Context and Problem Statement¶

The neural network requires special treatment because we use quaternions. The quaternion orientation format is not continuous.

Decision Drivers ¶

• Slow improvement on performance
• Simplicity

Considered Options¶

• Basis 3x3 matrix
• Quaternion
• Euler
• 6D Truncation 3x3 Basis

Decision Outcome¶

Choose 6D Truncation 3x3 basis because its theory is the simplest, and the paper mathematically proved it was accurate.

Positive Consequences ¶

• Better training gain
• The code is simpler to understand

Negative Consequences ¶

• Work to change to new representation

Pros and Cons of the Options ¶

Basis 3x3 matrix¶

Basis 3x3 matrix.

• Good, because continuous
• Bad, because it uses nine rather than six floats.
• Bad, because space inefficient

Quaternion¶

The status quo is the Quaternion.

• Good, because it is standard
• Good, because the status quo
• Bad, because it has poles.

Euler¶

• Bad, no.
• …

6D Truncation 3x3 Basis¶

A basis is a 3x3 matrix. The Basis is normalized.

//x_raw is the X Axis / first row of the basis
//y_raw is the Y Axis / second row of the basis.

// On the Continuity of Rotation Representations in Neural Networks
// arXiv:1812.07035
Basis compute_rotation_matrix_from_ortho_6d(Vector3 x_raw, Vector3 y_raw) {
    Vector3 x = x_raw.normalized();
    Vector3 z = x.cross(y_raw);
    z = z.normalized();
    Vector3 y = z.cross(x);
    Basis basis;
    basis.set_axis(Vector3::AXIS_X, x);
    basis.set_axis(Vector3::AXIS_Y, y);
    basis.set_axis(Vector3::AXIS_Z, z);
    return basis;
}

• Good, because it's space-efficient with six floats
• Bad, because it's not standard

Links ¶

• https://arxiv.org/pdf/1812.07035.pdf