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Bilateral Symmetry Analysis Module

Bilateral symmetry assessment is crucial for understanding movement disorders, rehabilitation progress, and motor control strategies.
This module analyzes coordination patterns using three complementary methods:

  • Bilateral Symmetry Index (BSI) — spatial mirror-symmetry measure.
  • Phase Synchronization (PLV) — temporal coordination of left–right movement.
  • Canonical Correlation Analysis (CCA) — multivariate correlation of trajectories.

Note

These methods assume approximately Gaussian noise and sufficient frame history.

Bilateral Symmetry Index (BSI)

The BSI quantifies spatial symmetry between left and right trajectories by comparing mirrored movements across the sagittal plane.

Input:
- Left and right 3D joint trajectories $$ L, R \in \mathbb{R}^{n \times 3} $$

Procedure:

  1. Mirror right trajectory across sagittal plane:
\[ R' = [-R_x, R_y, R_z] \]
  1. Compute absolute difference:
\[ D = |L - R'| \]
  1. Normalize by total magnitude:
\[ S = |L| + |R'| \]
  1. Relative asymmetry percentage:
\[ A = \frac{1}{n} \sum_i \frac{D_i}{S_i} \times 100 \]
  1. Symmetry index:
\[ \text{BSI} = 1 - \frac{A}{100} \]

Output:

\[ \text{BSI} \in [0, 1] \]

Tip

Higher BSI indicates more symmetric posture and movement.

Phase Synchronization (Analytic Signal Approach)

Phase synchronization quantifies the temporal relationship between left and right signals by analysing the constancy of their phase difference.
It is computed via Hilbert transform and expressed as Phase Locking Value (PLV).

Input:
Two 1D limb displacement signals $$ x(t), y(t) $$.

Procedure:

  1. Compute analytic signals:
\[ z_x(t) = x(t) + iH[x(t)], \]
\[ z_y(t) = y(t) + iH[y(t)] \]
  1. Extract instantaneous phases:
\[ \phi_x(t), \phi_y(t) \]
  1. Compute phase difference:
\[ \Delta \phi(t) = \phi_x(t) - \phi_y(t) \]
  1. Compute Phase Locking Value (PLV):
\[ \text{PLV} = \left| \frac{1}{N} \sum_t e^{i \Delta \phi(t)} \right| \]

Output:

\[ \text{PLV} \in [0, 1] \]

Tip

PLV > 0.7 → strong coupling
PLV 0.4–0.7 → moderate
PLV < 0.4 → weak coupling

Canonical Correlation Analysis (CCA)

CCA measures multivariate correspondence between left and right 3D trajectories.
It finds linear combinations that maximize correlation, revealing shared coordination structure.

Input:

\[ L, R \in \mathbb{R}^{n \times 3} \]

Procedure:

  1. Flatten coordinates into features.
  2. Apply one-component CCA:
\[ (U, V) = \text{CCA}(L, R) \]
  1. Compute canonical correlation:
\[ \rho = \text{corr}(U, V) \]

Output:

\[ \rho \in [0, 1] \]

Tip

  • 1.0–0.8: excellent coordination
  • 0.8–0.6: good
  • 0.6–0.4: moderate
  • <0.4: poor