dfMaker Details

Fast Scaling Mode (fast_scaling = TRUE)

When fast_scaling = TRUE, the transformation is simplified and uses only the pose keypoints (the first set of keypoints). This results in faster computation since it avoids extra references.

Steps:

1. Define the Origin (o_point):

   • Select a keypoint to serve as the origin \((0, 0)\) in the new coordinate system.
   • Denote the coordinates of the origin as \((x_{\text{origin}}, y_{\text{origin}})\).

2. Calculate the Primary Base Vector (\(\mathbf{v_i}\)):

   • Choose a keypoint (i_point) to define the primary base vector.
   • Compute: \[ \mathbf{v_i} = (x_i, y_i) - (x_{\text{origin}}, y_{\text{origin}}) \] where \((x_i, y_i)\) are the coordinates of i_point.

3. Compute the Scaling Factor (s):

   • The scaling factor is the x-component of \(\mathbf{v_i}\): \[ s = v_{i,x} \]
   • This factor scales the keypoints along the x-axis.

4. Apply the Transformation:

   • For each keypoint \((x, y)\), compute the transformed coordinates:

     \[ x' = \frac{x - x_{\text{origin}}}{s}, \quad y' = -\frac{y - y_{\text{origin}}}{s} \]

   • The y-coordinate is negated to adjust for coordinate system differences (e.g., image coordinates have the y-axis pointing downwards).

Summary Equation:

\[ \begin{cases} x' = \dfrac{x - x_{\text{origin}}}{s} \\ y' = -\dfrac{y - y_{\text{origin}}}{s} \end{cases} \]

Full Transformation Mode (fast_scaling = FALSE)

When fast_scaling = FALSE, the transformation uses both primary and secondary base vectors to perform a full affine transformation, which can handle rotations and scaling in both axes.

Additional Steps:

1. Calculate the Secondary Base Vector (\(\mathbf{v_j}\)):

   • If i_point and j_point are different:

     \[ \mathbf{v_j} = (x_j, y_j) - (x_{\text{origin}}, y_{\text{origin}}) \]

   • If i_point and j_point are the same (to maintain orthogonality):

     \[ \mathbf{v_j} = (-v_{i,y}, v_{i,x}) \]

     This computes a vector perpendicular to \(\mathbf{v_i}\).

2. Construct the Transformation Matrix (\(M_t\)):

   \[ M_t = \begin{pmatrix} v_{i,x} & v_{j,x} \\ v_{i,y} & v_{j,y} \end{pmatrix} \]

3. Compute the Inverse Transformation Matrix (\(M_t^{-1}\)) Using Cramer’s Rule:

   • Determinant:

     \[ \det(M_t) = v_{i,x} \cdot v_{j,y} - v_{j,x} \cdot v_{i,y} \]

   • Inverse Matrix:

     \[ M_t^{-1} = \frac{1}{\det(M_t)} \begin{pmatrix} v_{j,y} & -v_{j,x} \\ -v_{i,y} & v_{i,x} \end{pmatrix} \]

4. Apply the Transformation:

   • For each keypoint \((x, y)\), compute the relative position:

     \[ \begin{pmatrix} x_{\text{rel}} \\ y_{\text{rel}} \end{pmatrix} = \begin{pmatrix} x - x_{\text{origin}} \\ y - y_{\text{origin}} \end{pmatrix} \]

   • Transform the coordinates:

     \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = M_t^{-1} \cdot \begin{pmatrix} x_{\text{rel}} \\ y_{\text{rel}} \end{pmatrix} \]

Example Applied in the Function

Using transformation_coords = c(1, 1, 5, 5) and fast_scaling = TRUE:

• Origin (o_point = 1): Keypoint index 1.
• Primary Base Vector (i_point = 5): Keypoint index 5.
• Secondary Base Vector (j_point = 5): Same as i_point, so \(\mathbf{v_j}\) is perpendicular to \(\mathbf{v_i}\).

\[ \begin{cases} x' = \dfrac{x - x_{\text{origin}}}{v_{i,x}} \\ y' = -\dfrac{y - y_{\text{origin}}}{v_{i,x}} \end{cases} \]

Implications:

• The x-coordinate is scaled to establish a unit length along the x-axis.
• The y-coordinate is scaled and inverted to maintain proportion and adjust for coordinate orientation.