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<li class="toctree-l1 current"><a class="current reference internal" href="#">Covariance Estimation</a><ul>
<li class="toctree-l2"><a class="reference internal" href="#introduction">Introduction</a></li>
<li class="toctree-l2"><a class="reference internal" href="#gauge-invariance">Gauge Invariance</a></li>
<li class="toctree-l2"><a class="reference internal" href="#covariance"><code class="docutils literal"><span class="pre">Covariance</span></code></a></li>
<li class="toctree-l2"><a class="reference internal" href="#rank-of-the-jacobian">Rank of the Jacobian</a></li>
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<div class="section" id="covariance-estimation">
<span id="chapter-nnls-covariance"></span><h1>Covariance Estimation<a class="headerlink" href="#covariance-estimation" title="Permalink to this headline">¶</a></h1>
<div class="section" id="introduction">
<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
<p>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.</p>
<p>Let us consider the non-linear regression problem</p>
<div class="math">
\[y = f(x) + N(0, I)\]</div>
<p>i.e., the observation <span class="math">\(y\)</span> is a random non-linear function of the
independent variable <span class="math">\(x\)</span> with mean <span class="math">\(f(x)\)</span> and identity
covariance. Then the maximum likelihood estimate of <span class="math">\(x\)</span> given
observations <span class="math">\(y\)</span> is the solution to the non-linear least squares
problem:</p>
<div class="math">
\[x^* = \arg \min_x \|f(x)\|^2\]</div>
<p>And the covariance of <span class="math">\(x^*\)</span> is given by</p>
<div class="math">
\[C(x^*) = \left(J'(x^*)J(x^*)\right)^{-1}\]</div>
<p>Here <span class="math">\(J(x^*)\)</span> is the Jacobian of <span class="math">\(f\)</span> at <span class="math">\(x^*\)</span>. The
above formula assumes that <span class="math">\(J(x^*)\)</span> has full column rank.</p>
<p>If <span class="math">\(J(x^*)\)</span> is rank deficient, then the covariance matrix <span class="math">\(C(x^*)\)</span>
is also rank deficient and is given by the Moore-Penrose pseudo inverse.</p>
<div class="math">
\[C(x^*) = \left(J'(x^*)J(x^*)\right)^{\dagger}\]</div>
<p>Note that in the above, we assumed that the covariance matrix for
<span class="math">\(y\)</span> was identity. This is an important assumption. If this is
not the case and we have</p>
<div class="math">
\[y = f(x) + N(0, S)\]</div>
<p>Where <span class="math">\(S\)</span> is a positive semi-definite matrix denoting the
covariance of <span class="math">\(y\)</span>, then the maximum likelihood problem to be
solved is</p>
<div class="math">
\[x^* = \arg \min_x f'(x) S^{-1} f(x)\]</div>
<p>and the corresponding covariance estimate of <span class="math">\(x^*\)</span> is given by</p>
<div class="math">
\[C(x^*) = \left(J'(x^*) S^{-1} J(x^*)\right)^{-1}\]</div>
<p>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 <span class="math">\(S^{-1/2} f(x)\)</span> instead of just <span class="math">\(f(x)\)</span>,
where <span class="math">\(S^{-1/2}\)</span> is the inverse square root of the covariance
matrix <span class="math">\(S\)</span>.</p>
</div>
<div class="section" id="gauge-invariance">
<h2>Gauge Invariance<a class="headerlink" href="#gauge-invariance" title="Permalink to this headline">¶</a></h2>
<p>In structure from motion (3D reconstruction) problems, the
reconstruction is ambiguous upto a similarity transform. This is
known as a <em>Gauge Ambiguity</em>. 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 the work of
Kanatani & Morris <a class="reference internal" href="bibliography.html#kanatanimorris" id="id1">[KanataniMorris]</a>.</p>
</div>
<div class="section" id="covariance">
<h2><a class="reference internal" href="#_CPPv2N5ceres10CovarianceE" title="ceres::Covariance"><code class="xref cpp cpp-class docutils literal"><span class="pre">Covariance</span></code></a><a class="headerlink" href="#covariance" title="Permalink to this headline">¶</a></h2>
<p><a class="reference internal" href="#_CPPv2N5ceres10CovarianceE" title="ceres::Covariance"><code class="xref cpp cpp-class docutils literal"><span class="pre">Covariance</span></code></a> 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 cost functions compute
residuals such that their covariance is identity.</p>
<p>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. <a class="reference internal" href="#_CPPv2N5ceres10CovarianceE" title="ceres::Covariance"><code class="xref cpp cpp-class docutils literal"><span class="pre">Covariance</span></code></a>
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.</p>
</div>
<div class="section" id="rank-of-the-jacobian">
<h2>Rank of the Jacobian<a class="headerlink" href="#rank-of-the-jacobian" title="Permalink to this headline">¶</a></h2>
<p>As we noted above, if the Jacobian is rank deficient, then the inverse
of <span class="math">\(J'J\)</span> is not defined and instead a pseudo inverse needs to be
computed.</p>
<p>The rank deficiency in <span class="math">\(J\)</span> can be <em>structural</em> – columns
which are always known to be zero or <em>numerical</em> – depending on the
exact values in the Jacobian.</p>
<p>Structural rank deficiency occurs when the problem contains parameter
blocks that are constant. This class correctly handles structural rank
deficiency like that.</p>
<p>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.</p>
<blockquote>
<div><ol class="loweralpha simple">
<li>The rank deficiency arises from overparameterization. e.g., a
four dimensional quaternion used to parameterize <span class="math">\(SO(3)\)</span>,
which is a three dimensional manifold. In cases like this, the
user should use an appropriate
<a class="reference internal" href="nnls_modeling.html#_CPPv2N5ceres21LocalParameterizationE" title="ceres::LocalParameterization"><code class="xref cpp cpp-class docutils literal"><span class="pre">LocalParameterization</span></code></a>. Not only will this lead to better
numerical behaviour of the Solver, it will also expose the rank
deficiency to the <a class="reference internal" href="#_CPPv2N5ceres10CovarianceE" title="ceres::Covariance"><code class="xref cpp cpp-class docutils literal"><span class="pre">Covariance</span></code></a> object so that it can
handle it correctly.</li>
<li>More general numerical rank deficiency in the Jacobian requires
the computation of the so called Singular Value Decomposition
(SVD) of <span class="math">\(J'J\)</span>. 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.</li>
</ol>
</div></blockquote>
<p><a class="reference internal" href="#_CPPv2N5ceres10Covariance7OptionsE" title="ceres::Covariance::Options"><code class="xref cpp cpp-class docutils literal"><span class="pre">Covariance::Options</span></code></a></p>
<dl class="class">
<dt id="_CPPv2N5ceres10Covariance7OptionsE">
<span id="ceres::Covariance::Options"></span><em class="property">class </em><code class="descclassname">Covariance::</code><code class="descname">Options</code><a class="headerlink" href="#_CPPv2N5ceres10Covariance7OptionsE" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="member">
<dt id="_CPPv2N10Covariance7Options11num_threadsE">
<span id="ceres::Covariance::Options::num_threads__i"></span>int <code class="descclassname">Covariance::Options::</code><code class="descname">num_threads</code><a class="headerlink" href="#_CPPv2N10Covariance7Options11num_threadsE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">1</span></code></p>
<p>Number of threads to be used for evaluating the Jacobian and
estimation of covariance.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N10Covariance7Options34sparse_linear_algebra_library_typeE">
<span id="ceres::Covariance::Options::sparse_linear_algebra_library_type__SparseLinearAlgebraLibraryType"></span>SparseLinearAlgebraLibraryType <code class="descclassname">Covariance::Options::</code><code class="descname">sparse_linear_algebra_library_type</code><a class="headerlink" href="#_CPPv2N10Covariance7Options34sparse_linear_algebra_library_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">SUITE_SPARSE</span></code> Ceres Solver is built with support for
<a class="reference external" href="http://faculty.cse.tamu.edu/davis/suitesparse.html">SuiteSparse</a>
and <code class="docutils literal"><span class="pre">EIGEN_SPARSE</span></code> otherwise. Note that <code class="docutils literal"><span class="pre">EIGEN_SPARSE</span></code> is
always available.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N10Covariance7Options14algorithm_typeE">
<span id="ceres::Covariance::Options::algorithm_type__CovarianceAlgorithmType"></span>CovarianceAlgorithmType <code class="descclassname">Covariance::Options::</code><code class="descname">algorithm_type</code><a class="headerlink" href="#_CPPv2N10Covariance7Options14algorithm_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">SPARSE_QR</span></code></p>
<p>Ceres supports two different algorithms for covariance estimation,
which represent different tradeoffs in speed, accuracy and
reliability.</p>
<ol class="arabic">
<li><p class="first"><code class="docutils literal"><span class="pre">SPARSE_QR</span></code> uses the sparse QR factorization algorithm to
compute the decomposition</p>
<blockquote>
<div><div class="math">
\[\begin{split}QR &= J\\
\left(J^\top J\right)^{-1} &= \left(R^\top R\right)^{-1}\end{split}\]</div>
</div></blockquote>
<p>The speed of this algorithm depends on the sparse linear algebra
library being used. <code class="docutils literal"><span class="pre">Eigen</span></code>‘s sparse QR factorization is a
moderately fast algorithm suitable for small to medium sized
matrices. For best performance we recommend using
<code class="docutils literal"><span class="pre">SuiteSparseQR</span></code> which is enabled by setting
<code class="xref cpp cpp-member docutils literal"><span class="pre">Covaraince::Options::sparse_linear_algebra_library_type</span></code>
to <code class="docutils literal"><span class="pre">SUITE_SPARSE</span></code>.</p>
<p>Neither <code class="docutils literal"><span class="pre">SPARSE_QR</span></code> cannot compute the covariance if the
Jacobian is rank deficient.</p>
</li>
<li><p class="first"><code class="docutils literal"><span class="pre">DENSE_SVD</span></code> uses <code class="docutils literal"><span class="pre">Eigen</span></code>‘s <code class="docutils literal"><span class="pre">JacobiSVD</span></code> to perform the
computations. It computes the singular value decomposition</p>
<div class="math">
\[U S V^\top = J\]</div>
<p>and then uses it to compute the pseudo inverse of J’J as</p>
<div class="math">
\[(J'J)^{\dagger} = V S^{\dagger} V^\top\]</div>
<p>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.</p>
</li>
</ol>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N10Covariance7Options31min_reciprocal_condition_numberE">
<span id="ceres::Covariance::Options::min_reciprocal_condition_number__i"></span>int <code class="descclassname">Covariance::Options::</code><code class="descname">min_reciprocal_condition_number</code><a class="headerlink" href="#_CPPv2N10Covariance7Options31min_reciprocal_condition_numberE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <span class="math">\(10^{-14}\)</span></p>
<p>If the Jacobian matrix is near singular, then inverting <span class="math">\(J'J\)</span>
will result in unreliable results, e.g, if</p>
<div class="math">
\[\begin{split}J = \begin{bmatrix}
1.0& 1.0 \\
1.0& 1.0000001
\end{bmatrix}\end{split}\]</div>
<p>which is essentially a rank deficient matrix, we have</p>
<div class="math">
\[\begin{split}(J'J)^{-1} = \begin{bmatrix}
2.0471e+14& -2.0471e+14 \\
-2.0471e+14 2.0471e+14
\end{bmatrix}\end{split}\]</div>
<p>This is not a useful result. Therefore, by default
<a class="reference internal" href="#_CPPv2N5ceres10Covariance7ComputeERK6vectorI4pairIPKdPKdEEP7Problem" title="ceres::Covariance::Compute"><code class="xref cpp cpp-func docutils literal"><span class="pre">Covariance::Compute()</span></code></a> will return <code class="docutils literal"><span class="pre">false</span></code> if a rank
deficient Jacobian is encountered. How rank deficiency is detected
depends on the algorithm being used.</p>
<ol class="arabic">
<li><p class="first"><code class="docutils literal"><span class="pre">DENSE_SVD</span></code></p>
<div class="math">
\[\frac{\sigma_{\text{min}}}{\sigma_{\text{max}}} < \sqrt{\text{min_reciprocal_condition_number}}\]</div>
<p>where <span class="math">\(\sigma_{\text{min}}\)</span> and
<span class="math">\(\sigma_{\text{max}}\)</span> are the minimum and maxiumum
singular values of <span class="math">\(J\)</span> respectively.</p>
</li>
<li><p class="first"><code class="docutils literal"><span class="pre">SPARSE_QR</span></code></p>
<blockquote>
<div><div class="math">
\[\operatorname{rank}(J) < \operatorname{num\_col}(J)\]</div>
<p>Here <span class="math">\(\operatorname{rank}(J)\)</span> is the estimate of the rank
of <span class="math">\(J\)</span> returned by the sparse QR factorization
algorithm. It is a fairly reliable indication of rank
deficiency.</p>
</div></blockquote>
</li>
</ol>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N10Covariance7Options15null_space_rankE">
<span id="ceres::Covariance::Options::null_space_rank__i"></span>int <code class="descclassname">Covariance::Options::</code><code class="descname">null_space_rank</code><a class="headerlink" href="#_CPPv2N10Covariance7Options15null_space_rankE" title="Permalink to this definition">¶</a></dt>
<dd><p>When using <code class="docutils literal"><span class="pre">DENSE_SVD</span></code>, the user has more control in dealing
with singular and near singular covariance matrices.</p>
<p>As mentioned above, when the covariance matrix is near singular,
instead of computing the inverse of <span class="math">\(J'J\)</span>, the Moore-Penrose
pseudoinverse of <span class="math">\(J'J\)</span> should be computed.</p>
<p>If <span class="math">\(J'J\)</span> has the eigen decomposition <span class="math">\((\lambda_i,
e_i)\)</span>, where <span class="math">\(\lambda_i\)</span> is the <span class="math">\(i^\textrm{th}\)</span>
eigenvalue and <span class="math">\(e_i\)</span> is the corresponding eigenvector, then
the inverse of <span class="math">\(J'J\)</span> is</p>
<div class="math">
\[(J'J)^{-1} = \sum_i \frac{1}{\lambda_i} e_i e_i'\]</div>
<p>and computing the pseudo inverse involves dropping terms from this
sum that correspond to small eigenvalues.</p>
<p>How terms are dropped is controlled by
<cite>min_reciprocal_condition_number</cite> and <cite>null_space_rank</cite>.</p>
<p>If <cite>null_space_rank</cite> is non-negative, then the smallest
<cite>null_space_rank</cite> eigenvalue/eigenvectors are dropped irrespective
of the magnitude of <span class="math">\(\lambda_i\)</span>. 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 <cite>Covariance::Compute()</cite> will fail and return <cite>false</cite>.</p>
<p>Setting <cite>null_space_rank = -1</cite> drops all terms for which</p>
<div class="math">
\[\frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}\]</div>
<p>This option has no effect on <code class="docutils literal"><span class="pre">SPARSE_QR</span></code>.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N10Covariance7Options19apply_loss_functionE">
<span id="ceres::Covariance::Options::apply_loss_function__b"></span>bool <code class="descclassname">Covariance::Options::</code><code class="descname">apply_loss_function</code><a class="headerlink" href="#_CPPv2N10Covariance7Options19apply_loss_functionE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <cite>true</cite></p>
<p>Even though the residual blocks in the problem may contain loss
functions, setting <code class="docutils literal"><span class="pre">apply_loss_function</span></code> 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.</p>
</dd></dl>
<dl class="class">
<dt id="_CPPv2N5ceres10CovarianceE">
<span id="ceres::Covariance"></span><em class="property">class </em><code class="descclassname"></code><code class="descname">Covariance</code><a class="headerlink" href="#_CPPv2N5ceres10CovarianceE" title="Permalink to this definition">¶</a></dt>
<dd><p><a class="reference internal" href="#_CPPv2N5ceres10Covariance7OptionsE" title="ceres::Covariance::Options"><code class="xref cpp cpp-class docutils literal"><span class="pre">Covariance::Options</span></code></a> as the name implies is used to control
the covariance estimation algorithm. Covariance estimation is a
complicated and numerically sensitive procedure. Please read the
entire documentation for <a class="reference internal" href="#_CPPv2N5ceres10Covariance7OptionsE" title="ceres::Covariance::Options"><code class="xref cpp cpp-class docutils literal"><span class="pre">Covariance::Options</span></code></a> before using
<a class="reference internal" href="#_CPPv2N5ceres10CovarianceE" title="ceres::Covariance"><code class="xref cpp cpp-class docutils literal"><span class="pre">Covariance</span></code></a>.</p>
</dd></dl>
<dl class="function">
<dt id="_CPPv2N5ceres10Covariance7ComputeERK6vectorI4pairIPKdPKdEEP7Problem">
<span id="ceres::Covariance::Compute__vector:pair:doubleCP.doubleCP::CR.ProblemP"></span>bool <code class="descclassname">Covariance::</code><code class="descname">Compute</code><span class="sig-paren">(</span><em class="property">const</em> vector<pair<<em class="property">const</em> double *, <em class="property">const</em> double *>> &<em>covariance_blocks</em>, <a class="reference internal" href="nnls_modeling.html#_CPPv2N5ceres7ProblemE" title="ceres::Problem">Problem</a> *<em>problem</em><span class="sig-paren">)</span><a class="headerlink" href="#_CPPv2N5ceres10Covariance7ComputeERK6vectorI4pairIPKdPKdEEP7Problem" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute a part of the covariance matrix.</p>
<p>The vector <code class="docutils literal"><span class="pre">covariance_blocks</span></code>, 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.</p>
<p>Since the covariance matrix is symmetric, if the user passes
<code class="docutils literal"><span class="pre"><block1,</span> <span class="pre">block2></span></code>, then <code class="docutils literal"><span class="pre">GetCovarianceBlock</span></code> can be called with
<code class="docutils literal"><span class="pre">block1</span></code>, <code class="docutils literal"><span class="pre">block2</span></code> as well as <code class="docutils literal"><span class="pre">block2</span></code>, <code class="docutils literal"><span class="pre">block1</span></code>.</p>
<p><code class="docutils literal"><span class="pre">covariance_blocks</span></code> cannot contain duplicates. Bad things will
happen if they do.</p>
<p>Note that the list of <code class="docutils literal"><span class="pre">covariance_blocks</span></code> 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.</p>
<p>The return value indicates the success or failure of the covariance
computation. Please see the documentation for
<a class="reference internal" href="#_CPPv2N5ceres10Covariance7OptionsE" title="ceres::Covariance::Options"><code class="xref cpp cpp-class docutils literal"><span class="pre">Covariance::Options</span></code></a> for more on the conditions under which
this function returns <code class="docutils literal"><span class="pre">false</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="_CPPv2NK5ceres18GetCovarianceBlockEPKdPKdPd">
<span id="ceres::GetCovarianceBlock__doubleCP.doubleCP.doublePC"></span>bool <code class="descclassname"></code><code class="descname">GetCovarianceBlock</code><span class="sig-paren">(</span><em class="property">const</em> double *<em>parameter_block1</em>, <em class="property">const</em> double *<em>parameter_block2</em>, double *<em>covariance_block</em><span class="sig-paren">)</span> <em class="property">const</em><a class="headerlink" href="#_CPPv2NK5ceres18GetCovarianceBlockEPKdPKdPd" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the block of the cross-covariance matrix corresponding to
<code class="docutils literal"><span class="pre">parameter_block1</span></code> and <code class="docutils literal"><span class="pre">parameter_block2</span></code>.</p>
<p>Compute must be called before the first call to <code class="docutils literal"><span class="pre">GetCovarianceBlock</span></code>
and the pair <code class="docutils literal"><span class="pre"><parameter_block1,</span> <span class="pre">parameter_block2></span></code> OR the pair
<code class="docutils literal"><span class="pre"><parameter_block2,</span> <span class="pre">parameter_block1></span></code> must have been present in the
vector covariance_blocks when <code class="docutils literal"><span class="pre">Compute</span></code> was called. Otherwise
<code class="docutils literal"><span class="pre">GetCovarianceBlock</span></code> will return false.</p>
<p><code class="docutils literal"><span class="pre">covariance_block</span></code> must point to a memory location that can store
a <code class="docutils literal"><span class="pre">parameter_block1_size</span> <span class="pre">x</span> <span class="pre">parameter_block2_size</span></code> matrix. The
returned covariance will be a row-major matrix.</p>
</dd></dl>
<dl class="function">
<dt id="_CPPv2NK5ceres32GetCovarianceBlockInTangentSpaceEPKdPKdPd">
<span id="ceres::GetCovarianceBlockInTangentSpace__doubleCP.doubleCP.doublePC"></span>bool <code class="descclassname"></code><code class="descname">GetCovarianceBlockInTangentSpace</code><span class="sig-paren">(</span><em class="property">const</em> double *<em>parameter_block1</em>, <em class="property">const</em> double *<em>parameter_block2</em>, double *<em>covariance_block</em><span class="sig-paren">)</span> <em class="property">const</em><a class="headerlink" href="#_CPPv2NK5ceres32GetCovarianceBlockInTangentSpaceEPKdPKdPd" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the block of the cross-covariance matrix corresponding to
<code class="docutils literal"><span class="pre">parameter_block1</span></code> and <code class="docutils literal"><span class="pre">parameter_block2</span></code>.
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.</p>
<p>Compute must be called before the first call to <code class="docutils literal"><span class="pre">GetCovarianceBlock</span></code>
and the pair <code class="docutils literal"><span class="pre"><parameter_block1,</span> <span class="pre">parameter_block2></span></code> OR the pair
<code class="docutils literal"><span class="pre"><parameter_block2,</span> <span class="pre">parameter_block1></span></code> must have been present in the
vector covariance_blocks when <code class="docutils literal"><span class="pre">Compute</span></code> was called. Otherwise
<code class="docutils literal"><span class="pre">GetCovarianceBlock</span></code> will return false.</p>
<p><code class="docutils literal"><span class="pre">covariance_block</span></code> must point to a memory location that can store
a <code class="docutils literal"><span class="pre">parameter_block1_local_size</span> <span class="pre">x</span> <span class="pre">parameter_block2_local_size</span></code> matrix. The
returned covariance will be a row-major matrix.</p>
</dd></dl>
</div>
<div class="section" id="example-usage">
<h2>Example Usage<a class="headerlink" href="#example-usage" title="Permalink to this headline">¶</a></h2>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="kt">double</span> <span class="n">x</span><span class="p">[</span><span class="mi">3</span><span class="p">];</span>
<span class="kt">double</span> <span class="n">y</span><span class="p">[</span><span class="mi">2</span><span class="p">];</span>
<span class="n">Problem</span> <span class="n">problem</span><span class="p">;</span>
<span class="n">problem</span><span class="p">.</span><span class="n">AddParameterBlock</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">);</span>
<span class="n">problem</span><span class="p">.</span><span class="n">AddParameterBlock</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">2</span><span class="p">);</span>
<span class="o"><</span><span class="n">Build</span> <span class="n">Problem</span><span class="o">></span>
<span class="o"><</span><span class="n">Solve</span> <span class="n">Problem</span><span class="o">></span>
<span class="n">Covariance</span><span class="o">::</span><span class="n">Options</span> <span class="n">options</span><span class="p">;</span>
<span class="n">Covariance</span> <span class="nf">covariance</span><span class="p">(</span><span class="n">options</span><span class="p">);</span>
<span class="n">vector</span><span class="o"><</span><span class="n">pair</span><span class="o"><</span><span class="k">const</span> <span class="kt">double</span><span class="o">*</span><span class="p">,</span> <span class="k">const</span> <span class="kt">double</span><span class="o">*></span> <span class="o">></span> <span class="n">covariance_blocks</span><span class="p">;</span>
<span class="n">covariance_blocks</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">make_pair</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">));</span>
<span class="n">covariance_blocks</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">make_pair</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y</span><span class="p">));</span>
<span class="n">covariance_blocks</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">make_pair</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">));</span>
<span class="n">CHECK</span><span class="p">(</span><span class="n">covariance</span><span class="p">.</span><span class="n">Compute</span><span class="p">(</span><span class="n">covariance_blocks</span><span class="p">,</span> <span class="o">&</span><span class="n">problem</span><span class="p">));</span>
<span class="kt">double</span> <span class="n">covariance_xx</span><span class="p">[</span><span class="mi">3</span> <span class="o">*</span> <span class="mi">3</span><span class="p">];</span>
<span class="kt">double</span> <span class="n">covariance_yy</span><span class="p">[</span><span class="mi">2</span> <span class="o">*</span> <span class="mi">2</span><span class="p">];</span>
<span class="kt">double</span> <span class="n">covariance_xy</span><span class="p">[</span><span class="mi">3</span> <span class="o">*</span> <span class="mi">2</span><span class="p">];</span>
<span class="n">covariance</span><span class="p">.</span><span class="n">GetCovarianceBlock</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">covariance_xx</span><span class="p">)</span>
<span class="n">covariance</span><span class="p">.</span><span class="n">GetCovarianceBlock</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">covariance_yy</span><span class="p">)</span>
<span class="n">covariance</span><span class="p">.</span><span class="n">GetCovarianceBlock</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">covariance_xy</span><span class="p">)</span>
</pre></div>
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