Optimize on-manifold: Left or Right Plus in LocalParameterization::Plus()?

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f2020 l_

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Apr 22, 2021, 6:05:37 PM4/22/21

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Hi,

I am new to ceres solver. I have some question about writing LocalParameterization:

When I read the tutorials and other researchers' open source projects, I saw their LocalParameterization parts are different.

If using 7D quaternion + translation (qx qy qz qw tx ty tz) as global variables, and 6D Lie algebra (tx ty tz r1 r2 r3) as local variables, for the code:

-------------------------------------------------------------

class LocalParameterizationSE3 : public ceres::LocalParameterization
{
public:
  virtual ~LocalParameterizationSE3(){}
  virtual bool Plus
  (
    float const* T_raw, 
    float const* delta_raw, 
    float* T_plus_delta_raw
  ) const
  {
    Eigen::Map<Sophus::SE3f const> const pose_se3(T_raw);
    // Eigen::Map<Eigen::Matrix<float, 6, 1, Eigen::RowMajor> const> delta(delta_raw);
    Eigen::Map<Eigen::Vector6f const> delta(delta_raw);
    Eigen::Map<Sophus::SE3f> T_plus_delta(T_plus_delta_raw);
    T_plus_delta = pose_se3 * Sophus::SE3f::exp(delta);

    return true;
  }

private:
  virtual bool ComputeJacobian
  (
    float const* T_raw,
    float* jacobian_raw
  ) const
  {
    Eigen::Map<Sophus::SE3f const> pose_se3(T_raw);
    Eigen::Map<Eigen::Matrix<float, 7, 6, Eigen::RowMajor>> jacobian(jacobian_raw);
    jacobian = pose_se3.Dx_this_mul_exp_x_at_0();
    return true;
  }
}

-----------------------------------------------------------------

Q1: What time do I use left plus or right plus? Or I can choose either one that I prefer, and then write corresponding ComputeJacobian()?

left plus: out_T = exp(delta_T) * T

right plus: out_T = T * exp(delta_T)

Q2: Are the jacobians the same for right/left plus?

Q3: Is there any material about computing the jacobian of `T * exp(delta_T)` and `exp(delta_T) * T` ?

Sameer Agarwal

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Apr 23, 2021, 12:30:52 PM4/23/21

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you can choose left or right and then have the Jacobian accordingly.

On Thu, Apr 22, 2021 at 3:08 PM f2020 l_ <lanfeiy...@gmail.com> wrote:

Hi,
I am new to ceres solver. I have some question about writing LocalParameterization:
When I read the tutorials and other researchers' open source projects, I saw their LocalParameterization parts are different.

If using 7D quaternion + translation (qx qy qz qw tx ty tz) as global variables, and 6D Lie algebra (tx ty tz r1 r2 r3) as local variables, for the code: (Sorry I don't know how to insert code block in google group)

Q1: What time do I use left plus or right plus? Or I can choose either one that I prefer, and then write corresponding ComputeJacobian()?
left plus: out_T = exp(delta_T) * T
right plus: out_T = T * exp(delta_T)

Q2: Are the jacobians the same for right/left plus?

Q3: Is there any material about computing the jacobian of `T * exp(delta_T)` and `exp(delta_T) * T` ?

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Sameer Agarwal

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Apr 23, 2021, 12:51:50 PM4/23/21

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https://arxiv.org/abs/1107.1119

Is what you want to read.

Sameer

On Thu, Apr 22, 2021 at 2:59 PM f2020 l_ <lanfeiy...@gmail.com> wrote:

Hi,
I am new to ceres solver. I have some question about writing LocalParameterization:
When I read the tutorials and other researchers' open source projects, I saw their LocalParameterization parts are different.

If using 7D quaternion + translation (qx qy qz qw tx ty tz) as global variables, and 6D Lie algebra (tx ty tz r1 r2 r3) as local variables, for the code:

<pre style='color:#000000;background:#ffffff;'>class LocalParameterizationSE3 : public ceres::LocalParameterization
{
public:
virtual ~LocalParameterizationSE3(){}
virtual bool Plus
(
float const* T_raw,
float const* delta_raw,
float* T_plus_delta_raw
) const
{
Eigen::Map<Sophus::SE3f const> const pose_se3(T_raw);
// Eigen::Map<Eigen::Matrix<float, 6, 1, Eigen::RowMajor> const> delta(delta_raw);
Eigen::Map<Eigen::Vector6f const> delta(delta_raw);
Eigen::Map<Sophus::SE3f> T_plus_delta(T_plus_delta_raw);
T_plus_delta = pose_se3 * Sophus::SE3f::exp(delta);

return true;
}

private:
virtual bool ComputeJacobian
(
float const* T_raw,
float* jacobian_raw
) const
{
Eigen::Map<Sophus::SE3f const> pose_se3(T_raw);
Eigen::Map<Eigen::Matrix<float, 7, 6, Eigen::RowMajor>> jacobian(jacobian_raw);
jacobian = pose_se3.Dx_this_mul_exp_x_at_0();
return true;
}
}
</pre>

Q1: What time do I use left plus or right plus? Or I can choose either one that I prefer, and then write corresponding ComputeJacobian()?
left plus: out_T = exp(delta_T) * T
right plus: out_T = T * exp(delta_T)

Q2: Are the jacobians the same for right/left plus?

Q3: Is there any material about computing the jacobian of `T * exp(delta_T)` and `exp(delta_T) * T` ?

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Alex Stewart

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Apr 23, 2021, 3:19:32 PM4/23/21

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For Q2 no, they are not - Jose Blanco has a tech-report here that gives the forms for both left and right jacobians.

For Q3 yes, Ethan Eade has a tutorial here, Tim Barfoot's book is also a very good and thorough reference for this (and many other things).

To view this discussion on the web visit https://groups.google.com/d/msgid/ceres-solver/CABqdRUC4jJHzBxPgiGg552H6peJkr-zw6249SqLE9y_5XdTDOw%40mail.gmail.com.

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