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object-orientation.md

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Orientation Detection Using PCA

How Principal Component Analysis (PCA) can be used to determine the orientation of an object (often represented as a "blob" of pixel points) in an image.

1. Extracting the Points of the Blob

  • Segmentation / Thresholding: First, isolate the object of interest by applying segmentation techniques or thresholding to obtain a binary mask.
  • Pixel Extraction: Gather all $(x, y)$ coordinates for pixels that belong to the object (this collection of points is called the "blob").

2. Computing the Centroid

For a set of points $${(x_1, y_1), (x_2, y_2), \dots, (x_N, y_N)}$$

calculate the centroid $\bar{x}, \bar{y}$ as follows:

$$ \bar{x} = \frac{1}{N}\sum_{i=1}^{N} x_i,\quad \bar{y} = \frac{1}{N}\sum_{i=1}^{N} y_i. $$

This centroid represents the center of mass of the blob.

3. Centering the Points

Subtract the computed centroid from every point to get centered coordinates:

$$ (x_i', y_i') = (x_i - \bar{x},\ y_i - \bar{y}). $$

Centering is essential before calculating the covariance matrix.

4. Forming the Covariance Matrix

Using the centered points, form the 2D covariance matrix $\Sigma$:

$$ \Sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{bmatrix}, $$

where

$$ \sigma_{xx} = \frac{1}{N}\sum_{i=1}^{N} (x_i')^2,\quad \sigma_{yy} = \frac{1}{N}\sum_{i=1}^{N} (y_i')^2,\quad \sigma_{xy} = \sigma_{yx} = \frac{1}{N}\sum_{i=1}^{N} x_i' y_i'. $$

This matrix captures how the points are spread out around the centroid in different directions.

5. Computing Eigenvalues and Eigenvectors

Eigenvalue Problem: Solve for $\lambda$ and $v$ such that

$$ \Sigma , v = \lambda , v. $$

Interpretation:

  • The two eigenvalues, $\lambda_1$ and $\lambda_2$, indicate the variance of the blob along the directions of their corresponding eigenvectors, $v_1$ and $v_2$.
  • The principal eigenvector is the one associated with the largest eigenvalue ($\lambda_{\text{max}}$), as it indicates the direction of maximum variance. This direction is typically the object's long axis.

6. Interpreting the Results

  • The principal eigenvector, aligned with the largest spread of the data, represents the object's orientation.
  • For instance, if you have a pear-shaped object, the principal eigenvector points along its long axis (from the base to the tip).

Why This Works

  • Covariance Matrix: Encodes the spread (variance) of the blob in every direction.
  • PCA: Diagonalizing the covariance matrix (via eigenvalue decomposition) reveals the principal components – the directions in which the data variance is maximized and minimized.
  • Principal Eigenvector: The direction corresponding to the highest variance (largest eigenvalue) naturally aligns with the object's major (long) axis.