i & 2) {
+ covariance_blocks.push_back(all_covariance_blocks_[1]);
+ }
+
+ if (i & 4) {
+ covariance_blocks.push_back(all_covariance_blocks_[2]);
+ }
+
+ if (i & 8) {
+ covariance_blocks.push_back(all_covariance_blocks_[3]);
+ }
+
+ if (i & 16) {
+ covariance_blocks.push_back(all_covariance_blocks_[4]);
+ }
+
+ if (i & 32) {
+ covariance_blocks.push_back(all_covariance_blocks_[5]);
+ }
+
+ Covariance covariance(options);
+ EXPECT_TRUE(covariance.Compute(covariance_blocks, &problem_));
+
+ for (int i = 0; i < covariance_blocks.size(); ++i) {
+ const double* block1 = covariance_blocks[i].first;
+ const double* block2 = covariance_blocks[i].second;
+ // block1, block2
+ GetCovarianceBlockAndCompare(block1, block2, covariance, expected_covariance);
+ // block2, block1
+ GetCovarianceBlockAndCompare(block2, block1, covariance, expected_covariance);
+ }
+ }
+ }
+
+ void GetCovarianceBlockAndCompare(const double* block1,
+ const double* block2,
+ const Covariance& covariance,
+ const double* expected_covariance) {
+ const int row_begin = FindOrDie(column_bounds_, block1).first;
+ const int row_end = FindOrDie(column_bounds_, block1).second;
+ const int col_begin = FindOrDie(column_bounds_, block2).first;
+ const int col_end = FindOrDie(column_bounds_, block2).second;
+
+ Matrix actual(row_end - row_begin, col_end - col_begin);
+ EXPECT_TRUE(covariance.GetCovarianceBlock(block1,
+ block2,
+ actual.data()));
+
+ ConstMatrixRef expected(expected_covariance, 6, 6);
+ double diff_norm = (expected.block(row_begin,
+ col_begin,
+ row_end - row_begin,
+ col_end - col_begin) - actual).norm();
+ diff_norm /= (row_end - row_begin) * (col_end - col_begin);
+
+ const double kTolerance = 1e-5;
+ EXPECT_NEAR(diff_norm, 0.0, kTolerance)
+ << "rows: " << row_begin << " " << row_end << " "
+ << "cols: " << col_begin << " " << col_end << " "
+ << "\n\n expected: \n " << expected.block(row_begin,
+ col_begin,
+ row_end - row_begin,
+ col_end - col_begin)
+ << "\n\n actual: \n " << actual
+ << "\n\n full expected: \n" << expected;
+ }
+
+ double parameters_[10];
+ Problem problem_;
+ vector<pair<const double*, const double*> > all_covariance_blocks_;
+ map<const double*, pair<int, int> > column_bounds_;
+};
+
+
+TEST_F(CovarianceTest, NormalBehavior) {
+ // J
+ //
+ // 1 0 0 0 0 0
+ // 0 1 0 0 0 0
+ // 0 0 2 0 0 0
+ // 0 0 0 2 0 0
+ // 0 0 0 0 2 0
+ // 0 0 0 0 0 5
+ // -5 -6 1 2 3 0
+ // 3 -2 0 0 0 2
+
+ // J'J
+ //
+ // 35 24 -5 -10 -15 6
+ // 24 41 -6 -12 -18 -4
+ // -5 -6 5 2 3 0
+ // -10 -12 2 8 6 0
+ // -15 -18 3 6 13 0
+ // 6 -4 0 0 0 29
+
+ // inv(J'J) computed using octave.
+ double expected_covariance[] = {
+ 7.0747e-02, -8.4923e-03, 1.6821e-02, 3.3643e-02, 5.0464e-02, -1.5809e-02, // NOLINT
+ -8.4923e-03, 8.1352e-02, 2.4758e-02, 4.9517e-02, 7.4275e-02, 1.2978e-02, // NOLINT
+ 1.6821e-02, 2.4758e-02, 2.4904e-01, -1.9271e-03, -2.8906e-03, -6.5325e-05, // NOLINT
+ 3.3643e-02, 4.9517e-02, -1.9271e-03, 2.4615e-01, -5.7813e-03, -1.3065e-04, // NOLINT
+ 5.0464e-02, 7.4275e-02, -2.8906e-03, -5.7813e-03, 2.4133e-01, -1.9598e-04, // NOLINT
+ -1.5809e-02, 1.2978e-02, -6.5325e-05, -1.3065e-04, -1.9598e-04, 3.9544e-02, // NOLINT
+ };
+
+ Covariance::Options options;
+ options.use_dense_linear_algebra = false;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+
+ options.use_dense_linear_algebra = true;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+}
+
+
+TEST_F(CovarianceTest, ConstantParameterBlock) {
+ problem_.SetParameterBlockConstant(parameters_);
+
+ // J
+ //
+ // 0 0 0 0 0 0
+ // 0 0 0 0 0 0
+ // 0 0 2 0 0 0
+ // 0 0 0 2 0 0
+ // 0 0 0 0 2 0
+ // 0 0 0 0 0 5
+ // 0 0 1 2 3 0
+ // 0 0 0 0 0 2
+
+ // J'J
+ //
+ // 0 0 0 0 0 0
+ // 0 0 0 0 0 0
+ // 0 0 5 2 3 0
+ // 0 0 2 8 6 0
+ // 0 0 3 6 13 0
+ // 0 0 0 0 0 29
+
+ // pinv(J'J) computed using octave.
+ double expected_covariance[] = {
+ 0, 0, 0, 0, 0, 0, // NOLINT
+ 0, 0, 0, 0, 0, 0, // NOLINT
+ 0, 0, 0.23611, -0.02778, -0.04167, -0.00000, // NOLINT
+ 0, 0, -0.02778, 0.19444, -0.08333, -0.00000, // NOLINT
+ 0, 0, -0.04167, -0.08333, 0.12500, -0.00000, // NOLINT
+ 0, 0, -0.00000, -0.00000, -0.00000, 0.03448 // NOLINT
+ };
+
+ Covariance::Options options;
+ options.use_dense_linear_algebra = false;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+
+ options.use_dense_linear_algebra = true;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+}
+
+TEST_F(CovarianceTest, LocalParameterization) {
+ double* x = parameters_;
+ double* y = x + 2;
+
+ problem_.SetParameterization(x, new PolynomialParameterization);
+
+ vector<int> subset;
+ subset.push_back(2);
+ problem_.SetParameterization(y, new SubsetParameterization(3, subset));
+
+ // Raw Jacobian: J
+ //
+ // 1 0 0 0 0 0
+ // 0 1 0 0 0 0
+ // 0 0 2 0 0 0
+ // 0 0 0 2 0 0
+ // 0 0 0 0 0 0
+ // 0 0 0 0 0 5
+ // -5 -6 1 2 0 0
+ // 3 -2 0 0 0 2
+
+ // Global to local jacobian: A
+ //
+ //
+ // 1 0 0 0 0
+ // 1 0 0 0 0
+ // 0 1 0 0 0
+ // 0 0 1 0 0
+ // 0 0 0 1 0
+ // 0 0 0 0 1
+
+ // A * pinv((J*A)'*(J*A)) * A'
+ // Computed using octave.
+ double expected_covariance[] = {
+ 0.01766, 0.01766, 0.02158, 0.04316, 0.00000, -0.00122,
+ 0.01766, 0.01766, 0.02158, 0.04316, 0.00000, -0.00122,
+ 0.02158, 0.02158, 0.24860, -0.00281, 0.00000, -0.00149,
+ 0.04316, 0.04316, -0.00281, 0.24439, 0.00000, -0.00298,
+ 0.00000, 0.00000, 0.00000, 0.00000, 0.00000, 0.00000,
+ -0.00122, -0.00122, -0.00149, -0.00298, 0.00000, 0.03457
+ };
+
+
+ Covariance::Options options;
+ options.use_dense_linear_algebra = false;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+
+ options.use_dense_linear_algebra = true;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+}
+
+
+TEST_F(CovarianceTest, TruncatedRank) {
+ // J
+ //
+ // 1 0 0 0 0 0
+ // 0 1 0 0 0 0
+ // 0 0 2 0 0 0
+ // 0 0 0 2 0 0
+ // 0 0 0 0 2 0
+ // 0 0 0 0 0 5
+ // -5 -6 1 2 3 0
+ // 3 -2 0 0 0 2
+
+ // J'J
+ //
+ // 35 24 -5 -10 -15 6
+ // 24 41 -6 -12 -18 -4
+ // -5 -6 5 2 3 0
+ // -10 -12 2 8 6 0
+ // -15 -18 3 6 13 0
+ // 6 -4 0 0 0 29
+
+ // 3.4142 is the smallest eigen value of J'J. The following matrix
+ // was obtained by dropping the eigenvector corresponding to this
+ // eigenvalue.
+ double expected_covariance[] = {
+ 5.4135e-02, -3.5121e-02, 1.7257e-04, 3.4514e-04, 5.1771e-04, -1.6076e-02,
+ -3.5121e-02, 3.8667e-02, -1.9288e-03, -3.8576e-03, -5.7864e-03, 1.2549e-02,
+ 1.7257e-04, -1.9288e-03, 2.3235e-01, -3.5297e-02, -5.2946e-02, -3.3329e-04,
+ 3.4514e-04, -3.8576e-03, -3.5297e-02, 1.7941e-01, -1.0589e-01, -6.6659e-04,
+ 5.1771e-04, -5.7864e-03, -5.2946e-02, -1.0589e-01, 9.1162e-02, -9.9988e-04,
+ -1.6076e-02, 1.2549e-02, -3.3329e-04, -6.6659e-04, -9.9988e-04, 3.9539e-02
+ };
+
+ Covariance::Options options;
+ options.use_dense_linear_algebra = true;
+ options.null_space_rank = 1;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+
+ options.use_dense_linear_algebra = true;
+ options.null_space_rank = 0;
+ options.min_singular_value_threshold = sqrt(3.5);
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+}
+
+class RankDeficientCovarianceTest : public CovarianceTest {
+ protected:
+ virtual void SetUp() {
+ double* x = parameters_;
+ double* y = x + 2;
+ double* z = y + 3;
+
+ {
+ double jacobian[] = { 1.0, 0.0, 0.0, 1.0};
+ problem_.AddResidualBlock(new UnaryCostFunction(2, 2, jacobian), NULL, x);
+ }
+
+ {
+ double jacobian[] = { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 };
+ problem_.AddResidualBlock(new UnaryCostFunction(3, 3, jacobian), NULL, y);
+ }
+
+ {
+ double jacobian = 5.0;
+ problem_.AddResidualBlock(new UnaryCostFunction(1, 1, &jacobian), NULL, z);
+ }
+
+ {
+ double jacobian1[] = { 0.0, 0.0, 0.0 };
+ double jacobian2[] = { -5.0, -6.0 };
+ problem_.AddResidualBlock(
+ new BinaryCostFunction(1, 3, 2, jacobian1, jacobian2),
+ NULL,
+ y,
+ x);
+ }
+
+ {
+ double jacobian1[] = {2.0 };
+ double jacobian2[] = { 3.0, -2.0 };
+ problem_.AddResidualBlock(
+ new BinaryCostFunction(1, 1, 2, jacobian1, jacobian2),
+ NULL,
+ z,
+ x);
+ }
+
+ all_covariance_blocks_.push_back(make_pair(x, x));
+ all_covariance_blocks_.push_back(make_pair(y, y));
+ all_covariance_blocks_.push_back(make_pair(z, z));
+ all_covariance_blocks_.push_back(make_pair(x, y));
+ all_covariance_blocks_.push_back(make_pair(x, z));
+ all_covariance_blocks_.push_back(make_pair(y, z));
+
+ column_bounds_[x] = make_pair(0, 2);
+ column_bounds_[y] = make_pair(2, 5);
+ column_bounds_[z] = make_pair(5, 6);
+ }
+};
+
+TEST_F(RankDeficientCovarianceTest, MinSingularValueTolerance) {
+ // J
+ //
+ // 1 0 0 0 0 0
+ // 0 1 0 0 0 0
+ // 0 0 0 0 0 0
+ // 0 0 0 0 0 0
+ // 0 0 0 0 0 0
+ // 0 0 0 0 0 5
+ // -5 -6 0 0 0 0
+ // 3 -2 0 0 0 2
+
+ // J'J
+ //
+ // 35 24 0 0 0 6
+ // 24 41 0 0 0 -4
+ // 0 0 0 0 0 0
+ // 0 0 0 0 0 0
+ // 0 0 0 0 0 0
+ // 6 -4 0 0 0 29
+
+ // pinv(J'J) computed using octave.
+ double expected_covariance[] = {
+ 0.053998, -0.033145, 0.000000, 0.000000, 0.000000, -0.015744,
+ -0.033145, 0.045067, 0.000000, 0.000000, 0.000000, 0.013074,
+ 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
+ 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
+ 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,
+ -0.015744, 0.013074, 0.000000, 0.000000, 0.000000, 0.039543
+ };
+
+ Covariance::Options options;
+ options.use_dense_linear_algebra = true;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+
+ options.null_space_rank = 1;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+
+ options.null_space_rank = 2;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+
+ options.null_space_rank = 3;
+ ComputeAndCompareCovarianceBlocks(options, expected_covariance);
+}
+
+} // namespace internal
+} // namespace ceres
