Differentiate#

Overview#

The differentiate module in splineops provides a collection of algorithms for the computation of image differentials based on cubic B‑spline interpolation [1], [2]. By modeling a grayscale image as a continuous function reconstructed from its discrete samples, the module enables the accurate computation of derivatives. It offers several operations such as

  • Gradient Magnitude: the local rate of change of the intensity;

  • Gradient Direction: the direction along which the intensity changes most;

  • Laplacian:the sum of second-order derivatives;

  • Largest Hessian Eigenvalue: the maximal curvature;

  • Smallest Hessian Eigenvalue: the minimal curvature; and

  • Hessian Orientation: the principal orientation of the curvature.

Image Representation#

A grayscale image is modeled as the continuous function

\[f(x_1, x_2) = \sum_{k_1,k_2} c[k_1,k_2] \cdot \phi(x_1-k_1, x_2-k_2),\]

where \(\phi(x_1,x_2)\) is defined as the tensor-product cubic B‑spline

\[\phi(x_1,x_2) = \beta^{3}(x_1) \cdot \beta^{3}(x_2).\]

This formulation allows one to compute exact derivatives of the image by first determining the spline coefficients \(c[k_1,k_2]\).

Differentiation Operations#

Based on the spline representation, the module computes several differential operators. We define

\[\begin{split}f_1 \equiv \frac{\partial f}{\partial x_1},\\ f_2 \equiv \frac{\partial f}{\partial x_2},\\ f_{11}\equiv \frac{\partial^2 f}{\partial x_1^2},\\ f_{22}\equiv \frac{\partial^2 f}{\partial x_2^2},\\ f_{12}=f_{21} \equiv \frac{\partial^2 f}{\partial x_1 \partial x_2}.\end{split}\]

  • Gradient Magnitude: Computed as the Euclidean norm of the first derivatives

    \[\|\pmb{\nabla}f(x,y)\| = \sqrt{\bigl(f_1\bigr)^2 + \bigl(f_2\bigr)^2}.\]

  • Gradient Direction: The direction of the gradient is given by

    \[\theta(x,y) = \arctan\!\Bigl(\tfrac{f_2}{f_1}\Bigr).\]

  • Laplacian: A second-order operator that highlights regions of rapid intensity change as

    \[\Delta f(x,y) = f_{11} + f_{22}.\]

  • Largest Hessian Eigenvalue: The maximal eigenvalue of the Hessian matrix is given by

    \[\lambda_{\text{max}} = \tfrac12\Bigl(f_{11} + f_{22} + \sqrt{4f_{12}^2 + (f_{11} - f_{22})^2}\Bigr).\]

  • Smallest Hessian Eigenvalue: The minimal eigenvalue of the Hessian matrix is given by

    \[\lambda_{\text{min}} = \tfrac12\Bigl(f_{11} + f_{22} - \sqrt{4f_{12}^2 + (f_{11} - f_{22})^2}\Bigr).\]

  • Hessian Orientation: This operation returns the orientation that corresponds to the maximal second derivative, as

    \[\theta_H(x,y) = \pm \tfrac12 \arccos\!\Bigl(\tfrac{f_{11} - f_{22}}{\sqrt{4f_{12}^2 + (f_{11}-f_{22})^2}}\Bigr),\]

    where the sign is determined by the sign of the cross derivative \(f_{12}\).

Implementation Details#

The Differentials class implements these operations as follows: the input image is provided by its samples. We assume it to be a cubic B-spline and first determine its interpolation coefficients. The differential-based computations that we perform are then perfectly consistent with this continuously defined function. We finally build the output image by sampling the ideal, continuously defined intermediate result.

Differentiate Example#

References#