Hessian
The Hessian command computes a scalar map from the Hessian matrix of the image — the matrix of second-order partial derivatives. This captures local curvature and is useful for detecting blobs and ridges.
When to use
- Blob detection — use the Determinant mode to find circular, blob-like structures (e.g. cell nuclei, vesicles).
- Ridge detection — use the Eigenvalue modes to detect filamentous or tubular structures.
Parameters
| Parameter | Description |
|---|---|
| Mode | Which scalar feature to extract from the Hessian (see below) |
Modes
| Mode | Formula | Highlights |
|---|---|---|
| Determinant | $\det(H) = I_{xx}I_{yy} - I_{xy}^2$ | Blobs and corners |
| Eigenvalues X | Larger eigenvalue $\lambda_1$ | Maximum curvature; principal ridge axis |
| Eigenvalues Y | Smaller eigenvalue $\lambda_2$ | Secondary curvature; interest points when both eigenvalues are large |
Background
The Hessian matrix at pixel $(x,y)$ is:
$$ H = \begin{pmatrix} I_{xx} & I_{xy} \ I_{xy} & I_{yy} \end{pmatrix} $$
where $I_{xx}$, $I_{yy}$, $I_{xy}$ are second-order spatial derivatives of the image intensity.