DSO - AccumulatorApprox

Direct sparse odometry

Abstract

• This is a class to compute the H and b in the Gauss-Newton optimization. Related variables are
  • Intrinsic parameters (\(f_x, f_y, c_x, c_y\))
  • Relative pose (\(\boldsymbol{\xi}_{ji}\))
  • Relative photometric parameters (\(-e^{a_{ji}}, -b_{ji}\))

Variable

Notes

1. In the real implementation, the matrix \(\mathbf{H} = \begin{bmatrix}\mathbf{J}^{T} \mathbf{J} & \mathbf{b} \\ \mathbf{b}^{T} & \mathbf{e}^{T} \mathbf{e} \end{bmatrix} = \begin{bmatrix}\mathbf{H}_{\text{tl}} & \mathbf{H}_{\text{tr}} \\ \mathbf{H}_{\text{tr}}^{T} & \mathbf{H}_{\text{br}} \end{bmatrix}\) is divided into three parts.
   1. The first part is the top left 10 x 10.
      • \(\mathbf{H}_{\text{tl}} = \mathbf{J}_{\text{geo}}^{T}\mathbf{J}_{\text{geo}}\), where \(\mathbf{J}_{\text{geo}} = \frac{\partial \mathbf{e}}{\partial \begin{bmatrix} \mathbf{C} \\ \boldsymbol{\xi}_{ji}\end{bmatrix}}\).
      • Due to the symmetric property, there are only \(10 + 9 + 8 + \cdots + 1 = 55\) elements computed.
   2. The second part is the top right 10 x 3.
      • \(\mathbf{H}_{\text{tr}} = \begin{bmatrix}\mathbf{J}_{\text{geo}}^{T}\mathbf{J}_{\text{photo}} & \mathbf{J}_{\text{geo}}^{T} \mathbf{e}\end{bmatrix}\), where \(\mathbf{J}_{\text{photo}} = \frac{\partial \mathbf{e}}{\partial \mathbf{a}_{ji}}\).
   3. The third part is the bottom right 3 x 3.
      • \(\mathbf{H}_{\text{br}} = \begin{bmatrix}\mathbf{J}_{\text{photo}}^{T}\mathbf{J}_{\text{photo}} & \mathbf{J}_{\text{photo}}^{T} \mathbf{e} \\ (\mathbf{J}_{\text{photo}}^{T} \mathbf{e})^{T} & \mathbf{e}^{T} \mathbf{e} \end{bmatrix}\).
      • Due to the symmetric property, there are only \(3 + 2 + 1 = 6\) elements computed.

Function

void update()

/** \brief Compute the top left part of Hessian
 *
 *  Compute the Hessian ONLY related to intrinsic parameters and relative
 *  pose. Since it is a symmetric matrix, we only need to compute the upper
 *  right part of it.
*/
void update(const float* const x4, const float* const x6,
            const float* const y4, const float* const y6, const float a,
            const float b, const float c);

void updateTopRight()

void updateTopRight(const float* const x4, const float* const x6,
                    const float* const y4, const float* const y6,
                    const float TR00, const float TR10, const float TR01,
                    const float TR11, const float TR02, const float TR12);

void updateBotRight()

/** \brief Compute the bottom right part of Hessian
 *
 * Compute the Hessian and b ONLY related to photometric affine parameters
*/
void updateBotRight(const float a00, const float a01, const float a02,
                    const float a11, const float a12,
                    const float a22);

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